Measurement of spatial coherence of light beams with shadows and digital micromirror device

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Measurement of spatial coherence of light beams with shadows and

digital micromirror device

AMAR NATH GHOSH

Master of Science Thesis May 2016

Department of Physics and Mathematics

University of Eastern Finland

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Amar Nath Ghosh Measurement of spatial coherence of light beams with shadows and digital micromirror device, 46 pages University of Eastern Finland

Master’s Degree Programme in Photonics Supervisors Assoc. Prof. Tero Set¨al¨a

Dr. Henri Partanen

Abstract

In this thesis, a method for measuring the spatial coherence of a stationary, quasi- monochromatic source is presented. The considered light fields are partially spatially coherent, almost fully spatially coherent, and almost incoherent. This method is im- plemented through the comparison of two radiant intensities which are measured at the far field with and without a small obscuration at the test plane. The mutual intensity and the normalized degree of spatial coherence of the fields, at all pair of points having a common centroid with the obscuration, are measured simultaneously from the observed radiant intensities. Furthermore, the results obtained from this method are compared with Young’s two-pinhole method for partially spatially coher- ent, almost fully spatially coherent, and almost incoherent optical fields.

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Preface

I am grateful to my supervisors Assoc. Prof. Tero Set¨al¨a and Dr. Henri Partanen for their constant guidance and instructions during my thesis work. I am thankful to the department of Physics and Mathematics for providing me financial support during my Master studies.

I am thankful to Dr. Rahul Dutta for his valuable help and advice for writing the thesis. I also wish to thank my elder brothers Somnath, Subhajit, and Gaurav for spending some quality time with me, inside or outside the university. I am always thankful to my parents and sisters for their support and encouragement.

Joensuu, June 23, 2016 Amar Nath Ghosh

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Chapter I

Introduction

Light has properties of both wave’s and particle’s. The nature of light was extensively studied around 17th century. In 1690, Christian Huygens proposed light as a wave phenomenon [1, 2]. Later in 1704 Isaac Newton introduced a corpuscular theory that explained light as a combination of microscopic particles travelling along a straight line from the source [3, 4]. For more than 100 years, the particle theory dominated over the wave theory of light due to the superior status of Newton in the scientific community. Finally, in 1801, Thomas Young performed his well known experiment of interference of light [5] which could only be interpreted in terms of the wave theory of light. Coherence theory of light is closely related to Young’s experiment as shown later by Zernike who connected the visibility of interference fringes with the degree of spatial coherence [6].

Every light source is to some extent random in nature. This randomness of light could come either from the radiation due to spontaneous emission or from the chaotic medium through which the light passes. The random characteristics of light can be studied in terms of the field correlations which form the basis of optical coherence theory. Coherence is one of the fundamental properties of light. Two secondary light sources derived from a single source are said to be perfectly coherent if the waves emitted from them have a constant phase difference. Optical coherence theory deals with the statistical properties of light fields. An optical field is said to be stationary if its statistical properties do not depend on the choice of time origin, i.e., the properties depend only on the time difference. In the case of nonstationary optical fields, the statistical properties are time dependent [7]. The coherence properties of light can be classified to three domains: spatial, temporal, and spectral. The

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spatial coherence studies the correlation of the wave fields at two spatial points at a given time instant. Similarly the temporal (or spectral) coherence deals with the correlation of the fields at two time instants (or frequencies) at a single spatial point.

The coherence theory of stationary light sources is well understood and it has been extensively investigated recently. The investigation on the coherence theory was initiated by Verdet [8] and later it was carried out by Von Laue [9], Wiener [10], van Cittert [11], Zernike [6], Pancharatnam [12], Mandel, Agarwal, and Wolf [13, 14].

