Physical Optics Analysis of a

Fiber-Delivered Displacement

Interferometer

by

Richard C. G. Smith

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Master of Science

Supervised by

Jonathan D. Ellis

Institute of OpticsArts, Sciences and Engineering

Edmund A. Hajim School of Engineering and Applied Sciences

University of RochesterRochester, New York

2013

ii

Curriculum Vitae

The author was born in Beverly, Massachussetts. He attended College of Charleston,

and graduated with a Bachelors of Arts and Science degrees in Music and Physics,

respectively. He began doctoral studies in Optics at the University of Rochester

in 2008. He pursued his research in Optics under the direction of Jonathan D.

Ellis.

iii

Acknowledgments

I would like to acknowledge the assistance given to me in conducting this research.

Steven Gillmer was instrumental in assisting with experiments as well as design-

ing a prototype interferometer. Chen Wang developed a phase meter that was

vital to this research. The rest of the Precision Instrumentation Group were also

invaluable. Finally, the assisstance of the Wayne Knox Group, John Marciante

Group, and the Hopkins lab were also instrumental.

iv

Abstract

Optical interferometry has been used for over a century for precision metrology.

Advances in interferometry, such as the inclusion of a laser source, have increased

the precision of such interferometers. In this thesis, we examine a fiber-fed, hetero-

dyne, spatially separated, differential wavefront interferometer. The combination

of technology in this interferometer can allow easier and more robust alignment

of the system, reduce the complexity, and increase the precision.

Heterodyne interferometry has been examined in great depth, and was not

the primary focus of this research, but rather a tool that we will utilize. The

primary focus of this thesis is to examine the effects that the fibers will have

on the system, as well as what types of fibers are more suitable for this fiber-

fed system. Additionally, we will investigate the physics of a spatially separated,

differential wavefront interferometer.

Fibers can facilitate an easier alignment of metrology systems because they will

remain aligned if they are perturbed, unlike systems that are fed via free-space

optics. Additionally, the alignment of the source is decoupled from the actual in-

terferometer, which allows any faults in the alignment of either to be immediately

reduced to the actual source. Unfortunately, fibers can also introduce problems

into a heterodyne interferometer. The first is phase shifts due to any physical

or optical length change of the fibers. These changes can be due to the fiber

either moving, being subjected to an external stress, changing in temperature,

or any number of environmental perturbations. Since heterodyne interferometers

v

measure displacement in the form of phase changes, phase changes due to fiber

perturbations can be potentially problematic. We will show that, depending on

the heterodyne frequency, these effects are unimportant if these effects are con-

sidered in the design of an interferometer.

The polarization of the light exiting the fibers is important as well. The con-

trast ratio, and therefore the signal-to-noise ratio, of an interferometer is strongly

dependent on the relative polarization of the two beams interfering. In fibers,

the polarization can be altered by any of the perturbations mentioned above, so

it is important to quantify these effects and see if there is any way to mitigate

them. Our results show that these effects are significant in any fibers that are not

intended to maintain polarization and that efforts to mitigate the effects in other

types of fibers have so far been unsuccessful.

The polarization in a spatially separated interferometer is also important. Spa-

tially separated interferometers were initially introduced to eliminate periodic er-

ror that was present in traditional heterodyne displacement interferometers. pe-

riodic error can have many sources, and one of them was imperfect polarizations.

In this thesis, we present a theoretical description which shows the interferometer

investigated to be free from any periodic error caused by polarization.

Finally, we will investigate the physics of a differential wavefront interferom-

eter, which allows simultaneous measurements of three degrees of freedom, thus

potentially simplifying some metrology systems. Though the concept is a simple

one, the physical measurements made by these devices is much more complex,

since the input beams are Gaussian, rather than plane waves. We have developed

a model which agrees well with data from one such device.

vi

Table of Contents

Curriculum Vitae ii

Acknowledgments iii

Abstract iv

List of Tables viii

List of Figures ix

1 Introduction 1

1.1 Displacement Interferometry . . . . . . . . . . . . . . . . . . . . . 2

1.2 Mathematical Description . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Uncertainty in Displacement Interferometry . . . . . . . . . . . . 9

1.4 Periodic Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Motivation for Fiber-Coupling . . . . . . . . . . . . . . . . . . . . 19

2 Fiber Input 22

2.1 Perturbation-Induced Phase Shift . . . . . . . . . . . . . . . . . . 23

2.2 Perturbation-Induced Polarization Changes . . . . . . . . . . . . . 29

vii

3 Theoretical Investigations 38

3.1 Arbitrary Input Polarizations . . . . . . . . . . . . . . . . . . . . 38

3.2 Rectangular Detector for Differential Wavefront Interferometry . . 41

4 Conclusions and Future Work 61

4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Bibliography 64

A Mathematical Description of a Non-Spatially-Separated Displace-

ment Interferometer 69

viii

List of Tables

1.1 The changes in environmental parameters resulting in a change of

+1 part in 108 in the refractive index of air. . . . . . . . . . . . . 11

ix

List of Figures

1.1 An example setup of a displacement interferometer. . . . . . . . . 3

1.2 The interferometer configuration used in this thesis. . . . . . . . . 4

1.3 The interferometer configuration used in this thesis with all dimen-

sions labelled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 A schematic example of a spatially-separated displacement inter-

ferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 The cross-sections of a step-index and graded-index optical fiber. . 15

1.6 Cross-sectional view of several type of PM fiber. . . . . . . . . . . 19

1.7 The problem with free-space alignment. . . . . . . . . . . . . . . . 20

2.1 Setup for quantifying the perturbation-induced phase shifts for dif-

ferent fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 The maximum frequency resulting from every attempted perturba-

tion on every tested type of fiber. . . . . . . . . . . . . . . . . . . 26

2.3 The structure of the phase shift for one tap on different types of

fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 The frequency resulting from the phase shift of one tap on different

types of fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Setup for quantifying the perturbation-induced polarization changes

for different fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

x

2.6 Azimuthal angle of long-term fiber polarization stability. . . . . . 30

2.7 Ellipticity of long-term fiber polarization stability. . . . . . . . . . 31

2.8 The aziumthal angle of the polarization vectors of the light exiting

the fibers while moving the fibers. . . . . . . . . . . . . . . . . . . 32

2.9 The ellipticity of the polarization vectors of the light exiting the

fibers while moving the fibers. . . . . . . . . . . . . . . . . . . . . 32

2.10 The minimum dot product of the polarization vectors when moving

the fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.11 Necessary bend radius versus core diameter for a loss of 10 dB to

the LP11 mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.12 Setup for quantifying the perturbation-induced polarization changes

for tightly coiled fiber. . . . . . . . . . . . . . . . . . . . . . . . . 36

2.13 The minimum dot product of the polarization vectors when moving

both the coiled and uncoiled fibers. . . . . . . . . . . . . . . . . . 37

3.1 A schematic of a rectangular quadrant photodiode. . . . . . . . . 42

3.2 A simulation of the signal measured by our interferometer at several

beam waists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 A simulation of the signal measured by our interferometer at several

beam waists over a small rotation angle. . . . . . . . . . . . . . . 52

3.4 The measured rotation as a function of input rotation and beam

waist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 A map of the “sensitivity coefficient” versus detector size and beam

waist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6 A map of the effective length of a differential wavefront interferom-

eter designed with a range of detector sizes and beam waists. . . . 55

3.7 Beam 1 as incident on the quadrant photodiode. . . . . . . . . . . 56

xi

3.8 Beam 2 as incident on the quadrant photodiode. . . . . . . . . . . 57

3.9 The comparison of our ideal theoretical model with measured data. 58

3.10 The comparison of our theoretical model with measured data. . . 59

3.11 Recalculated yaw using our modeled effective length. . . . . . . . 60

1

1 Introduction

Optical interferometry has been used in precision metrology systems for over 125

years [Michelson and Morley, 1887]. Interferometers use the interference of waves

to determine the phase changes of those waves, which is usually caused by dis-

tance changes. The resolution of an interferometer is directly proportional to its

wavelength, which makes optical interferometers especially attractive. Early on,

the principle of homodyne interferometry has been used to measure displacements

at resolutions of 0.025 µm [Barker and Hollenbach, 1965]. However, these inter-

ferometry techniques have limitations due to the single-frequency light producing

detected signals that are dependnt on the amplitude of the signal, as well as be-

ing directionally insensitive due to the symmetry of that nonlinear function. The

solution to these issues has been heterodyne interferometers.

Heterodyne interferometers were introduced to increase the precision of the

optical path difference detection in surface figure interferometers [Massie et al.,

1979]. Heterodyne interferometers operate on the principle of two frequencies cre-

ating a beat frequency when interfered. This beat frequency can be measured and

used to compute the phase with a minimal dependence on the signal amplitude.

Heterodyne displacement interferometry is most frequently used for manufactur-

ing metrology [Steinmetz, 1990], and can be used for laser alignment [Muller et al.,

2005], and gravity wave detection [Schuldt et al., 2009]. Such interferometers are

2

desirable for their nanometer-level precision, nearly unlimited dynamic range, and

speed.

1.1 Displacement Interferometry

In order to measure such small displacements, displacement interferometry mea-

sures the phase change of heterodyne interference. An example of a displacement

interferometer is shown in Figure 1.1. A Zeeman split laser provides two optical

frequencies with orthogonal polarizations from the same laser. The polarizing

beamsplitter ensures that the nominal path taken by each beam is determined

by its polarization, which is different for each frequency. The beams are then

reflected back to the polarizing beam splitter after accruing different phase de-

pending on the location of the stationary and moving retroreflectors where they

are again reflected or transmitted according to their polarization and directed

towards the measurement photodetector. Since the polarizations are orthogonal,

linear polarizers are required to ensure interference.

Displacement interferometers rely on comparing the signal between the two

detectors. In order to effectively describe the physical effects, we will present a

description of these two signals. Though there will be more detailed derivations

later, we will present here the standard form of the ideal reference and measure-

ment signals [Wu et al., 2002]

ir (t) ∝ Ar cos (ωst) and (1.1)

im (t) ∝ Am cos (ωst+ φm) , (1.2)

where Ar and Am are the amplitudes of the signals, ωs is the angular split fre-

quency, in radians/s, and φm is the displacement-dependent phase, in radians, as

determined in Equation A.13. The measurement signal is typically multiplied by

3

Figure 1.1: An example setup of a displacement interferometer. After the beams

are combined at the reference detector, they are split by polarization to go to

different arms of the interferometer, accrue different amounts of phase, and

recombine at the measurement detector.

the in-phase and quadrature reference signal, filtered, and results in

I = B cos (φm) and (1.3)

Q = B sin (φm) , (1.4)

and after dividing Q by I and taking the arctangent, we get

arctan

(

Q

I

)

= φm +mπ ≈ mπ − 2~k · (~zm + ~zr) , (1.5)

where m is an integer, ~k is the free-space wave vector of the laser beam (after ap-

proximating both shifted beams as having the same wavelength), ~zm is the path

of the beam in the measurement arm, and ~zr is the path of the beam in the refer-

ence arm. Since recovering the value of m is impractical with this interferometer

configuration, the absolute phase is not measured, just the change in phase from

the starting point. If absolute position measurements must be made, a separate

system to determine “home” must be implemented.

4

1.2 Mathematical Description

ω0

StabilizedLaser

BS AOM, ω2

AOM, ω1

BS

BS

PDr

PDm

Mm

Mr

Figure 1.2: The interferometer configuration used in this thesis.

In this thesis, we use an interferometer configuration show in Figure 1.2 [Gillmer

et al., 2012]. There are a few fundamental differences from the configuration shown

in Figure 1.1, the most significant of which is that the beams are spatially sep-

arated. The main goal of this difference is to minimize the periodic error which

will be discussed in Section 1.4. In this section, we will derive the measured and

processed signal recorded by this interferometer.

