prof. hsuan-ting chang september 10, 2019, yuntech...

I. FOURIER OPTICSII. Optical Signal Processing

Prof. Hsuan-Ting Chang

September 10, 2019, YunTech University

Prof. Hsuan-Ting Chang I. FOURIER OPTICS II. Optical Signal Processing

Course Information

I Tuesday F, G, H (PM 1:10–4:00)

I Room: EL102

I PDF files of the PPT lectures can be downloaded inhttp:

//teacher.yuntech.edu.tw/htchang/OSP1061.htm


http://teacher.yuntech.edu.tw/htchang/OSP1061.htm

http://teacher.yuntech.edu.tw/htchang/OSP1061.htm

Instructor Information

I Instructor: Hsuan T. Chang, Professor

I Senior Member of OSA, SPIE, IEEE

I Phone: 05-5342601 ext. 4263

I Email: [email protected]

I URL:http://teacher.yuntech.edu.tw/htchang/

I Office: Room EN307, The Sixth EngineeringHall

I Office Hours: please make an appointment


http://teacher.yuntech.edu.tw/htchang/

Textbook

I J. Goodman, Introduction to Fourier Optics, 3rdEdition, Roberts & Company, Englewood Colorado2005


Reference Books

I Boone, Signal Processing using Optics, Oxford, 1998

I E. Hecht, Optics, 4nd Edition, Addison-Wesley, 2002

I F. Yu and I.C. Khoo, Principles of Optical Engineering, JohnWiley & Sons, 1990

I J. Gaskill, Linear Systems, Fourier Transforms, and Optics,John Wiley & Sons, 1978

I F. Yu, Optical Information Processing, John Wiley & Sons,1983

I F. Yu and S. Jutamulia, Optical Processing, Computing, andNeural Networks, John Wiley & Sons,1992

I F.Yu, S. Jutamulia, & S. Yin, Introduction to InformationOptics, Academic Press, 2001


International Journals:

I Applied Optics

I Journal of Optical Society of America, A

I Optics Communications

I Optics Letters

I Optics Express

I Optical Engineering

I IEEE/OSA Journal of Light Wave Technology

I More ...


International Conferences:

I SPIE Conferences

I OSA Conferences

I IEEE Conferences


Grading Policy:

I Midterm: 30%I Magazine article presentation: 20%

I Select an article in the provided magazines and make apresentation before the midterm exam

I Optics and Phonotics News (OPN) feature articles, OpticalSociety of America (OSA). Website:http://www.osa-opn.org/

I SPIE Newsroom technical articles, website:http://spie.org/x1004.xml

I IEEE Spectrum. Website: http://ieee.org/I Photonics Spectra and Biophotonics. Website:

http://www.photonics.com/

I Homework: 20%I Project and report: 30%

I (1) Select one paper from the assigned journals published in2018 ∼ 2019. (2) Present the papers you read (Provide yourPPT file)


http://www.osa-opn.org/

http://spie.org/x1004.xml

http://ieee.org/

http://www.photonics.com/

Covered topics:

1. Introduction

2. Analysis of 2D Signals and Systems

3. Foundations of Scalar Diffraction Theory

4. Fresnel and Fraunhofer Diffraction

5. Wave-Optics Analysis of Coherent OpticalSystems

6. Frequency Analysis of Optical Imaging Systems

7. Wavefront Modulation

8. Analog Optical Information Processing

9. Holography


Chapter 1 Introduction

1.1 Optics, Information, and Communication

1. Optics has gradually developed ever-closer ties with thecommunication and information sciences of electricalengineering.

2. Perhaps the strongest tie between the two disciplines liesin the similar mathematics which can be used to describethe respective system of interest – the mathematics ofFourier analysis and systems theory.


Chapter 2 Analysis of Two-Dimensional Signals andSystems

Contents:2.1 Fourier Analysis in Two Dimensions2.2 Local Spatial Frequency and Space-Frequency Localization2.3 Linear Systems2.4 Two-Dimensional Sampling Theory


2.1.1 Definition and Existence Conditions - 1

1. 2D Fourier transform of a function g of two independentvariables x , y is defined by

F{g} =

∫ ∞−∞

∫ ∞−∞

g(x , y) exp[−j2π(fXx + fY y)]dxdy , (2.1)

where fX and fY are referred to as spatial frequencies.