Many realistic light sources are partially coherent, i.e., not fully coherent nor completely incoherent. Therefore it is necessary to characterize the coherence properties of these sources. A significant number of methods do exist to measure the spatial coherence and the temporal (or spectral) coherence of light fields. The con- ventional Michelson interferometer is a standard way to evaluate the temporal coherence of stationary light sources [2]. In this thesis we focus on the measurement of the spatial coherence of stationary wave fields. The spatial coherence measurement was first performed in the classical experiment of Young’s two-pinhole setup where two pinholes are placed at two test points. Lights passing through the pinholes interfere with each other and produce interference fringes. The complex degree of spatial coherence of the light is then measured at those two points by detecting the position and the visibility of the interference fringes [6, 13]. The period of the fringe pattern depends on the separation of the pinholes. This dependence is removed in some other approaches of Young’s method by using two independent copies of the wavefront [15, 16]. In order to get the complex degree of spatial coherence for the entire field, each pinhole should be scanned independently along a transverse plane for all possible combinations of those test points. This procedure takes significant amount of time. The measurement time can be reduced if one can access many pairs of points simultaneously. This is the motivation behind the work presented in this thesis. Several experimental techniques have been proposed to optimize the measurement time, for example, a mask with a nonredundant array of pinholes al- lows us to measure the spatial coherence simultaneously for many pairs of points corresponding to any such pinholes [17]. More detailed approach has been proposed on the basis of the superposition of the two mutually displaced, reversed or rotated copies of the same wavefront [18–20].

In this thesis, we consider a unique method substituting the pinholes with a small obscuration which enables one to measure the spatial coherence simultaneously at

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many pairs of points whose center coincides with that of the obscuration. This can be achieved by studying the shadow of the obscuration at the far zone. Further, the results are compared with the standard Young’s two-pinhole method.

In Chapter 2, we consider the basic concept of spatial coherence in space-time domain. In addition, a method which uses shadows to measure the spatial coherence is described. This method is called shadow method which is presented here in both one dimension and two dimensions. Chapter 3 deals with the experimental setup and the corresponding measurement procedure is discussed briefly. The results of the shadow method and its comparison with Young’s two-pinhole method are also presented in Chapter 3. Finally, conclusions and some future aspects of the work are provided in Chapter 4.

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Chapter II

Optical coherence and its detection

The case of random fields for which the amplitudes and phases at two points are less than perfectly correlated leads us to the coherence theory of light [14]. Optical coherence theory is used to examine different realistic fields and random fluctuations in terms of the spatial and temporal correlations. In this work, all analysis is carried out using the scalar theory of light.

2.1 Coherence functions

In this section we define the basic concepts which are required to describe the spatial coherence properties of light. We begin with the complex analytic signal and then define the correlation function of stationary light in space-time domain [14]. The optical fields can be represented in terms of real quantities, but for mathematical simplicity we describe them with a complex representation which includes both the amplitude and phase of the fields.

Let us consider an analytic signal U(ρ, t) which represents the random optical field at position ρ and time t. We assume the optical field to be stationary and ergodic. The signal U(ρ, t) can be described by a Fourier integral with respect to the time variable [21] as

U(ρ, t) =

Z ∞ 0

U˜(ρ, ν) exp(−i2πνt) dν, (2.1) where ν is the frequency and

U˜(ρ, ν) =

Z ∞

−∞U(ρ, t) exp(i2πνt) dt. (2.2)

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We consider the fields at two different spatial points ρ1 and ρ2 at two instants of timet1 andt2. For stationary fields the coherence functions depend only on the time difference ∆t = t2 −t1. The correlation of the field fluctuations between the two spatial points in space–time domain is specified by the mutual coherence function (MCF) given as [14]

Γ(ρ1;ρ2,∆t) =hU^∗(ρ1, t)U(ρ2, t+ ∆t)it (2.3) where the asterisk denotes the complex conjugate and the angular brackets with subscripttdenote time averaging (or an ensemble average) of the function, generally defined as

hh(t)it= 1 2T

Z T

−T h(t) dt, (2.4)

where T represents the time period. If we put ρ1 = ρ2 = ρ and t1 = t2 = t in Eq. (2.3), the mean intensity of the field becomes

I(ρ) = Γ(ρ;ρ,0) =^D|U(ρ, t)|²^E. (2.5) The complex degree of coherence (normalized MCF) is defined as [14]

µ(ρ1;ρ2,∆t) = Γ(ρ1;ρ2,∆t)

qI(ρ1)I(ρ2). (2.6)

It can be shown that the absolute value of complex degree of spatial coherence satisfies the following inequalities

0≤ |µ(ρ1;ρ2,∆t)| ≤1. (2.7) The lower limit indicates complete incoherence, i.e., there is no correlation of the fields at the two points for any∆t, while the upper limit reflects full coherence. Any value between these two limits is possible and then the field is said to be partially coherent. For ∆t = 0, these arguments hold for spatial coherence.