In this interferometer, there are two beams and two detectors. Since each

beam is incident on each detector, we need a way to describe what is common

and what is different between the four combinations of electric field sources. The

beams will be designated “beam 1” and “beam 2,” were all properties of these

beams will have the subscript 1 and 2, respectively. For example, the angular

frequency, ω of beam 1 is described by the variable ω1. The two detectors will

be designated “r” for reference and “m” for measurement. The properties of the

beams at these detectors will be given a further subscript to identify the detector

at which this property is being described. For example, the phase, φ of beam 2

5

at the r detector is given by the variable φ2,r.

The electric fields due to each beam on the reference detector is given by

~E1,r = ~A1,rei(ω1t−φ1,r) and (1.6)

~E2,r = ~A2,rei(ω2t−φ2,r), (1.7)

where ~A’s are the vector magnitudes of the electric fields, ω1 and ω2 are the two

angular optical frequencies, and φ1 and φ2 are the phases accrued by propagation

through their arm of the system and will be explicitly defined shortly. Addition-

ally, the electric fields due to each beam on the measurement detector is given

by

~E1,m = ~A1,mei(ω1t−φ1,m) and (1.8)

~E2,m = ~A2,mei(ω2t−φ2,m), (1.9)

using the same variable conventions as above. Finally, we need to define the phases

accrued by each beam, which are given by

φ1,r = ~k1 · (n1~z1 + 2nr ~zr + nrd ~zrd) (1.10)

φ2,r = ~k2 · (n2~z2 + nrd~zrd) (1.11)

φ1,m = ~k1 · (n1~z1 + nmd~zmd) (1.12)

φ2,m = ~k2 · (n2~z2 +Nnm~zm + nmd~zmd) , (1.13)

where ~k1 is the wave vector with a magnitude of 2π/λ1 where λ1 is the wave-

length of beam one in vacuum and a direction parallel to the Poynting vector

of that beam, and ~k2’s magnitude is similarly 2π/λ2, with a direction given by

the Poynting vector of beam two. The variable z describes the distances of the

interferometer as shown in Figure 1.3, where z1 is the distance beam one travels

before the beamsplitter, z2 the distance beam two travels before the beamsplit-

ter, zrd the distance between the beamsplitter and the reference detector, zmd the

distance between the beamsplitter and the measurement detector, zr the distance

6

between the beamsplitter and the reference mirror, and zm the distance between

the beamsplitter and the measurement mirror. The refractive index, n, is assumed

constant over each part of the beam path. The N is the interferometer fold fac-

tor and describes how many times the beam passes through the distance under

measurement. In general, it can be large, but since our interferometer is a single

reflection, it is 2. This reflection is also why there is a factor of 2 multiplying zr.

ω2

ω1

PDm

BS

PDr

Mm

Mr

zr

zm

Fiber

collimators

z1

z2

zrd

zmd

Figure 1.3: The interferometer configuration used in this thesis with all dimen-

sions labelled.

The total electric field on each detector is given by adding the electric fields due

to both beams. From this result, we are able to approximate the intensities (I) at

the detectors by assuming monochromatic plane waves [Griffiths, 1998]. We can

then determine the signal from the detectors by assuming the current (i) from each

detector is proportional to the power on a detector, which is proportional to the

intensity at that detector, which is reasonable as long as the frequency response

of the detector is sufficiently high and that the intensity does not depend on any

7

in-plane variables. These signals are given by

ir ∝ Ir ≈cnǫ02

|E1,r + E2,r|2 and (1.14)

im ∝ Im ≈ cnǫ02

|E1,m + E2,m|2. (1.15)

By substituting equations 1.6, 1.7, 1.8, and 1.9 into equations 1.14 and 1.15, we

can calculate the intensity as

Ir =cnǫ02

[

|A1,r|2 + |A2,r|2 +(

~A1,r · ~A∗

2,rei[(ω1−ω2)t−(φ1,r−φ2,r)] + c.c.

)]

and (1.16)

Im =cnǫ02

[

|A1,m|2 + |A2,m|2 +(

~A1,m · ~A∗

2,mei[(ω1−ω2)t−(φ1,m−φ2,m)] + c.c.

)]

(1.17)

where c.c. denotes the complex conjugate of the preceding term, φ1,r − φ2,r and

φ1,m − φ2,m are given by

φ1,r − φ2,r = n1~k1 · ~z1 − n2

~k2 · ~z2 + nrd ~zrd ·(

~k1 − ~k2

)

+ 2nr~k1 · ~zr and (1.18)

φ1,m − φ2,m = n1~k1 · ~z1 − n2

~k2 · ~z2 + nmd ~zmd ·(

~k1 − ~k2

)

− 2nm~k2 · ~zm. (1.19)

Since the t = 0 point is arbitrary, we can begin our time scale when the phase

of Ir is zero. This change of variables gives us

Ir ≈cnǫ02

[

|A1,r|2 + |A2,r|2 +(

~A1,r · ~A∗

2,rei(ω1−ω2)t′ + c.c.

)]

and (1.20)

Im ≈ cnǫ02

[

|A1,m|2 + |A2,m|2 +(

~A1,m · ~A∗

2,mei[(ω1−ω2)t′−φm] + c.c.

)]

(1.21)

where φm is given by

φm = (nmd ~zmd − nrd ~zrd) ·(

~k1 − ~k2

)

− 2nm~k2 · ~zm − 2nr

~k1 · ~zr. (1.22)

In most situations, we are using a laser nominally at 633 nm with a split frequency

of 5 MHz, so |~k1| − |~k2| is given by

|~k1| − |~k2| = 2π

(

1

λ1

− 1

λ2

)

= 2πδν

c≈ 0.1 m−1, (1.23)

which is significantly less than |~k1| ≈ |~k2| ≈ 1 × 107 m−1. We are therefore able

to approximate

φm ≈ −2(

nm~k2 · ~zm + nr

~k1 · ~zr)

. (1.24)

8

By making a further approximation that all the amplitudes are real, the refrac-

tive index is constant in time, and then removing the DC component of current,

we are left with a very simple

ir,AC ∝ ~A1,r · ~A2,r

(

eiωst′ + c.c.)

= Ar cos (ωst′) and (1.25)

im,AC ∝ ~A1,m · ~A2,m

(

ei(ωst′−φm) + c.c.)

= Am cos (ωst′ − φm) , (1.26)

where ωs = ω1−ω2 is the angular split frequency, Ar = 2 ~A1,r · ~A2,r is the amplitude

of the reference signal and Am = 2 ~A1,m · ~A2,m is the amplitude of the measurement

signal.

In order to extract the phase from these signals, we must obtain in-phase and

quadrature signals by multiplying im,AC by ir,AC both with and without a π/2

phase shift, as follows

I = ArAm cos (ωst′) cos (ωst

′ − φm) and (1.27)

Q = ArAm cos (ωst′ − π/2) cos (ωst

′ − φm) (1.28)

which gives us

I = B [cos (φm) + cos (2ωst′ + φm)] and (1.29)

Q = B [cos (φm − π/2) + cos (2ωst′ + φm − π/2)] , (1.30)

where B = ArAm/2. We can then apply a low-pass filter to eliminate the terms

oscillating at 2ωs and we are left with

I = B cos (φm) and (1.31)

Q = B sin (φm) . (1.32)

We can then divide Equations 1.31 and 1.32 before taking the arctangent

arctan

(

Q

I

)

= arctan

[

B sin (φm)

B cos (φm)

]

= arctan [tan (φm)] = φm +mπ, (1.33)

where m is an integer, which allows us to determine φm wrapped from −π/2 to

π/2 using a displacement interferometer.

9

1.3 Uncertainty in Displacement Interferometry

There are a number of error sources in a fiber-coupled system. The only sources

that will be analyzed in depth in this thesis are those that are unique to fiber

systems. The goal for the noise caused by these systems is to be less than the

noise already present with existing technology, so that the addition of the fiber

coupling does not significantly affect the uncertainty of the total interferometer.

In order to establish what level of uncertainty added is acceptable, we must first

determine the uncertainty inherent in free-space systems. This will just be a basic

uncertainty calculation and will not include all error sources.

The sources of uncertainty in this ideal system can be analyzed after unwrap-

ping using standard algorithms, assuming the beam is propagating in the same

direction of the movement. When this is not true, an additional cosine error [Do-

iron and Stoup, 1997] will be present. If we look at Equation 1.24, we see that we

see that

φm ≈ −2(

nm~k2 · ~zm + nr

~k1 · ~zr)

, (1.34)

where k1 and k2 are the free-space wave vectors of the laser, shifted by the AOMs

and nm and nr are the refractive indices in the subscripted parts of the beam

path. This can be rearranged to solve for zm

zm = − φm

2nmk2− nrk1

nmk2zr ≈ − cφm

4πfn− zr (1.35)

and in this case, f is the laser frequency, and φm is the measured phase, which

has uncertainty due to both the resolution and any electronic noise of the mea-

surement system. Here, we have assumed the wave vectors and refractive indices

are approximately equal, since they are very close to equal. The uncertainty con-

tributed by them is much smaller since it scales with the length zr, which is much

smaller than than the factor that will multiply the identical uncertainties in the

other term. The standard form for combined uncertainty as outlined by NIST

10

[Taylor and Kuyatt, 1994] is given by

u2c(y) =

N∑

i=1

(

∂f

∂xi

)2

u2(xi) + 2N−1∑

i=1

N∑

j=i+1

∂f

∂xi

∂f

∂xj

u(xi, xj) (1.36)

where u(x) is the uncertainty in the measurement of x and f is the underlying

function. In this equation, the second term is only non-zero if the variables are

correlated, which we will assume they are not in this thesis. By stating the sum

explicitly and computing the derivative, we have

u2c(zm) = u2(zr)+

(

cφm

4πf 2n

)2

u2(f)+

(

c

4πfn

)2

u2(φm)+

(

cφm

4πfn2

)2

u2(n). (1.37)

This equation can be rearranged into a more useful form

u2c(zm) = u2(zr)+ (zm + zr)

2

[

u(f)

f

]2

+

[

cu2(φm)

4πfn

]2

+(zm + zr)2

[

u(n)

n

]2

(1.38)

which now states the overall uncertainty in our measurement in terms of the

relative uncertainty of each variable except zr, which contributes to uncertainty

directly, and φm, which contributes to uncertainty based on the conversion from

phase to distance. An important note is that this uncertainty scales with the

physical distance, even though the wrapping of the arctangent does not allow us

to measure that quantity directly as discussed in Section 1.1. As a result of this

issue, it is a good idea to minimize the length of the measurement arm. In most

bench top experiments with a piezo stage, zm + zr can be 1-5 cm, so for our noise

floor calculations, we will use a worst-case-scenario 5 cm for zm + zr.

The uncertainty of the refractive index of air was determined to be ±5 parts

in 108 [Estler, 1985]. For an uncertainty where any value within the upper and

lower bounds, a+ and a− respectively, is equally likely the uncertainty contributed

is determined to be u = a/√3 where a is (a+ − a−)/2 [Taylor and Kuyatt, 1994].

Therefore, u(n)/n = 5 · 10−8/√3 ≈ 2.89 · 10−8, with the uncertainty contributed

by that term being ≈ 1.4 nm. This uncertainty is in the ideal case of either

knowing or controlling the environment to a very high degree. The changes in

11

Table 1.1: The changes in environmental parameters resulting in a change of +1

part in 108 in the refractive index of air [Estler, 1985].

Parameter Nominal Value

Change for which ∆n = +1

part in 108

Pressure 101.3 kPa +3.73 Pa

Temperature 20.0◦C -0.01◦C

Humidity 40% -1%

Carbon dioxide concentration 340 ppm +67 ppm

environmental parameters that result in a change of +1 part in 108 in refractive

index are given in Table 1.1. This uncertainty is a best-case-scenario of metrology

in air. Unfortunately, the environmental parameters in the laboratory is neither

controlled nor measured with this precision, the results we present will have a

larger uncertainty.

