1

Fourier Transform ImagingSpectrometer at Visible

Wavelengths

Noah R. Block

Advisor: Dr. Roger Easton

Chester F. Carlson Center for Imaging Science

Rochester Institute of Technology

May 20, 2002

2

Abstract

The purpose of the experiment is to construct a device to calculate the spectrum of all the

pixels in the scene. The experimental setup is a Michelson interferometer that has been

modified to add a second reference beam illuminated with a known wavelength. The

additional light beam is used to compensate for errors in the motion of the movable

mirror due to the imprecise and unrepeatable motor. The reference beams error is used to

correct the object beams spectrum through error analysis.

The results for a two-dimensional scene are given as a three-dimensional graph

with the intensity of the spectrum displayed along the third axis. This method gives the

spectrum of an object at every location in it with up to nanometer resolution.

Table of Contents

Abstract

Objective

Background

Design of Experiment

Data Processing

Conclusions

Advancement

Appendix A

2

3

3

8

11

16

17

19

3

Objective

The purpose of the experiment is to construct a device to measure the spectrum of a two-

dimensional object. The Fourier transform imaging spectrometer previously constructed

by Eric Sztanko was modified and extended from a proof of concept to the point where

the spectrum of a high-intensity multi wavelength object may be measured.

Background

Michelson interferometry is a method for measuring the spectrum of a single source. A

basic Michelson interferometer can be seen in figure 1.

Figure 1: The object beam goes through the beam splitter and the amplitude gets halved in each arm(graphs on L1 and L2), and then recombines. Constructive and destructive interference result at the imageplane (the two graphs next to the camera represent constructive and destructive interference).

4

It is often necessary or desirable to measure the spectrum of every pixel in a scene

simultaneously. The way a Fourier transform imaging spectrometer (FTIS) works is by

capturing the fringe patterns created by the source going through the Michelson

interferometer. As M2 moves, the fringe pattern move (thus a single pixel locations gray

value oscillates from light to dark) at a speed proportional to the motors movement, the

wavelength of the fringe pattern can be calculated if the step size of the motor is known.

As the FTIS captures more images, a longer “window” records more oscillations giving a

better resolution (resolution is discussed later). Taking the Fourier transform of the

oscillations (also called interferogram) calculates the frequencies present in the

interferogram. The frequencies are then turned into their corresponding wavelengths and

the spectrum of the source is calculated. The Fourier transform calculates the frequencies

present in the signal. When a multicolor light is used, the wavelengths will overlap

creating a different fringe pattern than a single wavelength fringe pattern creates.

Sinusoids add according to equation 1.

†

cos(k1z - w1t) + cos(k2z - w2t) = 2 ⋅ cos(k1 - k2

2⋅ z) ⋅ cos( k1 + k2

2⋅ z - wt) (1)

As can be seen in Figure 1, the beam splitter divides the amplitude of the light

into (often equal) parts. The beams that travel paths L1 and L2 (path lengths may be

different) before being recombined at the beam splitter. If the relative path lengths differ,

then fringe patterns are produced by constructive and destructive interference. Mirror M2

on the motorized stage is set to move in the longitudinal direction (left and right in Figure

1). The CCD camera captures images of the interference pattern at preset time intervals

as M2 is moved. The change in the relative optical path lengths as M2 is moved produces

5

different interference patterns. The ensemble of recorded interferograms forms an

“interferogram cube” f(x,y,t). If zµt, then f(x,y,z) may be inferred. Each captured image

is “stacked” (Figure 2) to make an image cube, f(x,y,t).

Figure 2: Image “cube,” the number of samples represents the time as the motor takes the images, thenthe x and y coordinate of the image is shown.

The gray value of a specific pixel in every layer in the interferogram cube (figure 2) is

the interferogram of that specific scene pixel, and is the Fourier transform (FT) of the

spectrum. The Fourier transform of the interferogram is the spectrum of the object as can

be seen in figure 3.

6

Figure 3: The single pixel location of every image in the cube graphed, then the Fourier transform of thegraph is taken to find the frequencies present.

An interferogram is the oscillations caused by the wavelength of the source.

Figure 3 is a nice sinusoid because that is from a single wavelength laser. Since the

interferogram represents the signal from the source, the Fourier transform represents the

frequencies present from the source.