2.2 Spatial coherence measurement methods

Here we discuss a method to characterize spatial coherence of quasi-monochromatic fields by comparing two radiant intensities measured with and without a small obscuration in a test plane. In order to do so, we scan the obscuration over the test

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plane and the resultant radiant intensity is then measured at the far zone. At first, for simplicity, we introduce the dependence of the field on one transverse coordinate x (one-dimensional case). Then we generalize the field dependence to both xand y coordinates (two-dimensional case).

Let us consider a stationary, partially coherent, quasi-monochromatic, scattered, one-dimensional field which is denoted by U(x, t). According to the second-order coherence theory of stationary light, the coherence properties of the field at the plane z = 0can be described by the spatial domain correlation function named the mutual intensity which is defined by the ensemble average of the random fieldU(x, t) as

J(x1;x2) =hU^∗(x1, t)U(x2, t)i, (2.8) where the angular brackets are understood as ensemble averaging andx1,x2 are any two spatial points at the z = 0 plane.

We take the spectral amplitudes at the points x1 and x2 in the z = 0 plane to be U˜(x1, ν) and U(x˜ 2, ν), respectively. Then the spectral amplitudes at the points x^′1 and x^′2 at another plane distance z0 away (see Fig. 2.1) can be written in terms of the one-dimensional Fresnel integral [22, 23]

U˜(x^′_j, ν) = exp[i(¯kz0−π/4)]

√z0λ

Z ^∞

−∞

U(x˜ j, ν) exp

"

ik(x¯ ^′_j −xj)² 2z0

#

dxj, (2.9) where ¯k = 2π/λ represents the wavenumber, j ∈ (1,2), while λ and ν are the wavelength and frequency of light, respectively. From Eq. (2.9) we can write

DU˜^∗(x^′1, ν) ˜U(x^′2, ν^′)^E= 1 z0λ

Z Z ∞

−∞

DU˜^∗(x1, ν) ˜U(x2, ν^′)^Eexp

"

−ik(x¯ ^′1−x1)² 2z0

#

×exp

"

i¯k(x^′2−x2)² 2z0

#

dx1dx2. (2.10)

The cross-spectral density function, W(x^′1;x^′2, ν) which is a measure of the correlation between the spectral amplitudes at different spatial points x^′1, x^′2 and at frequency ν can be defined as

hU^∗(x^′1, ν)U(x^′2, ν^′)i=W(x^′1;x^′2, ν)δ(ν−ν^′), (2.11) whereδ is the Dirac delta function [24]. Using Eq. (2.11) in Eq. (2.10) we can write

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z

θ

z = 0

z0

x^′_j

xj

x

Figure 2.1: Illustration of notation related to the propagation of spectral amplitudes. The points x_j and x^′_j refer to the z = 0 and z = z0 plane, respectively, j= 1,2.

the propagation law for the cross-spectral density function as W(x^′1;x^′2, ν) = 1

z0λ

Z Z ∞

−∞W(x1;x2, ν) exp

"

−i¯k(x^′1−x1)² 2z0

#

×exp

"

i¯k(x^′2−x2)² 2z0

#

dx1dx2. (2.12)

Multiplying both sides of Eq. (2.12) with exp(−i2πντ1) and then integrating overν (0< ν <∞), we get

Z ∞

0 W(x^′1;x^′2, ν) exp(−i2πντ1) dν = ¯k 2πz0

Z ∞ 0

Z Z ∞

−∞W(x1;x2, ν)

×exp

(

−i2πν

"

τ1+ (x^′1 −x1)²

2z0c − (x^′2−x2)² 2z0c

#)

dx1dx2dν, (2.13) where c is the vacuum speed of light and τ1 is a parameter representing a time difference (as wee see shortly). From the generalized Wiener-Khintchine theorem [14]

we know that the cross-spectral density function and the mutual coherence function are related to each other by the following Fourier transform pair:

W(x^′1;x^′2, ν) =

Z ∞

−∞Γ(x^′1;x^′2, τ1) exp(i2πντ1) dτ1, (2.14a) Γ(x^′1;x^′2, τ1) =

Z ∞

0 W(x^′1;x^′2, ν) exp(−i2πντ1) dν. (2.14b)

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Using Eq. (2.14b) in Eq. (2.13), we get the propagation law for the mutual coherence function as follows