The uncertainty for laser frequency is primarily dependent on the laser source

producing the beams. We used a ThorLabs HRS015 Stabilized Helium Neon

Laser for all experiments in this thesis, which has a specified stability over one

minute of ±1 MHz. The distribution of this uncertainty is Gaussian, and the

uncertainty contributed by a normally distributed uncertainty is given by u ≈ a

where a is (a+−a−)/2 [Taylor and Kuyatt, 1994]. This results in a u(f) = 1 MHz.

When compared to the center frequency of ≈ 474 THz, u(f)/f ≈ 1.2 · 10−9. The

uncertainty contributed by this term is ≈ 0.11 nm.

The uncertainty added by the phase measurement is more difficult, as it is

dependent on the device used to measure the phase. Chen Wang, a colleague in

the group, is developing an FPGA board that has a target uncertainty of 20 pm,

so we will use that as we calculate the total uncertainty of the system.

Finally, the uncertainty due to the reference path length change is due to

12

thermal expansion of the interferometer. Our prototype interferometer is made of

Invar with a coefficient of thermal expansion of 1.2 × 10−6K−1 and the nominal

distance is 11 mm. If the temperature is known with the precision necessary to

know the refractive index to 1 part in 108, Table 1.1 gives a temperature of 0.01◦C.

Using this temperature to be consistent with the refractive index uncertainty,

we have a full width of a = 0.13 nm. This width is rectangular, as with the

uncertainty in refractive index, so the uncertainty is given by u = a/√3, or 76 pm.

Again, just like with the uncertainty in refractive index the temperature in the

laboratory is neither controlled nor measured with this precision, the results we

present will have a larger uncertainty.

When all of these sources of uncertainty are summed as determined in Equa-

tion 1.38, we arrive at an overall uncertainty of ≈ 1.4 nm because the uncertainty

in the refractive index of air is far more significant than the other sources of un-

certainty. The goal of this thesis is to present a way to fiber-couple a displacement

interferometer without significantly increasing the uncertainty. While this is a ba-

sic uncertainty estimate, it includes many of the common, large error sources that

contribute to measurement uncertainty and should give an representative basis

for comparison.

1.4 Periodic Error

Periodic error was initially presented in model by Quenelle [Quenelle, 1983] and

later experimentally verified by Sutton [Sutton, 1987]. Periodic error results from

non-ideal mixing of the two frequencies. The period is usually one wavelength

in optical path difference, but it can be a half of a wavelength for second-order

periodic error. Quenelle predicted that the upper limit on this error is 5 nm

peak-to-peak. This peak-to-peak amplitude gives us a standard deviation of ≈1.8 nm, which is similar to the method by which the uncertainty contributions

13

are calculated, which is larger than the uncertainty inherent in the system as

determined in Section 1.3 and would therefore contribute significantly to the the

uncertainty of a system. If the uncertainty due to other factors is reduced to

sufficiently close to the best-case uncertainty, periodic error becomes the limiting

factor in an interferometer system.

Figure 1.4: A schematic example of a spatially-separated displacement inter-

ferometer. The different frequencies are separated into different arms, rather

than polarizations, thus reducing the chance of frequency mixing which leads to

periodic error [Wu et al., 2002].

In a displacement interferometer using a Zeeman split laser, such as the setup

shown in Figure 1.1, polarization is the main source of frequency mixing. The

orthogonality of the two modes is not perfect, either due to the angle of linear

polarizations, or the amount of ellipticity present in both modes. Additionally,

the polarizing beamsplitter can induce problems with the polarization. Even

Brewster-type beamsplitters do not result in perfect polarization separation, even

14

when manufactured and aligned perfectly since only the reflected light is perfectly

polarized [Brewster, 1815]. Additionally, slight misalignments result in polariza-

tion, and therefore frequency, mixing. Finally, polarization mixing is possible in

the fiber delivery, and requires precise alignment of a specialized fiber in order to

minimize, but not eliminate, this error [Knarren et al., 2005].

One solution to these issues is the spatially separated interferometer. An

example of this setup is shown in Figure 1.4 [Wu et al., 2002]. Using spatially

separated beams and unconventional interferometer configurations, the frequency

leakage can be minimized and therefore periodic error can be reduced. In this

Thesis, we will be using a spatially-separated interferometer.

1.5 Optical Fibers

To begin the discussion on using optical fibers, we will first establish the theory

and notation we will be using. For step-index fibers with a cross-sectional profile

as shown in Figure 1.5, we will be using the solutions to Maxwell’s equations

for weakly-guiding fibers [Gloge, 1971]. In this case, a weakly guiding fiber is

described by the equation

∆ = (n1 − n2)/n2 ≪ 1. (1.39)

As a quick summary, and to present the notation used in this Thesis, we will

start with the equations

Ez =iZ0

k0n2i

∂Hx

∂yand (1.40)

Hz =i

k0Z0

∂Ey

∂x(1.41)

where ni is the refractive index of the material, as determined by the value of

r =√

x2 + y2, Ez and Hz are the electric and magnetic fields, respectively, k0

15

Figure 1.5: The cross-sections of a step-index and graded-index optical fiber. The

center core has a slightly higher refractive index than the surrounding cladding,

thus confining the light to the core. The step-index fiber has a ”‘top hat”’ profile

whereas the graded-index fiber has a polynomial profile in the core, with a flat

profile in the cladding [Agrawal, 2004].

is the free-space propagation constant, and Z0 is the plane-wave impedance in

vacuum. We then propose solutions of the form

Ey =Z0

ni

Hx = Ei

Jm(pr)/Jm(pa) r ≥ a

Km(qr)/Km(qa) r < a(1.42)

where Ei is the amplitude of the electric field p2 = n21k

20 − β2 and q2 = β2 − n2

2k20

with β being the propagation constant of the fiber mode. Jα(x) is a Bessel of

the first kind of order α with an argument of x and Kα(x) is a modified Bessel

function of the second kind, of order α with an argument of x. We then make

some approximations appropriate for weakly guiding fibers, apply the boundary

16

conditions and we arrive at the characteristic equation for the LP modes

p

[

Jm−1(ap)

Jm(ap)

]

= q

[

Km−1(aq)

Km(aq)

]

(1.43)

and from this eigenvalue equation, we can determine β. Note that for each integer,

m, multiple solutions may exist. These solutions are numbered by an integer n in

descending order of their β value to identify the LPmn mode.

The normalized frequency, or V-number, is frequently used to demonstrate

how many modes are supported in a fiber. It is given by the equation

V = k0a√

n21 − n2

2 = k0aNA (1.44)

where NA is the numerical aperture of the fiber, defined by the geometrical optics

total internal reflection condition

NA = n sin (θmax) =√

n21 − n2

2, (1.45)

where n is the external refractive index. Though this NA is defined paraxially,

the paraxial approximation is not typically valid for optical fibers. Also of note

is that the cutoff condition for single mode operation can be calculated using the

first root of the zero-order Bessel function, which gives V < 2.4048.

There are a few common type of fibers that will be used primarily in this

thesis. We will present some background on these fibers to facilitate discussion of

them later.

1.5.1 Single-Mode Fibers

As discussed above, single-mode fibers are fibers that only guide the fundamental

(LP00) mode. For the 633 nm wavelengths used in this thesis, we used a fiber with

an NA of 0.1 and mode field diameter of 4.3 µm. This fiber has a normalized

frequency of V = 2.1341, which is less than the single mode cutoff condition.

17

Coupling light into this type of fiber is difficult for two reasons. The first is

that the mode field area is quite small (< 20 µm2) and the second is that the

light must be coupled into the fundamental mode, since only that mode is guided.

This coupling is determined by the mode overlap integral, given by the equation

ηmn =|∫

E∗

1E2dA|2∫

|E1|2dA∫

|E2|2dA(1.46)

where ηmn is the amplitude of LPmn mode excited by an incident electric field E1

coupled into the fiber mode, Emn as described by Equation 1.42. Since ηmn gives

the percentage of the input amplitude, the percentage of power is given by the

η2mn [Snyder, 1969].

1.5.2 Multimode Fibers

Multimode fibers typically have both larger NAs and larger core diameters than

single-mode fibers. Increasing both of these parameters increases the V number

to well above the single mode cutoff condition. This results in a large number of

modes guided, each with a different spatial profile and propagation constant.

Multimode fiber is easier to couple into, not just because of a larger core area

(> 2000 µm2) because the total percentage of light coupled into the fiber is given

by the expression∑

m,n

η2mn (1.47)

where ηmn is given by Equation 1.46. If the fiber supports a reasonably large

number of modes (∼ 100), the completeness of the modal description implies that

most of the optical power incident on the core will be coupled into the fiber,

regardless of individual mode overlap.

Although it is easier to couple light into a multimode fiber, all of the different

modes propagate with different propagation constants and therefore accumulate

phase at different rates. This is known as modal dispersion and can result in

spatial or temporal information present in the fiber at the input being lost.

18

1.5.3 Graded-Index Fiber

Graded-index (GRIN) fibers are significantly different from the other fibers de-

scribed in this section, because the index of the core is a function of the position,

rather than a flat profile. The core of the fiber is has a power index profile given

by [Agrawal, 2004]

n(r) =

n1 [1−∆(r/a)α] r < a

n1 (1−∆) = n2 r ≥ a(1.48)

where α is the power of the GRIN fiber. For the simple case where α = 2, the

equation for all rays guided by this fiber is given by solving

d2r

dz2=

1

n

dn

dr(1.49)

which yields the solution

r = r0 cos (pz) + (r′0/p) sin (pz) (1.50)

where r0 and r′0 are the initial position and angle of the ray, respectively. This

shows that the rays all arrive simultaneously at periodic intervals. However, as

discussed in Section 1.5, the paraxial approximation is not valid for most fibers.

The actual description of the propagation constants is significantly more compli-

cated and not periodic [Ikuno, 1979].

1.5.4 Polarization Maintaining Fibers

Polarization maintaining fibers are a class of fiber designed to maintain polar-

ization through a large, intentional birefringence. This birefringence ensures that

unintentional polarization changes due to stresses or manufacturing defects are not

in phase with other polarization changes, and thus do not interfere constructively.

There are a number of ways to impart this birefringence, with the most common

being stress rods to take advantage of the stress-optic effect [Rogers, 2008]. In

19

this thesis, we use polarization maintaining fiber with panda-type circular stress

rods as shown in Figure 1.6.

Figure 1.6: Cross-sectional view of several type of PM fiber. The panda and

bow-tie fibers use stress rods of different shapes to create birefringence using the

stress optic effect, whereas the elliptical-clad fiber uses the geometry to create a

birefringence using the effective index.

1.6 Motivation for Fiber-Coupling

Despite the widespread use of displacement interferometry, some issues still re-

main and the intent of this thesis is to use fibers to mitigate some of them. As

more systems use displacement interferometers, there is a trend to use a single

heterodyne laser source for many axes, such as in lithography stage feedbacks

systems. This presents a coupled alignment problem for each interferometer axis,

as depicted in Figure 1.7, because drift and misalignment of steering components

affect each interferometer differently [Ellis et al., 2011].

20

Figure 1.7: Example of how a coupled alignment via beam steering optics can

adversely affect multiple interferometers, leading to needing a complete system

alignment that can be costly and time consuming.

For these applications, it is advantageous to have each individual interferom-

eter fiber-delivered, decoupling alignment between each interferometer axis. This

also has the added benefit of potentially reducing the size of the metrology frame

and loop in the system because beam steering components are eliminated.

There are many advantages to using a fiber-delivered interferometer system

in addition to the specifically multi-axis advantages mentioned above, including

relocation of heat sources (laser and electronics) far away from the system, decou-

pling interferometer alignment from source alignment, and remote sensing where

the interferometer may need to be repositioned, such as in optical probing. Fur-

thermore, operator errors can be mitigated since mechanical perturbations to the

fibers do not affect the alignment of the system and the alignment of fibers can

be done automatically by a commercial splicer.

21

The inclusion of optical fibers in such a system present new challenges, and

this thesis seeks to identify, quantify, and mitigate those challenges, as well as

producing a better physical description of displacement interferometers.