2. The inverse Fourier Transform of a function G (fX , fY ) isdefined as

F−1{G} =

∫ ∞−∞

∫ ∞−∞

G (fX , fY ) exp[j2π(fXx + fY y)]dfXdfY .

(2.2)



3. Sufficient conditions for the existence:(1) g must be absolutely integrable over the infinite (x , y)plane.(2) g must have only a finite number of discontinuities,maxima, and minima in any finite rectangle.(3) g must have no infinite discontinuities.



4. In some idealized mathematical functions, one or more of theabove conditions may be violated. For example, theDirac-delta function

δ(x , y) = limN→∞

N2 exp[−N2π(x2 + y 2)]. (2.3)

Other important examples are, for example, the functionsf (x , y) = 1, and f (x , y) = cos(2πfXx).

5. It is often possible to find a meaningful transform of functionsthat do not strictly satisfy the existence conditions, providedthose functions can be defined as the limit of a sequence offunctions that are transformable.



6. By transforming each member of function of the definingsequence, a corresponding sequence of transforms isgenerated, and we called the limit of this new sequence thegeneralized Fourier transform of the original function.



7. The Fourier transform of the function δ(x , y) can be obtainedfrom

F{N2 exp[−N2π(x2 + y 2)]} = exp[−π(f 2X + f 2

Y )

N2], (2.6)

and then

F{δ(x , y)} = limN→∞{exp[−π(f 2

X + f 2Y )

N2]} = 1. (2.7)


2.1.2 The Fourier Transform as a Decomposition – 1

1. Consider the inverse Fourier transform shown in Eq. (2.2). Wemay regard this expression as a decomposition of g(x , y) intoa linear combination of elementary functionsexp[j2π(fXx + fY y)].

2. For any particular frequency pair (fX , fY ) the correspondingelementary function has a phase that is zero or an integermultiple of 2π radians along lines described by

y = − fXfY

x +n

fY, (2.9)

where n is an integer.



3. This elementary function may be regarded as being “directed”in the (x , y) plane at an angle θ given by

θ = arctan(fYfX

). (2.10)

4. In addition, the spatial period (i.e., the distance betweenzero-phase lines) is given by (shown in Fig. 2.1)

L =1√

f 2X + f 2

Y

. (2.11)



Figure: 2.1


2.1.3 The Fourier Transform Theorems - 1

1. Linearity: F{αg + βh} = αF{g}+ βF{h}2. Similarity: If F{g(x , y)} = G (fX , fY ), then

F{g(ax , by)} = G (fXa,fYb

). (2.12)

3. Shift: If F{g(x , y)} = G (fX , fY ), then

F{g(x−a, y−b)} = G (fX , fY ) exp[−j2π(fXa+fY b)]}. (2.13)


2.1.3 The Fourier Transform Theorems - 2

4. Rayleigh’s (Parseval’s) theorem: IfF{g(x , y)} = G (fX , fY ), then∫ ∞

−∞

∫ ∞−∞|g(x , y)|2dxdy =

∫ ∞−∞

∫ ∞−∞|G (fX , fY )|2dfXdfY .

(2.14)

5. Convolution: If F{g(x , y)} = G (fX , fY ) andF{h(x , y)} = H(fX , fY ), then

F{∫ ∞−∞

∫ ∞−∞

g(ξ, η)h(x−ξ, y−η)dξdη} = G (fX , fY )H(fX , fY ).

(2.15)


2.1.3 The Fourier Transform Theorems -3

6. Autocorrelation: If F{g(x , y)} = G (fX , fY ), then

F{∫ ∞−∞

∫ ∞−∞

g(ξ, η)g ∗(ξ − x , η − y)dξdη} = |G (fX , fY )|2.

(2.16)Similarly,

F{|g(x , y)|2} =

∫ ∞−∞

∫ ∞−∞

G (ξ, η)G ∗(ξ − x , η − y)dξdη.

(2.17)

7. Fourier integral: At each point of continuity of g ,

FF−1{g(x , y)} = F−1F{g(x , y)} = g(x , y). (2.18)


2.1.4 Separable Functions - 1

1. g(x , y) = gX (x)gY (y) and g(r , θ) = gR(r)gΘ(θ).