Γ(x^′1;x^′2, τ1) = k¯ 2πz0

Z Z ∞

−∞Γ

"

x1;x2, τ1+(x^′1−x1)²

2z0c − (x^′2−x2)² 2z0c

#

dx1dx2. (2.15) From the properties of the envelope representation we can write the following

Γ

"

x1;x2, τ1+(x^′1−x1)²

2z0c − (x^′2−x2)² 2z0c

#

≈Γ(x1;x2, τ1)

×exp

(

−ik¯

"

(x^′1−x1)²

2z0 − (x^′2−x2)² 2z0

#)

. (2.16) Inserting Eq. (2.16) into Eq. (2.15) leads to

Γ(x^′1;x^′2, τ1) = ¯k 2πz0

Z Z ∞

−∞Γ(x1;x2, τ1) exp

(

−ik¯

"

(x^′1 −x1)²

2z0 − (x^′2−x2)² 2z0

#)

dx1dx2. (2.17) Assuming that |τ1| ≪1/∆ν, where∆ν is the frequency band, the mutual intensity J(x1;x2) and the mutual coherence function are related by the expression [14]

Γ(x1;x2, τ1)≈J(x1;x2) exp(−i2π¯ντ1). (2.18) In order to extract the mutual intensity from the mutual coherence function, we assume equal time correlation, i.e., we setτ1 = 0in Eq. (2.18), to obtain the mutual intensity as follows

J(x1;x2) = Γ(x1;x2,0). (2.19) Furthermore using Eq. (2.19), Eq. (2.17) takes the form

J(x^′1;x^′2) = ¯k 2πz0

Z Z ∞

−∞J(x1;x2)

×exp

"

−i¯k x^′21 −x^′1x1+x²1−x^′22 + 2x^′2x2−x^′22

2z0

!#

dx1dx2. (2.20) Next we consider the case that the points x^′1, x^′2 are located far from the z = 0 plane. If we assume x^′1 =x^′2 =x^′0 andλz0 ≫x²1, x²2, the mutual intensity, expressed in Eq. (2.20), can then be written as

J(x^′0;x^′0) = k¯ 2πz0

Z Z ∞

−∞J(x1;x2) exp

(

−ik¯

"

(x2 −x1)x^′0

z0

#)

dx1dx2. (2.21)

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In the exponential term of Eq. (2.21), x^′0/z0 = tanθ = sinθ =p, where p is named as directional variable and θ is the observation angle (see Fig. 2.1). The mutual intensity of the far field using Fraunhofer diffraction formula [25] can be written as

J(p) = k¯ 2πz0

Z Z ^∞

−∞J(x1;x2) exp[−ik(x¯ 2−x1)p] dx1dx2. (2.22) So the radiant intensity, defined asI(p) = z0J(p), becomes

I(p) = ¯k 2π

Z Z ∞

−∞J(x1;x2) exp[−i¯k(x2−x1)p] dx1dx2. (2.23) Next we perform a coordinate transformation from x1 and x2 to the centroid coordinate (¯x) and difference coordinate (x^′) which are given by

¯

x= x1+x2

2 , (2.24)

x^′ =x2−x1. (2.25)

The Jacobian related to this transformation is equal to one. We can thus express the coordinatesx1, x2 in terms of the centroid and difference coordinates such that x1 = (¯x−x^′/2) and x2 = (¯x+x^′/2). Then the radiant intensity can be written as

I(p) = ¯k 2π

Z Z ∞

−∞

J(¯¯x;x^′) exp(−ikx¯ ^′p) dx^′d¯x, (2.26) where

J(¯¯x;x^′) =J(¯x− x^′

2; ¯x+ x^′

2). (2.27)

The radiant intensity I(p) can be measured either in the far zone or by bringing the far field at the focus of a Fourier transforming lens which is implemented in our experimental setup.

A mask with an amplitude function A(x−x0) is next introduced at the z = 0 plane where x0 is a lateral displacement as shown in Fig. 2.2. Denoting the field after the mask by U_A(x), the mutual intensity can be written as

J_A(x1;x2) =hU_A^∗(x1)U_A(x2)i,

=A^∗(x1−x0)hU^∗(x1)U(x2)iA(x2−x0),

=A^∗(¯x−x^′/2−x0)hU^∗(x1)U(x2)iA(¯x+x^′/2−x0),

=A^∗ τ− x^′ 2

!

hU^∗(x1)U(x2)iA τ +x^′ 2

!