22

2 Fiber Input

The first challenge added by using optical fibers is in the delivery of the optical

signals to the metrology system. There are two types of challenges presented

and we have identified, quantified, and tested potential solutions for these ex-

tra challenges. The first is perturbation-induced phase shifts and the second is

perturbation-induced polarization changes.

We will primarily test three types of fiber in this thesis: ThorLabs P1-630A-

FC-2 single mode fiber (SM), Thor-Labs P1-630PM-FC-2 polarization maintaining

fiber (PM), and ThorLabs M31L02 GRIN multimode fiber (MM). As mentioned

in Section 1.5, launching into SM and PM fibers presents difficulties due to a

small core area as well as coupling into the fundamental mode. PM fiber presents

another issue because the polarization is only unchanged if light is launched into

one of the two eigen-polarizations of the fiber, which are usually linearly polarized

along the birefringent axes. MM fiber does not present these concerns, but it does

present others as will be presented shortly. We specifically used ThorLabs P1-

630A-FC-2 single mode fiber, Thor-Labs P1-630PM-FC-2 PM fiber, and ThorLabs

M31L02 graded index (GRIN) multimode fiber

We tested a number of situations to produce typical perturbations encoun-

tered in real-world metrology situations as well as applying no perturbations to

23

investigate the general stability of each fiber to environmental fluctuations in a

laboratory. The three perturbations we applied were lightly tapping the fiber

with a screwdriver to simulate something bumping into the fiber (tapping), ap-

plying heat by breathing on it (thermal), and gently moving the fiber to simulate

the interferometer moving on a stacked stage (move). These three perturbations

should provide an accurate representation of the suitability of the fibers under test.

The experimental setup was not isolated from normal laboratory environmental

fluctuations during these tests.

2.1 Perturbation-Induced Phase Shift

The first challenge is that of a perturbation-induced phase shift. Any mechanical

or thermal changes to the fiber will change the refractive index of that fiber, and

therefore also change the propagation constant of the light traveling in that fiber

[Hocker, 1979; Stone, 1988]. The result of this propagation constant change is a

time-dependent phase change in the light exiting the fiber. Since we are focusing

on spatially-separated beams, this phase change is not necessarily equal for both

input frequencies. In a system where the reference detector is placed before the

input fibers, this would result in the system measuring a false displacement based

on the severity and duration of the perturbation to the fiber. As such, this places

a restriction on the design where we must place the reference detector after the

input fibers.

Even taking this restriction into account, these perturbations can cause extra

challenges. A perturbation-induced, time-dependent phase change in the fiber

that launches the light of frequency ω1 into the will have an effect that is derived

similarly to the work for the ideal case shown in Section 1.2. Since we know

most of the phase terms will cancel out, we can present the electric fields on each

24

detector with those phases already removed,

~E1,r = ~A1,rei[ω1t−φf (t)], (2.1)

~E2,r = ~A2,reiω2t, (2.2)

~E1,m = ~A1,mei[ω1t−φf (t)], (2.3)

~E2,m = ~A2,mei(ω2t−φm), (2.4)

where all variables are the same as those defined in Section 1.2 except φf (t)

which is the phase change caused by the perturbation of the fiber. Similarly to

the derivation of Equations 1.25 and 1.26, we can show that the AC part of the

current on the detectors is given by

ir,AC ∝ Ar cos [ωst′ − φf (t

′)] and (2.5)

im,AC ∝ Am cos [ωst′ − φf (t

′)− φm] , (2.6)

where again all variables are the same as those defined in Section 1.2. It is

important to note at this point that these signals are the actual detected signals,

which then need to be processed. As such, the total frequency of the detected

signals for the reference and measurement detector fr and fm respectively, given

by

fr =1

2π

[

ωs −d

dtφf (t

′)

]

and (2.7)

fm =1

2π

{

ωs −d

dt[φf (t

′)− φm]

}

(2.8)

must be within the bandwidth of the full detection and processing system. Finally,

we can derive the processed in-phase and quadrature signals by multiplying the

signals together both with and without a π/2 phase shift added to ir,AC

I = B {cos [φm] + cos [2ωst′ − 2φf (t

′)− φm]} and (2.9)

Q = B {cos [φm − π/2] + cos [2ωst′ − 2φf (t

′)− φm]} , (2.10)

25

before using a low-pass filter to eliminate the terms with a 2ωs frequency, as long

as the time derivative of the perturbation-induced phase shift is not large enough

to reduce the instantaneous frequency of the undesired 2ωs term to below the

low-pass filter’s cutoff frequency. From this derivation, we can see that there

are two ways in which problems can arise from this phase shift. The first is by

increasing the frequency of the signal at the detector until it exceeds the frequency

response of our electronics, therefore causing aliasing. The second is by lowering

the frequency of the term that should be filtered out in both the in-phase and

quadrature signals until it is below the low-pass filter’s cutoff frequency. These

can be mitigated by ensuring a high frequency response for the electronics and

using a large split frequency so that the instantaneous frequency of the 2ωs term,

given by

fr,2ω =1

2π

[

2ωs −d

dt2φf (t

′)

]

and (2.11)

fm,2ω =1

2π

{

2ωs −d

dt[2φf (t

′)− φm]

}

(2.12)

is larger than the cutoff frequency, even with the maximum time derivative of

the perturbation-induced phase change. However, this further increases the re-

quired bandwidth of the electronics, and higher bandwidth electronics are more

expensive, so we quantified this phase shift.

We used the setup in Figure 2.1 to measure this phase shift, . A linearly

polarized optical beam from a single mode stabilized HeNe laser was separated

into two different paths, where each beam was frequency up-shifted using sepa-

rate acousto-optic modulators (AOMs) driven at slightly different RF frequencies,

resulting in a constant, known heterodyne frequency of 5 MHz. This single-pass

interferometer directly measures the phase change of the light going through the

different types of fiber as compared to a reference that propagates in free-space in

the same way that a displacement interferometer measures the phase caused by a

moving stage.

26

Figure 2.1: Setup for quantifying the perturbation-induced phase shifts for

different fibers. The goal is to determine the conditions and components that

are needed for fiber launching and outcoupling collimating to ensure that a

manageable phase shift can be maintained despite external perturbations on the

fiber.

None Thermal Move Tapping10

0

101

102

103

104

105

Max

imum

Fre

quen

cy (

Hz)

None Thermal Move Tapping10

0

101

102

103

104

105

Max

imum

Fre

quen

cy (

Hz)

MMPMSM

Figure 2.2: The maximum frequency resulting from every attempted perturbation

on every tested type of fiber.

The maximum frequency shift recorded with no perturbations intentionally

applied to the fiber is shown in Figure 2.2. The standard deviation of the frequency

27

shift with no perturbations applied is approximately 15.5 Hz for each fiber. This

noise level is the same for all three fibers. Additionally, there are no significant

differences to either the peak or standard deviation (approximately 15.5 Hz for all

fibers) of the noise level when applying heat to the fiber as shown in Figure 2.2.

The recorded phase while lightly tapping on the fibers with a screwdriver is

in Figure 2.3. The instantaneous frequency shift for one characteristic event is

shown in Figure 2.4 and the maximum frequency shift is shown in Figure 2.2.

That tapping the fiber resulted in the largest frequency shift is to be expected

because, though the phase shift in Figure 2.3 is small, the time scale over which it

changes is also very small. Since the frequency shift is the time derivative of the

phase shift, even small changes over a shorter time scale result in large frequency

shifts. The PM fiber has a significantly larger peak frequency shift than the other

fiber types, which had very similar maxima. The cause of this is suspected to be

the stress rods in the panda fiber acting as springs, causing the initial strike to

ring out for longer, thus potentially interacting with other perturbations.

0 2 4 6 8 10−3

−2

−1

0

1

Time (ms)

Fal

se d

ispl

acem

ent (

µm

)

0 2 4 6 8 10−3

−2

−1

0

1

Time (ms)

Fal

se d

ispl

acem

ent (

µm

) MMPMSM

Figure 2.3: The structure of the phase shift for one tap on different types of fiber.

The axes have been offset for clarity.

28

0 50

−60

−40

−20

0

20

40

60

Time (ms)

Fre

quen

cy (

kHz)

0 50

−60

−40

−20

0

20

40

60

Time (ms)

Fre

quen

cy (

kHz)

−20

−15

−10

−5

0

5

10

15

Equ

ival

ent v

eloc

ity (

mm

/s)

−20

−15

−10

−5

0

5

10

15

Equ

ival

ent v

eloc

ity (

mm

/s)

MMPMSM

Figure 2.4: The frequency resulting from the phase shift of one tap on different

types of fiber. The axes have been offset for clarity

The peak frequency shift that results from moving the fiber slowly is shown in

Figure 2.2. In this test, the PM fiber results in the smallest peak frequency shift,

with the MM fiber resulting in the highest peak frequency shift.

These frequency shifts are all below 100 kHz, which is small compared to the

split frequency of 5 MHz. Furthermore, the frequency response required by this

system measuring a lithography stage moving at a velocity v ∼1 m/s is given by

f =1

2π

(

ωs +d

dtφm

)

=1

2π

(

ωs +2πnv

λ0

)

≈ 6.6 MHz (2.13)

is also much larger than the maximum perturbation-induced, so it does not af-

fect the necessary frequency response. Therefore, in this system, the phase shift

induced by these perturbations can be ignored. In a system that used a smaller

split frequency, the effects of these perturbations could affect the requirements of

the electronics.

29

Figure 2.5: Setup for quantifying the perturbation-induced polarization changes

for different fibers. The goal is to determine the conditions and components that

are needed for fiber launching and outcoupling collimating to ensure that a man-

ageable polarization changes can be maintained despite external perturbations

on the fiber. The polarizer before the fiber is to ensure linearly polarized light is

entering the fiber and the polarimeter is a ThorLabs PAX5710.

2.2 Perturbation-Induced Polarization Changes

As shown in Equations 1.20 and 1.21, the amplitude of the interference term di-

rectly depends on the dot product of the input polarizations. If the dot product of

those polarizations decreases, then the signal strength decreases. Unfortunately,

as mentioned in Section 1.5, the polarization of light going through most fiber

types is subject to polarization changes due to perturbation-dependent birefrin-

gence. The modal dispersion results in the birefringence affecting the different

modes differently since the birefringence is based on the difference in effective in-

dex of the modes, rather than the material index. This fact, in addition to the

mode mixing discussed in Section 2.1, led us to expect a larger polarization shift

for MM fiber than for SM fiber, as well as expecting nearly no polarization shift

for PM fiber for the reasons discussed in Section 1.5.4.

30

0 0.2 0.4 0.6 0.8 1−90

−45

0

45

Time (hours)

Pol

ariz

atio

n an

gle

(deg

rees

)

0 0.2 0.4 0.6 0.8 1−90

−45

0

45

Time (hours)

Pol

ariz

atio

n an

gle

(deg

rees

) MMSMPM

Figure 2.6: The aziumthal angle of the polarization vectors of the light exiting

the fibers while applying no perturbations over the course of one hour.

We used the setup in Figure 2.5 to measure these polarization shifts, . We

applied the same perturbations as applied in Section 2.1 while measuring the po-

larization output in real-time. Additionally, we measured the long-term stability

of the polarization output of each fiber.

The long-term stability azimuthal angle and ellipticity can be seen in Fig-

ures 2.6 and 2.7, respectively. Aside from a small asymptotic drift in the single

mode fiber, the stability of the polarization through each fiber is very good without

any intentional perturbations.

After applying all perturbations, the largest polarization variation was caused

by moving the fiber. This is expected, since though the birefringence induced by

bending is relatively small [Ulrich et al., 1980], that birefringence is multiplied by

the distance through which the beam propagates to obtain the phase lag between

the polarizations [Rogers, 2008]. Therefore, the net phase lag caused by a simple

movement will be the largest of all perturbations. The azimuthal angle and ellip-

31

0 0.2 0.4 0.6 0.8 1−40

−30

−20

−10

0

10

Time (hours)

Elli

ptic

ity (

degr

ees)

0 0.2 0.4 0.6 0.8 1−40

−30

−20

−10

0

10

Time (hours)

Elli

ptic

ity (

degr

ees)

MMSMPM

Figure 2.7: The ellipticity of the polarization vectors of the light exiting the

fibers while applying no perturbations over the course of one hour.

ticity of the polarization of the light exiting each of the fibers when moving the

fibers is shown in Figures 2.8 and 2.9, respectively.