Use of an interferometer (instead of a spectrometer) benefits from “Fellgett’s

Advantage” (also known as the multiplex advantage) where there is an increase in the

accuracy of interferometry over spectrometry by a factor of (N/2)1/2, where N is the

number of samples taken. Fellgett’s advantage is only true if all other errors are the same

for the interferometer and spectrometer. (Thorne, 1989). Finding the spectrum of a scene

by interferometry is more computationally intensive than finding the spectrum of a single

source from a spectrometer because the Fourier transform of each interferogram must be

calculated, while a spectrometer finds the spectrum directly

Two types of interferograms that can be made with this apparatus are a one-sided

and two-sided interferogram. The advantages of using a two-sided interferogram are that

the zero point distance (ZPD) does not have to be exact because the interferogram is

symmetrical. This means that there will not be phase errors due to an inaccurate ZPD.

The other benefits are that thermal or electronic drift will not affect the results, and if the

7

alignment of the interferometer is lost, the interferogram will not be symmetric, thus it

can be seen immediately if the experiment has to be redone. The disadvantage of using a

two-sided interferogram is that the maximum optical path difference (OPD) is halved and

hence, the resolution is reduced. (Bell, 1972). Due to the motor that is available, a two-

sided interferogram will be used and doubling the number of images in the image cube

will solve the loss of resolution.

One limitation of the resolution is the amount of oscillations that are recorded (the length

of the RECT function). The longer the total distance L2-L1, the better the spectral

resolution, i.e., wavelengths that are very close together are more easily resolved. One

method to reduce noise, take longer exposures, that is usually employed cannot be used

because of limitations in the performance of the stepper motor. The minimum step size

that the manufacturer says the motor can do with precision is 200nm, which corresponds

to an optical path difference (OPD) of 400nm. Therefore, the available sampling interval

would let the spectrometer resolve wavelengths of up to 800nm due to the nyquest

frequency. To bypass this problem, the motor is sub stepped to the manufacturers

specifications, this decreases the precision in the motor movements. Another problem is

that the motor drifts with respect to time. The motor was found to drift at an almost

constant velocity with minor variations that do not increase as time increases but seems to

be related to a periodic pattern. The average velocity is about 3.6nm/sec. The graph of the

velocity of the motor when power was given to the motor, but no move command was

given can be seen in appendix A. The velocity can be determined because the signal is

from the reference beam, thus the peak-to-peak distance is 632.8 nm and each interval is

one second apart.

8

Therefore, experiments that require long times to complete are less accurate. The

exposure time of the CCD must be relatively short so that the fringes do not move

significantly during the exposure. This required that the object be sufficiently bright to

ensure an adequate signal-to-noise ratio.

The ultimate goal of the experiment is to have the object beam of the spectrometer

a white-light (tungsten) source because produces the full visual spectrum. However due

to the fact that the spectrometer acts like a bandpass filter (in the wavelengths ranging

from 400-700 nm), the coherence length of a visual fringe pattern decreases as the width

of a bandpass filter increases. A RECT tunrs into a Sinc function in Fourier space, as the

RECT gets longer, the Sinc gets narrower. At a RECT of infinite length, the Sinc turns

into a delta function. This means that the coherence length of a white-light source is

small, it is approximately 1 micron.

Experimental Designs and Methods

Part One: Designing the Experiment

The original design with a single source was deficient because of the inaccuracies of the

motor. It was found that the movements were not repeatable, thus the original concept of

creating an index that could be referred to for motor step size was not possible. The

solution used is to calibrate the system during every experimental run. The system was

calibrated by adding a second source that acts as a reference beam at a known

wavelength, 632.8nm for a red HeNe laser. ReferenceBeam

Object Beam

CollimatingLens

Mirror onmotor

Beamsplitter

Stationarymirrors

9

Figure 4:The experimental setup with the reference beam and object beam

Both reference and object beams reflect from the moving mirror and are incident on the

CCD array at the same time during the same run. This removes the position error from

the calculation. The object beam is calibrated using the signal error from the reference

beam. The only apparatus traversed by the beams that are not identical are the stationary

beam splitters and mirrors. Since the positions are constant, they contribute no additional

position errors.

To decrease the size of the images that were taken, the CCD array had to be

binned. Without binning, each image was approximately 2 MBytes, with binning (4

pixels x 4 pixels binned down to1 pixel), the size went down to about 125 kBytes. Then

with cropping part of the array, the size of the image was further decreased to 77 kBytes.