, (2.28)

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far field

mask ^θ z

x0

z = 0 x

IA(p;x0)

Figure 2.2: Depiction of notation related to the propagation of the mutual intensity to the far zone after introducing a maskA(x−x0)at thez= 0plane.

where τ = ¯x−x0 and the subscript A refers to the mask. Inserting Eq. (2.28) into Eq. (2.27) and developing Eq. (2.26) leads to the radiant intensity in the presence of the mask

IA(p;x0) = k¯ 2π

ZZ ^∞

−∞A^∗ τ − x^′ 2

!

A τ +x^′ 2

!

J¯(x0+τ;x^′) exp(−ikx¯ ^′p) dx^′dτ, (2.29) wherex0 on the left-hand side emphasizes that the mask is centered atx0. Different measurement methods are based on the choice of the mask. Next we describe three coherence measurement methods with different mask functions and proceed further with the best one.

2.2.1 Aperture method in one dimension

Like some other coherence measurement techniques [26], we use a sharp or apodized window for the mask function A(x) in order to isolate the coherence properties of the beam at different locations. For example, we can consider Shack-Hartmann wavefront sensor as a special case of this technique where the window is the aperture of each lenslet of a microlens array [27]. The coherence function J(x¯ 0 +τ;x^′) can be extracted from the radiant intensity IA(p;x0), but it is difficult for those points whose separations are greater than the windows extent. In order to explain this we

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expandJ(x¯ 0+τ;x^′)as a Taylor series around the aperture centroidx0as follows [24]

J(x¯ 0+τ;x^′) = ¯J(x0;x^′) + ∂J¯

∂x¯(x0;x^′)(x0+τ −x0)

1! + ∂²J¯

∂x¯²(x0;x^′)(x0 +τ −x0)² 2!

+∂³J¯

∂x¯³(x0;x^′)(x0+τ−x0)³ 3! +...,

=

∞

X

n=0

∂ⁿJ¯

∂x¯ⁿ(x0;x^′)τⁿ

n!. (2.30)

Substituting Eq. (2.30) into Eq. (2.29), we get IA(p;x0) = k¯

2π

∞

X

n=0

Z Z ∞

−∞A^∗ τ − x^′ 2

!

A τ +x^′ 2

!∂ⁿJ¯

∂x¯ⁿ(x0;x^′)τⁿ

n! exp(−i¯kx^′p) dx^′dτ,

= k¯ 2π

∞

X

n=0

Z ∞

−∞

∂ⁿJ¯

∂x¯ⁿ(x0;x^′)Aⁿ(x^′) exp(−i¯kx^′p) dx^′, (2.31) where

Aⁿ(x^′) =

Z ∞

−∞

τⁿ

n!A^∗ τ −x^′ 2

!

A τ +x^′ 2

!

dτ. (2.32)

For a real and symmetric aperture function, we getAn(x^′) = 0whenn is odd. If we neglect the higher order even terms in Eq. (2.31) and approximate this equation to its leading term n= 0, results in

IA(p;x0) = k¯ 2π

Z ∞

−∞

J(x¯ 0, x^′)A0(x^′) exp(−i¯kx^′p) dx^′. (2.33) Therefore a basic estimate of J(x¯ 0;x^′) can be retrieved by performing the inverse Fourier transform [28] of Eq. (2.33) as

J¯(x0;x^′)A⁰(x^′)≈

Z ∞

−∞IA(p;x0) exp(ikx¯ ^′p) dp , (2.34) where

A⁰(x^′) =

Z ∞

−∞A^∗ τ − x^′ 2

!

A τ+ x^′ 2

!

dτ, (2.35)

is the autocorrelation of the aperture function. For a rectangular opening of width w, the aperture function is A(x) = rect(x/w) [28]. The autocorrelation A⁰(x^′) is

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obtained from the overlapping area of two rectangular functions (see Fig. 2.3) as follows

A⁰(x^′) =w− |x^′|. (2.36) The function A⁰(x^′) is non-zero within the limit −w ≤ x^′ ≤ +w and otherwise zero as illustrated in Fig. 2.4. The same treatment for a Gaussian aperture function is shown in Ref. [29]. Therefore it is obvious from Eq. (2.35) that for the pair of points having a separation |x^′| larger than the width of aperture function, we can not recover the coherence functionJ¯(x0;x^′). This is a fundamental limitation of the aperture method.