In order to quantify the loss of signal that would be possible, we calculated

the minimum of the dot product of the polarization Stokes vectors at each time

point with every other time point, which is to say

f(t) = minτ

[

~S(t) · ~S(τ)]

(2.14)

where ~S(t) is the Stokes vector at time t and τ is a time variable which contains

the same values as t. This function shows what the minimum dot product (out

of a maximum of unity) of the polarization would be for each time point when

compared to all other time points taken over the duration of the test, which gives

us the potential worst-case signal level over the test period. The application of this

function to the data taken while moving the fiber gives us results in Figure 2.10

The PM fiber ensures that the polarization overlap is relatively high at all

32

0 4 8 12 16 20−90

−45

0

45

90

Time (s)

Pol

ariz

atio

n an

gle

(deg

rees

)

0 4 8 12 16 20−90

−45

0

45

90

Time (s)

Pol

ariz

atio

n an

gle

(deg

rees

) MMSMPM

Figure 2.8: The aziumthal angle of the polarization vectors of the light exiting

the fibers while moving the fibers. The movement was not identical for each fiber.

0 4 8 12 16 20−50

−40

−30

−20

−10

0

10

20

30

Time (s)

Elli

ptic

ity (

degr

ees)

0 4 8 12 16 20−50

−40

−30

−20

−10

0

10

20

30

Time (s)

Elli

ptic

ity (

degr

ees)

MMSMPM

Figure 2.9: The ellipticity of the polarization vectors of the light exiting the

fibers while moving the fibers. The movement was not identical for each fiber.

33

0 4 8 12 160

0.2

0.4

0.6

0.8

1

Time (s)

Min

imum

Dot

Pro

duct

0 4 8 12 160

0.2

0.4

0.6

0.8

1

Time (s)

Min

imum

Dot

Pro

duct MM

PMSM

Figure 2.10: The minimum dot product of the polarization vectors when moving

the fibers. The function used to generate these results is shown in Equation 2.14.

times, despite the perturbation. This is the situation for which PM fiber was

designed, so its superior performance is to be expected. The SM fiber performed

significantly better than the MM fiber, and had a minimum overlap of 0.01, with

an average overlap of 0.47. The MM fiber performed very poorly, with a minimum

overlap of 6.05 · 10−6 and an average overlap of 0.03.

This experiment shows that PM fiber is by far the superior fiber for maintaining

contrast. SM fiber might be acceptable, if the jacketing were stiffened to reduce

the severity of the bends, since the magnitude of the bend-induced birefringence

is inversely proportional to the square of the bend radius. MM fiber is certainly

not acceptable for use in a displacement interferometer. Unfortunately, SM fiber

is not a significant improvement for coupling light into the system like there is

with multimode fiber.

Since the improvements between PM and SM are small, but those between PM

and MM are large, it would be advantageous to find some way in which MM fibers

34

would be acceptable. We needed to determine the cause of the larger polarization

changes in the multimode fiber to dertimine a solution. With the modal dispersion

discussed in Section 1.5.2, we suspected that the different propagation constant

the fiber modes resulted in a different polarization change for each mode, which

was then coherently combined on the output to result in a large net polarization

change. In order to test this case, we needed a way to reduce the number of modes

in the MM fiber.

2.2.1 Bend-Induced Losses

Bend-loss has been used so that multimode fibers operate in a single mode for

high-powered laser applications [Koplow et al., 2000]. The concept of bend loss is

that the refractive index profile of the fiber is changed by the geometry of the bend,

resulting in a profile that is lossier to higher-order modes than the fundamental

mode [Marcuse, 1976a; Marcuse, 1976b]. The predicted mode field deformation

from this index profile has been experimentally verified in single mode [Bao et al.,

1983], few-moded [Verrier and Goure, 1990], and large mode area fibers [Smith

et al., 2012]. The loss coefficient for each mode is given by the equation

2αmn =

√πp2e−

23(q3/β2)R

emq3/2V 2√RKm−1(aq)Km+1(aq)

(2.15)

where p, q, β, a, m and V are defined in Section 1.5, Kα(x) again is the modified

Bessel function of the second kind of order α with an argument of x, and R is the

radius of the bend and em is given by

em =

1, m = 0

2, m 6= 0,(2.16)

where 2αmn is the power per unit length loss of the LPmn mode, 2α = δP/P ,

while α is the amplitude-loss coefficient. When integrated over a length, L, the

35

10 30 50 70 90 110 130 150 170 1900

10

20

30

40

50

60

70

Core diameter (µm)

Ben

d ra

dius

req

uire

d (m

m)

10 30 50 70 90 110 130 150 170 1900

10

20

30

40

50

60

70

Core diameter (µm)

Ben

d ra

dius

req

uire

d (m

m)

0.1 NA0.2 NA0.22 NA

Figure 2.11: Necessary bend radius versus core diameter for a loss of 10 dB to

the LP11 mode. Three common fiber NAs were included for comparison.

power remaining in the LPmn mode, assuming no mode mixing, is given by

Pmn(L) = P0,mne−2αmnL (2.17)

where P0,mn is the power at the input. In order to compare the results of a bent

fiber to those of a straight fiber, we must first identify a fiber that could be bent

enough to cause an appropriate amount of loss to the higher-order modes. Since

the LP11 mode has the closest propagation constant to the fundamental mode,

all modes that are higher-order than it will experience more loss than it will.

Therefore, we decided to induce 10 dB of loss to the LP11 mode by bending the

fiber over a bent length of 0.5 m so that only a relatively short piece of fiber is

needed. The bend radius necessary for this loss for a range of core diameters and

NAs is shown in Figure 2.11.

Finding a fiber that could bend tightly enough to achieve an acceptable first-

order mode loss without breaking was important. Most fiber manufacturers either

36

Figure 2.12: Setup for quantifying the perturbation-induced polarization changes

for tightly coiled fiber. In this setup, the fiber is coiled tightly around an

aluminum rod with a radius of 40 mm. This coil is intended to remove the

higher-order modes to test the polarization effects on the LP01 mode only. The

polarizer before the fiber is to ensure linearly polarized light is entering the fiber

and the polarimeter is a ThorLabs PAX5710.

listed minimum long-term and short-term bend radii, or responded with those

specifications when asked. NuFern produces a fiber with an NA of 0.12 and a

core radius of 100 µm (MM-S105/125-12A). The bend radius for this fiber to

have 10 dB of loss to the LP11 mode is 23 mm. The minimum long-term bend

radius for this fiber is 25 mm, but the minimum short-term bend radius is 12 mm.

Therefore, this fiber would be suitable for running these tests in the same setup

as shown in Figure 2.5.

These tests produced the results shown in Figure 2.13. These results were

directly contradictory with our hypothesis, because though the minimum overlap

of the coiled fiber was larger (1.95 · 10−6 compared to 4.48 · 10−7), the mean

of the overlap of the uncoiled fiber was larger (0.0087 compared to 0.0015) and

the maximum was clearly larger (0.3266 compared to 0.0119). One explanation

37

0 4 8 12 160

0.1

0.2

0.3

Time (s)

Min

imum

Dot

Pro

duct

0 4 8 12 160

0.1

0.2

0.3

Time (s)

Min

imum

Dot

Pro

duct

UncoiledCoiled

Figure 2.13: The minimum dot product of the polarization vectors when moving

both the coiled and uncoiled fibers. The function used to generate these results

is shown in Equation 2.14.

could be that the large and inconsistent birefringence induced by the stress of

coiling the fiber at the end of the propagation could result in amplification of any

polarization changes. Another is that the 10 dB loss of the second-order mode was

not enough to offset the increased mode mixing due to the stresses and macro-

bends. Finally, it could be that the modal dispersion is not the primary cause of

the lower performance of multimode fiber.

38

3 Theoretical Investigations

In this chapter, we eliminate polarization as a source of periodic error in a spa-

cially separated interferometer. We also investigate how differential wavefront

interferometry differs when using Gaussian beams versus plane waves.

3.1 Arbitrary Input Polarizations

In general, no polarization is perfectly linear as described in Section 1.2. Both

~E1 and ~E2 can be elliptical. This has been shown to cause periodic error in

interferometers where the two frequencies are launched in different polarizations

[Wu and Deslattes, 1998]. In this Section, we will show that there is no periodic

error caused by polarizations that are not ideally linear in a spatially-separated

interferometer. We begin with a description of our incoming beams using the

formalism of Jones vectors as

~E = | ~E|(

aei(ωt−kz+α)x+ bei(ωt−kz+β)y)

(3.1)

where a2+b2 = 1, x and y are the unit vectors, and α and β are the phase delays

between the X- and Y-axes. In this plane wave, the polarization is completely

determined by a, b, α, and β. In our displacement interferometer, we have two

39

input beams which we will approximate as plane waves whose full polarization-

dependent mathematical description going into the system is given by

~E1 = | ~E1|(

a1ei(ω1t−k1z+α1)x+ b1e

i(ω1t−k1z+β1)y)

and (3.2)

~E2 = | ~E2|(

a2ei(ω2t−k2z+α2)x+ b2e

i(ω2t−k2z+β2)y)

, (3.3)

where the subscripts are for the two separated beams of different optical frequen-

cies. In the case of our fiber-fed interferometer, we will place the x and y along

the PM birefringent axes with the Y-axis as the axis along which the stress was

applied. Our first assumption is that the two fiber axes are the same, which can

be ensured by keying and precisely aligning the output couplers. After the beam

propagates through the interferometer system, the electric fields caused by each

beam on the reference detector is given by

~E1,r = | ~E1,r|(

a1,rei(ω1t+α1,r)x+ b1,re

i(ω1t+β1,r)y)

and (3.4)

~E2,r = | ~E2,r|(

a2,rei(ω2t+α2,r)x+ b2,re

i(ω2t+β2,r)y)

, (3.5)

and the electric fields caused by each beam on the measurement detector is given

by

~E1,m = | ~E1,m|(

a1,mei(ω1t+α1,m)x+ b1,me

i(ω1t+β1,m)y)

and (3.6)

~E2,m = | ~E2,m|(

a2,mei(ω2t+φm+α2,m)x+ b2,me

i(ω2t+φm+β2,m)y)

. (3.7)

As above, we are assuming that the optical paths within the interferometer

are approximately equal and we can therefore set our arbitrary t = 0 s point to

be that which sets our initial optical path phase at 0, as in Section 1.2 and the

only deviation from that φm from the interferometer to the measurement mirror

40

and back as given in Equations 1.24. The intensity on each detector is given by

Ir ∝(

~E1,r + ~E2,r

)

·(

~E1,r + ~E2,r

)

∗

= | ~E1,r|2 + | ~E2,r|2

+2| ~E1,r|| ~E2,r|a1,ra2,r cos (ωst+∆αr)

+2| ~E1,r|| ~E2,r|b1,rb2,r cos (ωst+∆βr) (3.8)

Im ∝(

~E1,m + ~E2,m

)

·(

~E1,m + ~E2,m

)

∗

= | ~E1,m|2 + | ~E2,m|2

+2| ~E1,m|| ~E2,m|a1,ma2,m cos (ωst+∆αm − φm)

+2| ~E1,m|| ~E2,m|b1,mb2,m cos (ωst+∆βm − φm) , (3.9)

where ∆αl = α1,l − α2,l and ∆βl = β1,l − β2,l. After applying a high pass filter to

remove the DC terms and combining some constants, we get a simpler

Ir ∝ Ir [ar cos (ωst+∆αr) + br cos (ωst+∆βr)] and (3.10)

Im ∝ Im [am cos (ωst+∆αm − φm) + bm cos (ωst+∆βm − φm)] , (3.11)

where Il = | ~E1,l|| ~E2,l|, al = a1,la2,l/2 and bl = b1,lb2,l/2. We then generate an

in-phase signal and apply a low pass filter to remove the 2ωs terms

I ∝ IrIm

{

aram2

cos [∆αr −∆αr + φm] +brbm2

cos [∆βr −∆βr + φm]

}

+ IrIm

{

arbm2

cos [∆αr −∆βr + φm] +ambr2

cos [∆βr −∆αm + φm]

}

(3.12)

and then a quadrature signal in the same way after shifting phase of the reference

measurement by π/2

Q ∝ IrIm

{

aram2

sin [∆αr −∆αr + φm] +brbm2

sin [∆βr −∆βr + φm]

}

+ IrIm

{

arbm2

sin [∆αr −∆βr + phim] +ambr2

sin [∆βr −∆αm + φm]

}

.