2. F{g(x , y)} = FX{gX}FY {gY }.


2.1.4 Separable Functions - 2

3. The Fourier transform of a general function separable in polarcoordinates can be expressed as an infinite sum of weightedsum of weighted Hankel transforms

F{g(r , θ)} =∞∑

k=−∞

ck(−j)k exp(jkφ)Hk{gR(r)}, (2.22)

where

ck =1

2π

∫ 2π

0

gΘ(θ) exp(−jkθ)dθ

and Hk{} is the Hankel transform operator of order k , definedby

Hk{gR(r)} = 2π

∫ ∞0

rgR(r)Jk(2πrρ)dr . (2.23)

Here Jk is the kth-order Bessel function of the first kind.Prof. Hsuan-Ting Chang I. FOURIER OPTICS II. Optical Signal Processing

2.1.5 Functions with Circular Symmetry: Fourier BesselTransforms - 1

1. Circular symmetric function g(r , θ) = gR(r). Most opticalsystems have precisely this type of symmetry.

2. The transformation from rectangular to polar coordinates inboth the (x , y) and the (fX , fY ) planes are

r =√

x2 + y 2 x = r cos θ

θ = arctan(y

x) y = r sin θ

ρ =√

f 2X + f 2

Y fX = ρ cosφ

φ = arctan(fYfX

) fY = ρ sinφ.

3. We write the transform as a function of both radius and angle,

F{g(r , θ)} = G0(ρ, φ).


2.1.5 Functions with Circular Symmetry: Fourier BesselTransforms - 2

4. The Fourier transform of a circularly symmetric function isitself circularly symmetric and can be found by performing the1D manipulation.

G0(ρ, φ) = G0(ρ) = 2π

∫ ∞0

rgR(r)J0(2πrρ)dr . (2.31)

5. It’s accordingly referred to as the Fourier-Bessel transform,or alternatively as the Hankel transform of zero order.

6. Inverse transform:

gR(r) = 2π

∫ ∞0

ρG0(ρ)J0(2πrρ)dρ.

There is no difference between the transform and theinverse-transform operations.


2.1.6 Some Frequently Used Functions and Some UsefulFourier Transform Pairs - 1

1. All functions are shown in Figure 2.2.

2. Rectangle function

rect(x) =

1, |x | > 12

12, |x | = 1

20, otherwise

3. Sinc function

sinc(x) =sin(πx)

πx

4. Signum function

sgn(x) =

1, x > 00, x = 0,−1, x < 0



5. Triangle function

Λ(x) =

{1− |x |, |x | ≤ 10, otherwise

6. Comb function

comb(x) =∞∑

n=−∞

δ(x − n)

7. Circle function (Figure 2.3)

circ(√

x2 + y 2) =

1,√

x2 + y 2 < 112,√

x2 + y 2 = 10, otherwise

8. Table 2.1 Transform pairs for some functions separable inrectangular coordinates.



Figure: 2.2Prof. Hsuan-Ting Chang I. FOURIER OPTICS II. Optical Signal Processing


9. To find the transform of the circular symmetric function, let’suse the Fourier-Bessel transform.

B{circ(r)} = 2π

∫ 1

0

rJ0(2πrρ)dr

10. Using a change of variables r ′ = 2πrρ and the identity∫ x

0

ξJ0(ξ)dξ = xJ1(x),

B{circ(r)} =1

2πρ2

∫ 2πρ

0

r ′J0(r ′)dr ′ =J1(2πρ)

ρ(2.35)


2.2 Local Spatial Frequency and Space-FrequencyLocalization - 1

1. Consider the general case of complex-valued functions. Anysuch function can be represented in the form

g(x , y) = a(x , y) exp[jφ(x , y)] (2.36)

where a(x , y) is a real and nonnegative amplitude distribution,while φ(x , y) is a real phase distribution.

2. Assume that the amplitude distribution a(x , y) is a slowlyvarying function of (x , y). We here concentrate on thebehavior of the phase function φ(x , y).

3. The local spatial frequency of the function g as a frequencypair (flX , flY ) is defined as

flX =1

2π

∂

∂xφ(x , y), flY =

1

2π

∂

∂yφ(x , y).



4. Consider the function g(x , y) = exp[j2π(fXx + fY y)]representing a simple linear-phase exponential of frequencies(fX , fY ) (single Fourier component). We obtainflX = fX , flY = fY . The local frequencies reduce to thefrequencies of that component, and are constant over theentire (x , y) plane.