With the approximately 8000 images that are needed for nanometer resolution, the

10

equation to get the resolution of this system from the number of images in the cube, as

can be seen in figure 7, is

7436.7*(#images)-1.0227 (1)

Thus with a depth of 8000 images, a 2 MByte size image would come to about 16

GBytes worth of data for a single experimental run. That is just not practical, by binning

it down, the size for a run was about 600 MBytes.

Figure 5: The smaller pixels on the left diagram are grouped to adjacent pixels making them act as onesingle larger pixel that is shown on the right.

The object that was used for the final result was a four-quadrant color square with

red, green, blue, and yellow quarters. It was printed on a transparency and placed

between the collimating lens and beam splitter. The object can be seen in figure 6.

11

Figure 6: The four-color object that the experiment was conducted with. The object beam illuminated theobject (printed on a transparency) and the resulting fringe patterns were recorded. From upper left cornergoing in clockwise order: green, red, yellow, and blue.

Part Two: Data Processing

A simple percent error method is used to find the error of the reference source. The step

size of the motor must be known to determine the distance of each data point. Equation 2

shows how to find the step size.

(2)

Where lReference is known and Period(# images) is the number of images in the image cube

that make up one full period of the reference beam. To try and reduce the amount of

error, the length of every period was found and averaged together to try and minimize the

effects of the drift and off imprecise stepping. Next, equation 3 calculates the resolution

of the system, this tells the how small of a difference in frequencies can be found.

(3)

†

Dx = lReference(nm)Period(# images)

†

Dn = 1N ⋅ Dx

12

As can be seen, as N (the depth of the interferogram cube) increases, Dn decreases, thus

better resolution. The actual frequency is then calculated by multiplying change in

frequency above by the k index (the location on the frequency axis) in the frequency

domain as seen in equation 4.

(4)

The wavelength is then found by equation 5. (5)

n has units of nm-1, thus the inverse is nm, which is the wavelength.

The percent error of the reference beam is then calculated using equation 6.

(6)

Then using the error of the reference beam, the wavelengths originally calculated for the

object beam are then corrected by rearranging equation 6 to get equation 7.

(7)

The results that were calculated by this method were either very close (within one or two

nanometers) or dead on. The only wavelengths used to test this were the same (632.8

nm), and a green HeNe with wavelengths at 543.5 and 594.1 nm. The relationship

between the size of the image cube (depth, proportional to number of images) and

resolution can be seen in figure 7.

†

n = k ⋅ Dn

†

l =1n

†

% error =lReference - lexperimental

lReference⋅100

†

lcorrected =-100 ⋅ lexperimental% error -100

13

Figure 7: The relationship between the depth of the image cube and the resolution the system will have(at 50 nm step size intervals)

Figure 7 shows that approximately 8000 images were needed in this setup to obtain a

spectral resolution of 1nm where the average OPD was fifty nanometers. Thus, a

translation distance of approximately 400 mm was necessary to obtain sufficient

resolution.

To find the spectrum of a two-dimensional object, the Fourier transform of each

interferogram was evaluated at each pixel location in the image.

14

Figure 8: The diagram shows how the object relates to the interferogram, and then how the interferogramrelates to the spectrum of the signal visualized in a two-dimensional graph.

For visualization purposes, the spectrum is separated into three graphs,

blue (400 £ l £ 500 nm), green (501 £ l £ 600 nm), and red (601 £ l £ 700 nm). The

spectrum could be displayed in any number of ways if a specific wavelength or band of

wavelengths was desired to be observed by making minor changes in the program. The

three figures (9-11) show the presence of a signal in the lower right corner, this is noise

of unknown source. Figure 9 shows no signal in the blue region, as expected. The only

signal in figure11 is upper left and is due to the reference beam. Figure 10 shows a

15

pattern that correlates to the object. This noise in figure 10 and 11 could be attributed to

stray light from reflections in the optics.

Figure 9: Blue graph (400-500 nm) Figure 10: Green graph (501-600 nm)

Figure 11: Red graph (600-700 nm)

The reason why the only detected signal is in the green region is because the object was

illuminated by a two-color laser (green and yellow), so all wavelengths will be in the

green spectrum (501-600 nm). The target attenuated the amount of green light. As can be

seen in figure 11, only a small segment of the reference beam is incident upon the image

16

plane. However a space (approximately five millimeters in this experiment) is needed

between the reference and object beams so that scattered light will not interfere with each

other. Any signal detected in the “dead area” can be attributed to stray light, as can be

seen in figure 10.