(3.13)

Since there are no polarizing optics in the system and the alignment must be

accurate for other reasons, it is a reasonable assumption that the polarization does

41

not change in the system. Since a, b, α, and β completely determine polarization,

as discussed above, all of the subscripts are equal, which is to say ar = am = a,

br = bm = b, ∆αr = α1,r − α2,r = α1 − α2 = α1,m − α2,m = ∆αm, and ∆βr =

β1,r − β2,r = β1 − β2 = β1,m − β2,m = ∆βm. Therefore, we can write the in phase

and quadrature signals much more simply as

I ∝ ImIr

[

a2

2+ ab cos (∆α−∆β) +

b2

2

]

cos (φm) and (3.14)

Q ∝ ImIr

[

a2

2+ ab cos (∆α−∆β) +

b2

2

]

sin (φm) (3.15)

which can be combined as in Equation 1.33 to obtain

arctan

(

Q

I

)

= arctan

ImIr

[

a2

2+ ab cos (∆α−∆β) + b2

2

]

ImIr[

a2

2+ ab cos (∆α−∆β) + b2

2

]

sin (φm)

cos (φm)

= φm+mπ,

(3.16)

which gives us the desired φm since the equal amplitude coefficients cancel. Note

that, even if there are phase changes that are different along the two axes of the

fiber, they are equal for the in phase and quadrature signals, and therefore cancel

in the final arctangent. Thus, interferometers with spatially separated beams are

free from any periodic error due to imperfect polarization, independent of the

input polarization of either beam.

3.2 Rectangular Detector for Differential Wave-

front Interferometry

Differential wavefront interferometry is a relatively new technology originally used

to align beams to more precise angles than could be accomplished previously

[Muller et al., 2005]. This technology has been adapted to measure the angular

deflection of a measurement point along two degrees of freedom simultaneously

with the displacement of that measurement point [Schuldt et al., 2009]. This

42

Figure 3.1: A schematic of a rectangular quadrant photodiode. The quadrants

are labeled A through D, the width and height of the full detector is 2w and 2h,

respectively.

simultaneous measurement of the displacement along with the angular deflection

with a single measurement beam can vastly reduce the complexity of metrology

systems that could require three or more measurement beams to do the same

thing. The concept is that a quadrant photodiode is used as the measurement

detector, and phase shifts are measured independently for each quadrant. These

phase shifts are different from quadrant to quadrant if there is some angle at which

the beam is deflected.

For a plane wave, the electric fields at the measurement detector is given by

E1 = |E1|eiω1t and (3.17)

E2 = |E2|ei[ω2t+φm+k(2βx+2αy)] (3.18)

where φm is given by Equation 1.24, α is the angle that the measurement point

is deflected in the y-axis (pitch), and β is the angle that the measurement point

is deflected in the x-axis (yaw). They are both multiplied by two because if the

43

mirror at the measurement point is deflected by α, then the reflected beam will

be at twice that angle to the incident beam. The intensity on the detector is then

given by

I = |E1|2 + |E2|2 + |E1||E2|ei[ωst−φm−k(2βx+2αy)] + c.c. (3.19)

Unlike in Section 1.2, this intensity depends on both x and y, which are in-

plane variables. Therefore, the integration for each photodiode must be carried

out to determine the power, which is proportional to the signal. These integrals

are given by

A ∝∫ w

0

∫ h

0

dydxI(x, y), (3.20)

B ∝∫ w

0

∫ 0

−h

dydxI(x, y), (3.21)

C ∝∫ 0

−w

∫ 0

−h

dydxI(x, y), and (3.22)

D ∝∫ 0

−w

∫ h

0

dydxI(x, y), (3.23)

where the limits are the dimensions of the detector in Figure 3.1. The integrations

can be carried out to give

A ∝ 1

k2αβei[ωst+φm−k(βw+αh)] sin (kβw) sin (kαh) + c.c. (3.24)

B ∝ 1

k2αβei[ωst+φm−k(βw−αh)] sin (kβw) sin (kαh) + c.c. (3.25)

C ∝ 1

k2αβei[ωst+φm−k(−βw−αh)] sin (kβw) sin (kαh) + c.c. and (3.26)

D ∝ −1

k2αβei[ωst+φm−k(−βw+αh)] sin (kβw) sin (kαh) + c.c. (3.27)

where the amplitude is decreased for larger angles (either pitch or yaw), as well

as the phase changing over the range of motion. After processing to calculate the

44

in-phase and quadrature signals for quadrant A, we get

IA ∝ B

k2αβ

[

− cos (−φm + 2kβw + 2kαh) + cos (−φm + 2kαh)

+ cos (−φm + 2kβw)− cos (−φm)]

and (3.28)

QA ∝ B

k2αβ

[

− sin (−φm + 2kβw + 2kαh) + sin (−φm + 2kαh)

+ sin (−φm + 2kβw)− sin (−φm)]

(3.29)

where B = ArAm/4 with both Ar and Am defined in Section 1.2. We then divide

the quadrature signal by the in-phase signal and take the arctangent to recover

both φm, which contains the displacement as well as α and β. Though the results

of the arctangent are not obvious, Section 3.2.1 will discuss a quicker numerical

way to recover the appropriate angle.

However, the optical beams in our system are not plane waves. Therefore, we

must consider the case of a Gaussian beam incident on a rectangular quadrant

detector as defined in Figure 3.1 where each quadrant is a w x h photodiode with

the center of the full detector being defined as the origin. Each beam incident on

the quadrant detector is assumed to be a fundamental-order Gaussian as described

by

E(r) = |E| ω0

ω(z)e−

r2

ω(z)2−ik r2

2R(z)−ikz+iζ(z)+iωt

, (3.30)

where r is the distance from the origin, ω(z) is the beam waist, which is defined

as the distance from the origin to 1/e electric field amplitude or 1/e2 intensity, ω0

is the minimum beam waist (defined as w(0)), R(z) is the radius of curvature of

the wavefront, ζ(z) is the Gouy phase shift. The Gaussian parameters are given

45

by

ω(z) = ω0

√

√

√

√

(

1 +

(

z

zR

)2)

, (3.31)

R(z) = z

[

1 +(zRz

)2]

, and (3.32)

ζ(z) = arctan

(

z

zR

)

, (3.33)

where zR is the Rayleigh range given by

zR =πω2

0

λ. (3.34)

In this section, we will assume that the reference detector still has the same

signal as given by Equation 1.25, so we will only concentrate on the signal on each

quadrant of the measurement detector. The two beams incident on this detector

are given by

E1(x, y) = |E1|ω0,1

ω(z1)exp

[

−(x− δx1)2 + (y − δy1)

2

ω(z1)2− ik

(x− δx1)2 + (y − δy1)

2

2R(z2)

+ iζ(z1) + iω1t

]

and (3.35)

E2(x, y) = |E2|ω0,2

ω(z2)exp

{

−(x− δx2)2 + (y − δy2)

2

ω(z2)2− ik

(x− δx2)2 + (y − δy2)

2

2R(z2)

+iζ(z2) + iφm+iω2t+ ik [2β(x− δx2) + 2α(y − δy2)]

}

(3.36)

where all non-Gaussian variables are the same as in Equations 3.17 and 3.18. The

electric field given on the detector is the sum of these two fields, and the intensity

is given by

I(x, y) ≈ cnǫ02

|E1(x, y) + E2(x, y)|2 ∝ A21e

−2(x−δx1)

2+(y−δy1)2

ω(z1)2

+ A22e

−2(x−δx2)

2+(y−δy2)2

ω(z2)2 + A1A2 exp

{

−[

(x− δx1)2 + (y − δy1)

2]

( 1

ω(z1)2+

ik

2R(z1)

)

}

exp

{

−[

(x− δx2)2 + (y − δy2)

2]

(

1

ω(z2)2− ik

2R(z2)

)}

∗ exp[

i(

ζ(z1)− ζ(z2))]

exp {ik [2β(x− δx2) + 2α(y − δy2) + ωst− φm]}+ c.c.

(3.37)

46

where A1 = |E1|ω0,1/ω(z1) and A2|E2|ω0,2/ω(z2). Also, δx1 is the distance along

the x-axis that the centroid E1 is from the center of the detector, δy1 is the distance

along the y-axis that the centroid E1 is from the center of the detector, and δx2

and δy2 are the same for E2. These distances are given by

δx1 = x0,1, (3.38)

δy1 = y0,1, (3.39)

δx2 = x0,2 + 2βzm,d, and (3.40)

δy2 = y0,2 + 2αzm,d, (3.41)

where x0,1 is the initial misalignment along the X-axis of the centroid of E1, y0,1

is the initial misalignment along the Y-axis of the centroid of E1, x0,2 and y0,2

are the same for E2, and zm,d is the distance from the measurement mirror to the

detector. As before, we need to integrate the intensity to determine the signal on

each detector. These integrations for the signal on each photodiode are given by

Equations 3.20, 3.21, 3.22, and 3.23. In carrying out these integrations, we used

∫

e−(ax2+2bx+c)dx =

1

2

√

π

ae

b2−aca erf

(√ax+

b√a

)

+ const. (3.42)

[Abramowitz and Stegun, 1964, p. 303] and Equation 3.37 can be written in the

form given by this equation

I(x, y) ∝ D.C.+ A1A2 exp

{

−(

1

ω(z1)2+

1

ω(z2)2+

ik

2R(z1)− ik

2R(z2)

)

x2

+ 2

[

ikβ +

(

1

ω(z1)2+

ik

2R(z1)

)

δx1 +

(

1

ω(z2)2− ik

2R(z2)

)

δx2

]

x

}

∗ exp{

−(

1

ω(z1)2+

1

ω(z2)2+

ik

2R(z1)− ik

2R(z2)

)

y2

+ 2

[

ikα +

(

1

ω(z1)2+

ik

2R(z1)

)

δy1 +

(

1

ω(z2)2− ik

2R(z2)

)

δy2

]

y

}

∗ exp[

−(

1

ω(z1)2+

ik

2R(z1)

)

(

δx21 + δy21

)

−(

1

ω(z2)2− ik

2R(z2)

)

(

δx22 + δy22

)

]

∗ exp [i (ζ(z1)− ζ(z2) + ωst− φm − 2kβδx2 − 2kαδy2)] + c.c., (3.43)

47

where a is the term multiplying both x2 and y2, the b term is different for each

direction, and the the c term is all of the terms that contain neither x nor y

dependence. The D.C. terms will be filtered out, so they have been collected and

ignored. We can then integrate this equation in both x and y to determine the

power on the A quadrant

A ∝∫ w

0

∫ h

0

dxdyI(x, y) = D.C.+A1A2

2

√

π1

ω(z1)2+ 1

ω(z2)2+ ik

2R(z1)− ik

2R(z2)

∗ exp[

−(

1

ω(z1)2+

ik

2R(z1)

)

(

δx21 + δy21

)

−(

1

ω(z2)2− ik

2R(z2)

)

(

δx22 + δy22

)

]

∗ exp [i (ζ(z1)− ζ(z2) + ωst− φm − 2kβδx2 − 2kαδy2)]

∗ exp

[

ikβ +(

1ω(z1)2

+ ik2R(z1)