5. Consider a “finite chirp” function,

g(x , y) = exp[jπβ(x2 + y 2)]rect(x

2LX)rect(

y

2LY).

The local frequencies can be expressed as

flX = βxrect(x

2LX), flY = βy rect(

y

2LY).



6. In this case the local frequencies do depedent on location inthe (x , y) plane, within a rectangle of dimensions 2LX × 2LY .

7. From the Fourier transform of this function (Figure 2.4), weconclude that the local spatial frequency has provided a good(but not exact) indication of where the significant values ofthe Fourier spectrum will occur. However, local spatialfrequencies are not the same entity as the frequencycomponents of the Fourier spectrum.


2.3 Linear Systems

1. A system is defined to be a mapping of a set of inputfunctions into a set of output functions.

2. Let the mathematical operator S{} represent a system.Then

g2(x , y) = S{g1(x , y)},

where g1(x , y) and g2(x , y) represent the input andoutput functions, respectively.


2.3.1 Linearity and the Superposition Integral - 1

1. A system is linear if the superposition property is obeyed for allinput functions p and q and all complex constants a and b:

S{ap(x1, y1) + bq(x1, y1)} = aS{p(x1, y1)}+ bS{q(x1, y1)}.

2. A simple decomposition of the input function is offered by theshifting property of the δ function

g1(x1, y1) =

∫ ∫ ∞−∞

g1(ξ, η)δ(x1 − ξ, y1 − η)dξdη.

3. g1 is a linear combination of weighted and displaced δfunction; the elementary functions are just these δ functions.



4. To find the system response to g1,

g2(x2, y2) = S{∫ ∫ ∞

−∞g1(ξ, η)δ(x1 − ξ, y1 − η)dξdη}.

Now regarding the number g1(ξ, η) as simply a weightingfactor,

g2(x2, y2) =

∫ ∫ ∞−∞

g1(ξ, η)S{δ(x1 − ξ, y1 − η)}dξdη.

5. Let the symbol h(x2, y2; ξ, η) denote the response of thesystem at point (x2, y2) of the output space to a δ functioninput at coordinate (ξ, η) of the input space,

h(x2, y2; ξ, η) = S{δ(x1 − ξ, y1 − η)}.



6. The function h is called the impulse response (or in optics, thepoint spread function) of the system. Now,

g2(x2, y2) =

∫ ∫ ∞−∞

g1(ξ, η)h(x2, y2; ξ, η)dξdη.

7. This fundamental expression, known as the superpositionintegral, demonstrates the very important fact that a linearsystem is completely characterized by its response to unitimpulses.

8. For the case of a linear imaging system, the effects of imagingelements (lenses, stops, etc.) can be fully described byspecifying the (possibly complex-valued) images of pointsources located throughout the object field.


2.3.2 Invariant Linear Systems: Transfer Functions - 1

1. A linear imaging system is space-invariant (isoplanatic) if itsimpulse response h(x2, y2; ξ, η) depends only on the distances(x2 − ξ) and (y2 − η). For such a system, we can write

h(x2, y2; ξ, η) = h(x2 − ξ, y2 − η).

Then

g2(x2, y2) =

∫ ∫ ∞−∞

g1(ξ, η)h(x2 − ξ, y2 − η)dξdη, (2.49)

which is a 2D convolution of the object function g1(x1, y1)with the impulse response h of the system.

2. The short hand notation for a convolution relation g2 = g1⊗ h.


2.3.2 Invariant Linear Systems: Transfer Functions - 2

3. Transforming both sides of (2.49), the spectra G2(fX , fY ) andG1(fX , fY ) are seen to be related by

G2(fX , fY ) = H(fX , fY )G1(fX , fY ),

where H is the Fourier transform of the impulse response

H(fX , fY ) =

∫ ∫ ∞−∞

h(ξ, η) exp[−j2π(fX ξ + fY η)]dξdη.

4. The function H , called the transfer function of the system,indicates the effects of the system in the “frequency domain.”

5. The mathematical term eigenfunction is used for a functionthat retains its original form after passage through a system.Thus we see that the complex-exponential functions are theeigenfunctions of linear, invariant systems.