It should be noted that figures 9-11 have a resolution of about seven nanometers

because the interferogram cube had a depth of 1024 images. In the current form, the

thickness of the a two-dimensional interferogram cube cannot have more than a depth of

1024 images because the computer system (Sun Blade 1000 was used) cannot create

arrays large enough for a depth of 8000 (8192 actually used so that a fast Fourier

transform could be applied). The one-dimensional case can calculate image cubes with

depths of 8192 giving resolutions of under a nanometer.

Conclusions

It was found that the correct spectrum of an object could be found using a sub-standard

stepper motor (one that does not have repeatability at such small step sizes). The only

negative aspect of adding the second beam is that the size of the object that can be

imaged has to be decreased so that both beams can fit into the image plane (CCD array).

The addition of the reference beam to calibrate the apparatus during every

experimental run solved the problem of motor drift. It was found that “windowing”

actually makes the results worse when there are numerous sinusoids in a sample. Since

there are so many sinusoids toward the edge where the value goes either to zero, or close

to zero (depending on the type of windowing technique that is used), it fails to read some

of the oscillations, thus gives a false frequency. To try and decrease the prevalence of

noise from the results, a thresholding method was used. When the data was processed,

17

only signals that had a high enough intensity were used, through refining the data

processing, it is probably possible to use the noise from the reference beam to subtract

from the object beam. This is probably possible because the noise is systemic noise, thus

both signals should have the same noise.

Advancement

The experiments to date have demonstrated the successful implementation of the imaging

spectrometer. However, several aspects exist that could be improved during further

research. The program used to process the data could be upgraded, e.g. to support larger

arrays. The program can be changed by performing the calculations on an entire row or

column at once, continuing until it goes through the entire two-dimensional image.

Another way would be to lower the threshold that was being used, the way that might

done would be to find all of the spectral bands in the reference beam other than the

reference wavelength and subtract them from the object wavelengths.

To determine the full spectrum of an object, a white-light source is necessary. A

tungsten light is usually used for this because its spectral curve lies in the visible region.

The problem with a white-light fringe pattern is that it has a coherence length of only

about one micron. For nanometer resolution in this system, the minimum OPD is

approximately 400 microns. A possible way to bypass this problem would be to make the

mirror go back and forth every two 250 microns (OPD = 500 microns), thus keeping the

fringe pattern coherent. If the OPD goes past 1 micron then the fringe pattern will

disappear making it impossible to find the spectrum. That will most likely cause a lot of

other frequencies in the reference and object beams, but like mentioned above, if the

added frequency is systemic, then hopefully it will be the same for the reference and

18

object beams, thus using the same method aforementioned, find the superfluous

frequencies in the reference beam, then subtract them from the object.

If the above suggestions can be implemented, a full functioning Fourier transform

spectrometer will be a reality. The experiment up to this point has created a foundation

that solved some problems that were encountered, but others need to be overcome to

bring it to fruition.

19

Appendix A

Drift Velocity of Motor

Power applied to the motor when no movement command was issued. The drift velocity

of the motor seems to be fairly constant.

20

Resources

Albergotti, J. C. “Fourier Transform Spectroscopy Using a Michelson Interferometer.”American Journal of Physics Vol 40 (1972): 1070-1076.

Bell, Robert J. Introductory Fourier Transform Spectroscopy. New York, NY: Academic Press, 1972.

Berkey, Donald Kieth. “An Undergraduate Experiment in Fourier-TransformSpectrometry.” American Journal of Physics Vol 40 (1972): 267-270.

Dorrer, C., N. Belabas, J-P. Likforman, and M. Joffre. “Experimental implementation ofFourier-transform spectral interferometry and its application to the study ofspectrometers.” Applied Physics B, Lasers and Optics Vol B70 (2000): S99-S107.

Gingras, D.J. “Spectrum Estimation of FT-IR Data with Sampling Errors.” SPIEVol. 1145 (1989): 181-185.

Mertz, Lawrence. Transformations in Optics. New York: John Wiley & Sons, 1965.

Thorne, Anne. “High resolution Fourier transform spectroscopy in the ultra-violet.” SPIEVol. 1145 (1989): 43-47.