)

δx1 +(

1ω(z2)2

− ik2R(z2)

)

δx2

]2

1ω(z1)2

+ 1ω(z2)2

+ ik2R(z1)

− ik2R(z2)

∗ erf[√

1

ω(z1)2+

1

ω(z2)2+

ik

2R(z1)− ik

2R(z2)x

+ikβ +

(

1ω(z1)2

+ ik2R(z1)

)

δx1 +(

1ω(z2)2

− ik2R(z2)

)

δx2

√

1ω(z1)2

+ 1ω(z2)2

+ ik2R(z1)

− ik2R(z2)

]∣

∣

∣

∣

∣

w

0

∗ exp

[

ikα +(

1ω(z1)2

+ ik2R(z1)

)

δy1 +(

1ω(z2)2

− ik2R(z2)

)

δy2

]2

1ω(z1)2

+ 1ω(z2)2

+ ik2R(z1)

− ik2R(z2)

∗ erf[√

1

ω(z1)2+

1

ω(z2)2+

ik

2R(z1)− ik

2R(z2)y

+ikα +

(

1ω(z1)2

+ ik2R(z1)

)

δy1 +(

1ω(z2)2

− ik2R(z2)

)

δy2√

1ω(z1)2

+ 1ω(z2)2

+ ik2R(z1)

− ik2R(z2)

]∣

∣

∣

∣

∣

h

0

+ c.c., (3.44)

where the power on the other quadrants is given simply by changing the limits of

integration to those in Equations 3.21, 3.22, and 3.23. An important note in the

way that the complex conjugate can be performed is that the complex conjugate

of an error function is the same as an error function of the complex conjugate of

48

the argument, which is to say

erf∗ (z) = erf (z∗) , (3.45)

where z is an arbitrary complex number of the form z = x+ iy where both x and

y are arbitrary real numbers [Abramowitz and Stegun, 1964, p. 297].

Equation 3.44 can be simplified by making two approximations. The first is

that the real part of the x2 term is much larger than the imaginary part. Despite

being multiplied by the wave vector, the difference in the radius of curvature is very

small. For the 5 cm offset discussed in Section 1.3 and w0 of 1.5 mm, 1/w(z1)2 +

1/w(z2)2 > 103|k/2R(z1)−k/2R(z2)| between 0 and 1 m from the minimum beam

waist. Even for a 100 cm offset, 1/w(z1)2 + 1/w(z2)

2 > 102|k/2R(z1)− k/2R(z2)|for the same parameters. The second is that, with the above parameters again

the same, ω(z1) ≈ ω(z2), since ω(z1) > 105|ω(z1) − ω(z2)| for a 5 cm offset and

ω(z1) > 103|ω(z1) − ω(z2)| for a 100 cm offset. Finally, we calculate that δx

and δy for both beams are very small compared to the detector size. Assuming

misalignments result in initial beam distances ∼100 µm, the range of rotation is

200 µrad, and zm,d is ∼0.1 m, we still only centroid displacements ∼100 µm. If

we apply these approximations, we get a much simpler version of Equation 3.44

A ∝∫ w

0

∫ h

0

dxdyI(x, y) = D.C.+A1A2

4ω(z1)

2ei(ζ(z1)−ζ(z2)+ωst−φm)

exp

(−k2β2ω(z1)2

2

)

erf

( √2x

ω(z1)+

ikβω(z1)√2

)∣

∣

∣

∣

∣

w

0

exp

(−k2α2ω(z1)2

2

)

erf

( √2x

ω(z1)+

ikαω(z1)√2

)∣

∣

∣

∣

∣

h

0

+ c.c, (3.46)

where the scaling of the phase now only depends on the size of each beam and the

detector. This form is much easier to understand, though we will use the more

exact version for all calculations in this thesis.

49

3.2.1 Numerical Simulation

Before determining the total signal for each quadrant, as well as the calculated

angle, we examined the equation to simplify the required computer simulation.

After the D.C. components are filtered out, we have an equation of the form

A = fA(z, α, β)eiωst + f ∗(z, α, β)e−iωst (3.47)

which we then apply the same modifications of the reference signal as in Equa-

tions 1.27 and 1.28

IA =1

2

(

eiωst + e−iωst) [

fA(z, α, β)eiωst + f ∗(z, α, β)e−iωst

]

and (3.48)

QA =1

2

(

ei(ωst−π/2) + e−i(ωst−π/2)) (

fA(z, α, β)eiωst + f ∗(z, α, β)e−iωst

)

(3.49)

which are the in-phase and quadrature signals for the A quadrant. After carrying

out this multiplication and filtering out the signals at 2ωs, we have

IA =1

2[fA(z, α, β) + f ∗(z, α, β)] and (3.50)

QA =i

2[fA(z, α, β)− f ∗(z, α, β)] (3.51)

which is equivalent to

IA = Re [fA(z, α, β)] and (3.52)

QA = − Im [fA(z, α, β)] (3.53)

and since we are then taking the arctangent, we can see that we get

φA = − arctanIm [fA(z, α, β)]

Re [fA(z, α, β)]= −θA +mπ (3.54)

where φA is the angle of the complex quantity described by fA(z, α, β) = |fA(z, α, β)|eiθA .

We can apply these techniques to the plane wave case discussed in Equa-

tions 3.24, 3.25, 3.26, and 3.27 where we can recover the angles just as the arct-

angent would to get

50

φA = −φm + k (βw + αh) (3.55)

φB = −φm + k (βw − αh) (3.56)

φC = −φm + k (−βw − αh) and (3.57)

φD = −φm + k (−βw + αh) . (3.58)

To convert these into physical displacements, we divide them by the factor

in Equation 1.24 with the approximation that the refractive index is 1 and wave

vectors are equal, we have z = φ/ (2k). Therefore, we have each of our physical

displacements as

zA = − (zm + zr) +(βw + αh)

2(3.59)

zB = − (zm + zr) +(βw − αh)

2(3.60)

zC = − (zm + zr) +(−βw − αh)

2and (3.61)

zD = − (zm + zr) +(−βw + αh)

2. (3.62)

We can determine the displacement by summing all displacements and dividng

by 4zA + zB + zC + zD

4= − (zm + zr) (3.63)

whereas the angles are determined by subtracting the right from the left then

dividing by the total detector length for yaw and the top from the bottom then

dividing by the total detector height for pitch

zA + zB − zC − zD2w

= β and (3.64)

zA + zD − zB − zC2h

= α (3.65)

which gives us an ideal, plane wave way of calculating the angles which we can

the apply to our Gaussian case to determine the difference.

51

Since our approximation in Equation 3.46 shows that the most significant effect

is from different size beam waists, we simulated the measurements that would be

recorded with three different beam waists over an angular ramp, using the length

from the plane wave assumption. These simulations are shown in Figure 3.2 and

show that the effect on the scaling of the measured data is significant. Addition-

ally, our model predicts that our system will lose sensitivity with any beam waist

at ≈100 µrad. Though the measurements made by our system are not linear, for

smaller rotation angles they can be approximated as linear, as seen in Figure 3.3.

0 50 100 150 2000

50

100

150

200

Mirror Rotation (µrad)

Mea

sure

d R

otat

ion

(µrad

)

0 50 100 150 2000

50

100

150

200

Mirror Rotation (µrad)

Mea

sure

d R

otat

ion

(µrad

)

ω0 = 1.5 mm

ω0 = 2 mm

ω0 = 2.5 mm

Ideal

Figure 3.2: A simulation of the signal measured by our interferometer at several

beam waists. The three beam waists result in a different scaling of the measured

rotation. All three also lose sensitivity at larger rotation angles, with the largest

oscillating as the angle increases.

Since this instrument is intended to measure small angles, we focused on the

linear regime. We simulated the measured rotation for many different beam waists

between 1 mm and 4 mm. The detector size was fixed at the size of the detector

we used in the laboratory (2.5 mm x 2.5 mm square pixels). These results are

shown in Figure 3.4 and we see that the detected rotation angle increases mono-

52

0 10 20 30 40 50 600

20

40

60

Mirror Rotation (µrad)

Mea

sure

d R

otat

ion

(µrad

)

0 10 20 30 40 50 600

20

40

60

Mirror Rotation (µrad)

Mea

sure

d R

otat

ion

(µrad

)

ω0 = 1.5 mm

ω0 = 2 mm

ω0 = 2.5 mm

Ideal

Figure 3.3: A simulation of the signal measured by our interferometer at several

beam waists over a small rotation angle. The three beam waists result in a

different scaling of the measured rotation. Over this range of rotation angles, our

system is very close to linear.

tonically, though not linearly with beam waist. This behavior is expected from

approximation shown in Equation 3.46.

These scaling factors, mainly due to the beam waist size, are because the

Gaussian nature of the beam weights the center of the spot more than the edges,

therefore reducing the active area of the detector. This results in an “effective

length” to be applied to the difference instead of the detector length in Equa-

tions 3.64 and 3.65. These effective dimensions will result in the correct angle,

due to the linearity shown in Figure 3.3 and can be calculated in the same manner

zA + zB − zC − zDw

≈ β and (3.66)

zA + zD − zB − zCh

≈ α (3.67)

where w is the effective width of the detector and h is the effective height of the

detector. In Figure 3.5, we compare the effective length to the actual length,

53

Figure 3.4: The measured rotation as a function of input rotation and beam

waist. The rotation measured for each beam waist is approximately linear.

Additionally, the ratio of measured angle to input angle increases monotonically,

but not linearly with beam waist.

resulting in the factor by which the actual length of the detector would have to

be multiplied to obtain the effective length.

In addition to obtaining the multiplicative coefficient between the actual and

effective lengths, we can determine the effective length, as shown in Figure 3.6.

Knowing the effective length for a range of design parameters allows us to design

an interferometer with a lower sensitivity to noise. In general, the noise for each

channel will have a strong dependence on neither detector length nor beam waist,

54

Detector size (mm)

Bea

m w

aist

(m

m)

1 1.5 2 2.5 3 3.5 4

1

1.5

2

2.5

3

3.5

4

Detector size (mm)

Bea

m w

aist

(m

m)

1 1.5 2 2.5 3 3.5 4

1

1.5

2

2.5

3

3.5

4 0

0.2

0.4

0.6

0.8

1

Figure 3.5: A map of the “sensitivity coefficient” versus detector size and beam

waist. This “sensitivity coefficient” is the multiplicative factor that would need

to be applied to the actual detector dimension to arrive at the effective dimension

versus both detector size and beam waist. To create this figure, we have assumed

a square detector.

since the first-order uncertainty does not depend on these parameters. Therefore,

a larger effective length will minimize the effect this noise will have on the overall

rotation measurement. As shown in Figure 3.6, a larger detector size and beam

waist is desired, though the frequency response of a larger detector is smaller, and

the intensity of a larger beam waist is smaller.

3.2.2 Experimental Confirmation

To experimentally test our theoretical description, we needed to determine the

values of the parameters in Equation 3.44. The most important parameter is the

55

Detector size (mm)

Bea

m w

aist

(m

m)

1 1.5 2 2.5 3 3.5 4

1

1.5

2

2.5

3

3.5

4 1.5

2

2.5

3

3.5

4

4.5

5

Detector size (mm)

Bea

m w

aist

(m

m)

1 1.5 2 2.5 3 3.5 4

1

1.5

2

2.5

3

3.5

4

Figure 3.6: A map of the effective length of a differential wavefront interferometer

designed with a range of detector sizes and beam waists.

beam waist. We used F280APC-B fiber collimators, which cite a theoretical beam

diameter (2ω0) of 3.4 mm, but a beam divergence of 0.015◦, which implies a beam

diameter of 3.1 mm since the equation for a Gaussian beam divergence is

θ =λ

πω0

(3.68)

where θ is half the full angle, in radians.