2.4 Two-Dimensional Sampling Theory

1. To represent a function g(x , y) by an array of its sampledvalues taken on a discrete set of points in the (x , y) plane,if these samples are taken sufficiently close to each other,g can be reconstructed with considerable accuracy bysimple interpolation.

2. For a particular class of functions (known as bandlimitedfunctions) the reconstruction can be accomplishedexactly, provided only that the interval between samples isnot greater than a certain limit.


2.4.1 The Whittaker-Shannon Sampling Theorem - 1

1. Consider a rectangular lattice of samples of the function g

gs(x , y) = comb(x

X)comb(

y

Y)g(x , y).

2. (Figure 2.5) The sampled function thus consists of an array ofδ functions, spaced at intervals of width X in the x directionand width Y in the y direction.

3. The spectrum Gs of gs can be found by convolving thetransform of comb(x/X )comb(y/Y ) with the transform of g ,or

Gs(fX , fY ) = F{comb(x

X)comb(

y

Y)} ⊗ G (fX , fY ).



Figure: 2.5



4. With some mathematical derivation,

Gs(fX , fY ) =∞∑

n=−∞

∞∑m=−∞

G (fX −n

X, fY −

m

Y). (2.53)

Evidently the spectrum of gs can be found simply by erectingthe spectrum of g about each point (n/X ,m/Y ) in the(fX , fY ) plane (Figure 2.6b).

5. If X and Y are sufficiently small, then the separations 1/Xand 1/Y of the various spectral islands will be great enough toassure that the adjacent regions do not overlap. Thus therecovery of the original spectrum G from Gs can beaccomplished exactly by passing the sampled function gsthrough a linear invariant filter that transmits the term(n = 0,m = 0) of Eq. (2.53) without distortion.


2.4.1 The Whittaker-Shannon Sampling Theorem - Fig2.6



6. To determine the maximum allowable separation betweensamples, let 2BX and 2BY represent the widths in the fX andfY directions, respectively, of the smallest rectangle.

7. Separation of the spectral regions is assured if

X ≤ 1

2BXand Y ≤ 1

2BY.

8. The maximum spacings of the sampling lattice for exactrecovery of the original function are thus (2BX )−1 and(2BY )−1.

9. There is one transfer function that will always yield the desiredresults regardless of the shape R,

H(fX , fY ) = rect(fX

2BX)rect(

fY2BY

).



10. The exact recovery of G from Gs

Gs(fX , fY )rect(fX

2BX)rect(

fY2BY

) = G (fX , fY ).

11. The equivalent identity in the space domain is

[comb(x

X)comb(

y

Y)g(x , y)]⊗ h(x , y) = g(x , y), (2.56)

where h is the impulse response of the filter,

h(x , y) = F−1{rect(fX

2BX)rect(

fY2BY

)}

= 4BXBY sinc(2BXx)sinc(2BY y).



12. Noting that

comb(x

X)comb(

y

Y)g(x , y)

= XY∞∑

n=−∞

∞∑m=−∞

g(nX ,mY )δ(x − nX , y −mY ),

13. Eq. (2-56) becomes

g(x , y) = 4BXBYXY∞∑

n=−∞

∞∑m=−∞

g(nX ,mY )×

sinc[2BX (x − nX )]sinc[2BY (y −mY )].



14. Finally, when the sampling intervals X and Y are taken tohave their maximum allowable values, the identity becomes

g(x , y) =∞∑

n=−∞

∞∑m=−∞

g(n

2BX,

m

2BY)×

sinc[2BX (x − n

2BX)]sinc[2BY (y − m

2BY)]. (2.57)

15. Whittaker-Shannon sampling theorem: exact recovery of abandlimited function can be achieved from an appropriatelyspaced rectangular array of its sampled values.


2.4.2 Space-Bandwidth Product

1. Space-bandwidth product of the function g : If g is ofsignificant value only in the region −LX ≤ x < LX ,−LY ≤ y < LY , and if g is sampled, in accord with thesampling theorem, on a rectangular lattice with spacings(2BX )−1, (2BY )−1 in the x and y directions, respectively,then the total number of significant samples required torepresent g(x , y) is seen to be

M = 16LXLYBXBY .


Homework assignments

Problems: 2.3, 2.7, 2.11


prof. hsuan-ting chang september 10, 2019, yuntech...

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