To determine the exact beam waists, we used a CCD to take images of the

beams at the detector position. We then filtered the images to remove the high

spatial frequencies caused by interference effects from the CCD by applying a

Fourier filter to them. Finally, we used the second moment method to determine

the beam waist [Siegman, 1998]. The processed images are presented in Figures 3.7

and 3.8. The CCD was not moved between when the image of beam 1 and beam

56

2 were captured, so these images also allowed us to calculate the initial relative

offset between the beams. The second moment method calculated that the average

beam waist ((2ωx+2ωy)/2) for beam 1 was 3.0 mm and beam 2 was also 3.0 mm.

x camera dimension (mm)

y ca

mer

a di

men

sion

(m

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5

0

0.5

1

1.5

x camera dimension (mm)

y ca

mer

a di

men

sion

(m

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5

0

0.5

1

1.5

Figure 3.7: Beam 1 as incident on the quadrant photodiode. The image has been

smoothed to eliminate interference effects by applying a Fourier filter.

Both beams are only approximately Gaussian, though their exact structure is

not know from their intensity pattern alone. Beam 2 in Figure 3.8 is slightly bi-

modal, which may indicate the presence of higher-order modes, which is unlikely

from 5 m long PM fiber, or aberrations from either the fiber tip or the collimator.

These deviations from Gaussian were ignored for the purpose of this modeling

and we used these beam waists and measured the distances necessary to calculate

all of the parameters for the first-order, non-aberrated Gaussian. To qualify our

57

x camera dimension (mm)

y ca

mer

a di

men

sion

(m

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5

0

0.5

1

1.5

x camera dimension (mm)

y ca

mer

a di

men

sion

(m

m)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5

0

0.5

1

1.5

Figure 3.8: Beam 2 as incident on the quadrant photodiode. The image has been

smoothed to eliminate interference effects by applying a Fourier filter.

system, we compared it to a Renishaw ML10 commercial system. This system has

been tested and is traceable to the national standards, so we have assumed that

the measurements made by the system are accurate to within allowable refractive

index errors and alignment. When generating the expected measurements, we

used the data collected by the Renishaw system as the actual rotation angle.

The results of these measurements compared to the expected results of our

measurements are shown in Figure 3.9. There was not complete agreement be-

tween this idealized model and the data we recorded. Therefore, we added wave-

front noise that would be expected from the optics in our system. The mirrors are

58

quoted as λ/10 at 633 nm surface flatness, so we added a low-spatial-frequency

phase noise with an RMS of λ/10 and numerically calculated what our system

would measure. This simulation is compared with the measured rotation angle in

Figure 3.10. When we included this non-ideal noise, our simulation agreed well

with our data with no fitting parameters.

0 2 4 6 8 10 12 14

−100

−80

−60

−40

−20

0

t (s)

Mea

sure

d R

otat

ion

(µrad

)

0 2 4 6 8 10 12 14

−100

−80

−60

−40

−20

0

t (s)

Mea

sure

d R

otat

ion

(µrad

)

RenishawSimulatedMeasured

Figure 3.9: The comparison of our ideal theoretical model with measured data.

This model simulating ideal, Gaussian beams does not agree completely with our

measurements.

The effective length for these beams, with this alignment, is 2.1 mm, deter-

mined by comparing the output of the simulation to the input. Notably, this effec-

tive length is smaller than the diameter of each Gaussian beam. A re-calculation

of the yaw from our measured data, as well as the error between that data and

the measurements from the Renishaw is shown in Figure 3.11. The error in our

recalculated yaw does not scale with the rotation of the stage. The range of the

error is ∼5 µrad, peak-to-peak. In the lab, the uncertainty in refractive index is

only ∼1 part in 107. A rigorous calculation of the uncertainty can be carried out

as described in Section 1.3, but our estimate of the uncertainty in refractive index

59

0 2 4 6 8 10 12 14−100

−80

−60

−40

−20

0

t (s)

Mea

sure

d R

otat

ion

(µrad

)

0 2 4 6 8 10 12 14−100

−80

−60

−40

−20

0

t (s)

Mea

sure

d R

otat

ion

(µrad

)

RenishawSimulatedMeasured

Figure 3.10: The comparison of our theoretical model with measured data. The

model and the measured data both agree very well, and are both different from

the measurements from the Renishaw.

is approximate, so we will merely carry out an order of magnitude calculation.

This results in an uncertainty in displacement of ∼10 nm (since the nominal zr

was closer to 20 cm than the 5 cm discussed in Section 1.3). With an effective

length of 2.1 mm, uncertainty of that magnitude on only one of the detectors

would result in an uncertainty in angle of ∼5 µrad, and there are four detectors

that could be subjected to this uncertainty. Thus, errors of the magnitude seen

in Figure 3.11 are to be expected.

60

0 2 4 6 8 10 12 14−100

−80

−60

−40

−20

0

t (s)

Mea

sure

d R

otat

ion

(µrad

)

0 2 4 6 8 10 12 14−100

−80

−60

−40

−20

0

t (s)

Mea

sure

d R

otat

ion

(µrad

) Renishaw

Measured

−10

−5

0

5

10

Err

or (µ

rad)

−10

−5

0

5

10

Err

or (µ

rad)

Error

Figure 3.11: Recalculated yaw using our modeled effective length. There is good

agreement between the two systems, and the error does not depend on rotation

angle.

61

4 Conclusions and Future Work

We have presented work in three main areas critical to a fiber-delivered, hetero-

dyne, spatially separated, differential wavefront interferometer. The importance

and final conclusions will be discussed here.

The suitability of fibers for delivery of the optical beams is dependent on both

the perturbation-induced frequency shifts and the polarization. The perturbation-

induced frequency shifts can be mitigated by designing the interferometer such

that both the reference and the measurement detectors are located after the fiber

propagation. This design ensures that, since the optical beam that propagated

through each fiber is incident on each detector, and therefore any perturbation-

induced frequency shifts would be common to both signals and ultimately cancel.

We still must quantify the maximum frequency possible, to ensure that it would

not be higher than the frequency response of the measurement system, and with

a maximum frequency shift of 100 kHz recorded, any system intended to measure

split frequencies at ∼ 1 MHz would not be affected by those perturbations.

The polarization data is not as promising. The polarization maintaining fiber

was unaffected by polarization perturbations, but it was the only fiber that was

completely reliable. Single mode fiber could potentially work with some modifi-

cations, but multimode fiber was unsuitable and attempts to identify the cause of

62

and mitigate the polarization problems have been unsuccessful so far.

We also investigated polarization as a theoretical error source. In traditional

heterodyne displacement interferometers, theoretical investigations of polarization

have led to the discovery of sources of the error known as periodic error. Spa-

tially seperated interferometers have been introduced as a way to eliminate many

of those sources of periodic error. Our theoretical investigation of the specific

interferometer used in this thesis shows that there should be no periodic error

stemming from polarization in any interferometer of similar design.

The last subject covered in this thesis was the physics and resulting angular

sensitivity of a differential wavefront interferometer fed by both ideal and non-

ideal Gaussian beams. A differential wavefront interferometer can measure two

orthogonal angular deflections simultaneously with overall displacement using one

laser beam and a quadrant measurement detector. Since we both predicted and

observed a difference in sensitivity from what would be expected with a plane

wave input, we developed a model that accurately predicted the sensitivity and

provides a method to calculate the correct angular displacement from the mea-

surements made by the interferometer. This model will be critical to calibrate any

such interferometers with only a CCD image of both beams at the measurement

detector.

4.1 Future Work

The results in Section 2.2.1 bear more investigation to determine the cause of the

poor performance of the multimode fiber, and the even worse performance of the

coiled multimode fiber. There are also additional ways to eliminate other variables.

We could taper multimode fiber to single mode size and splice it to either single

mode fiber or even polarization maintaining fiber to investigate whether that is

a possible solution to increase the ease and efficiency of launching light into the

63

fiber while maintaining interference in the system.

In Section 3.2.2, our numerical simulation matched well with our experimental

data. However, the numerical simulation was very sensitive to the exact surface

figure, which was generated randomly. Additionally, Figures 3.7 and 3.8 are not

Gaussian, as assumed in our theoretical description. A wavefront sensor would

allow us to determine the shape of electric field of each beam on our measurement

detector, and confirm the numerical simulations, as well as allow us to create a

more accurate analytic solution. Additionally, we would need to test this model

for a larger range of parameters.

64

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Opt., 37(28):6696–6700, Oct 1998.

69

A Mathematical Description of

a Non-Spatially-Separated

Displacement Interferometer

We begin the same way as the derivation covered in Section 1.2, by defining the

electric fields due to each beam on the reference detector is given as

~E1,r = ~A1,rei(ω1t−φ1,r) and (A.1)

~E2,r = ~A2,rei(ω2t−φ2,r), (A.2)

where ~A’s are the vector magnitudes of the electric fields, ω1 and ω2 are the two

angular optical frequencies, and φ1 and φ2 are the phases accrued by propagation

through their arm of the system and will be explicitly defined shortly. Addition-

ally, the electric fields due to each beam on the measurement detector is given

by

~E1,m = ~A1,mei(ω1t−φ1,m) and (A.3)

~E2,m = ~A2,mei(ω2t−φ2,m), (A.4)

70

using the same variable conventions as above. However, the phases accrued by

each beam are different, and given by

φ1,r = ~k1 · (n1~z1 + nsd~zsd) (A.5)

φ2,r = ~k2 · (n1~z1 + nsd~zsd) (A.6)

φ1,m = ~k1 · (n1~z1 + n2~z2 + 2nr ~zr + nd ~zd) (A.7)

φ2,m = ~k2 · (n1~z1 + n2~z2 +Nnm~zm + nd ~zd) , (A.8)

where ~k1 is the wave vector with a magnitude of 2π/λ1 where λ1 is the wave-

length of beam one in vacuum, and a direction parallel to the Poynting vector of

that beam and ~k2’s magnitude is similarly 2π/λ2, with a direction given by the

Poynting vector of beam two. The z’s describe the distances of the interferometer,

where z1 is the distance the beams travel before the first beamsplitter, zsd is the

distance from the first beamsplitter to the reference detector, z2 the distance the

beams travel between the two beamsplitters, zr is the distance beam one travels

to, from, and in the reference retroreflector (labelled RR1 in Figure 1.1), zm is

the distance beam two travels to, from, and in the measurement retroreflector

(labelled RR2 in Figure 1.1), and zr is the distance from the second beamsplitter

to the measurement detector. The n’s describe the refractive index (assumed con-

stant) over the path described by the same subscripts. The N is the fold factor

of the interferometer and describes how many times the beam passes through the

distance under measurement. In general, it can be large, but since our interfer-

ometer is a single reflection, it is 2. This reflection is also why there is a factor of

2 multiplying zr.

Since the notation is the same, it is easy to see that Equations 1.16 and 1.17

are still true with φ1,r − φ2,r and φ1,m − φ2,m now being given by

φ1,r − φ2,r = n1~z1 ·(

~k1 − ~k2

)

+ nsd~zsd ·(

~k1 − ~k2

)

(A.9)

φ1,m − φ2,m = n1~z1 ·(

~k1 − ~k2

)

+ n2~z2 ·(

~k1 − ~k2

)

+nd ~zd ·(

~k1 − ~k2

)

+ 2nr ~zr · ~k1 − 2nm~zm · ~k2. (A.10)

71

Since, as already discussed, |~k1|−|~k2| ≪ |~k1| ≈ |~k2|, we can approximate φ1,r−φ2,r

and φ1,m − φ2,m as

φ1,r − φ2,r = 0 (A.11)

φ1,m − φ2,m = 2nr ~zr · ~k1 − 2nm~zm · ~k2. (A.12)

Since φ1,r − φ2,r = 0, no change of variables is required, and φm is given directly

by

φm = φ1,m − φ2,m = 2nr ~zr · ~k1 − 2nm~zm · ~k2 ≈ 2nk (zr − zm) (A.13)

and similarly, we can arrive at φm after computing the in-phase and quadrature

signals, and applying a low-pass filter. The difference in this interferometer com-

pared to the one derived in Section 1.2 is that interferometer measures the sum

of the phase accrued through the reference and measurement arm, whereas this

interferometer measures the difference of those two phases.