0

1 1

x^′ x^′/2

−x^′/2

−x^′

w

Figure 2.3: Overlap of two rectangular functions with widthw and height 1 centered at −x^′/2and +x^′/2.

0 1 2

-2 -1

x^′/w

−x^′/w

A⁰(x^′)

Figure 2.4: Autocorrelation function A⁰(x^′) for a rectangular aperture represented byA(x) =rect(x/w).

2.2.2 Binary phase-mask method in one dimension

In this section, we discuss a binary phase-mask method which is used to overcome the fundamental limitation of the aperture method [30]. In this method, instead

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of an aperture, a binary transparent phase mask is employed at z = 0 plane and the difference of two radiant intensities measured with and without the mask is calculated. This is implemented by subtracting Eq. (2.29) from Eq. (2.26) as follows:

∆I(p;x0) = I(p)−IA(p;x0),

= k¯ 2π

Z Z ∞

−∞

J¯(¯x;x^′) exp(−i¯kx^′p) dx^′d¯x− ¯k 2π

Z Z ∞

−∞A^∗ τ− x^′ 2

!

A τ+ x^′ 2

!

×J(x¯ 0+τ;x^′) exp(−ikx¯ ^′p) dx^′dτ. (2.37) Substituting x¯=x0+τ and Eq. (2.30) into Eq. (2.37), we get

∆I(p;x0) = ¯k 2π

Z Z ^∞

−∞

∞

X

n=0

τⁿ n!

∂ⁿJ¯

∂x¯ⁿ(x0;x^′) exp(−ikx¯ ^′p) dx^′dτ

− ¯k 2π

Z Z ∞

−∞

∞

X

n=0

τⁿ n!

∂ⁿJ¯

∂x¯ⁿ(x0;x^′)A^∗ τ −x^′ 2

!

×A τ +x^′ 2

!

exp(−i¯kx^′p) dx^′dτ,

= ¯k 2π

Z ∞

−∞

∞

X

n=0

∂ⁿJ¯

∂x¯ⁿ(x0;x^′) exp(−i¯kx^′p)

×

Z ∞

−∞

τⁿ n!

"

1−A^∗ τ −x^′ 2

!

A τ +x^′ 2

!#

dτdx^′,

= ¯k 2π

∞

X

n=0

Z ∞

−∞

∂ⁿJ¯

∂x¯ⁿ(x0;x^′) ¯An(x^′) exp(−ikx¯ ^′p) dx^′, (2.38) where

A¯ⁿ(x^′) =

Z ∞

−∞

τⁿ n!

"

1−A^∗ τ− x^′ 2

!

A τ+ x^′ 2

!#

dτ. (2.39)

After performing the inverse Fourier transform of Eq. (2.38), we obtain

∞

X

n=0

∂ⁿJ¯

∂x¯ⁿ(x0;x^′) ¯Aⁿ(x^′) =

Z ∞

−∞∆I(p;x0) exp(i¯kx^′p) dp. (2.40) As the binary phase mask is transparent with a single discontinuity [2], we consider it to be a signum function A(x) = sgn(x). In this case, the value of the term A^∗τ −^x2^′

Aτ +^x₂^′ in Eq. (2.39) is either 1 or -1. In order to obtain a non-zero value ofA¯ⁿ(x^′), the mask function product in Eq. (2.39) has to be equal to -1, which

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is possible ifτ is within the limit −|x^′|/2≤τ ≤ |x^′|/2. Under these conditions, we get

A¯ⁿ(x^′) = 1 n!

Z ^|

x′|

2

−^|^x^′|

2

τⁿ[1−(−1)] dτ,

=

1 2

n

+

−1 2

n |x^′|ⁿ⁺¹

(n+ 1)!. (2.41)

From Eq. (2.41) it is clear thatA¯ⁿ(x^′) = 0for odd n. If we neglect the higher order even terms and approximate the series in Eq. (2.39) by its leading term n = 0, we get

J¯(x0;x^′) ¯A⁰(x^′)≈

Z ∞

−∞∆I(p;x0) exp(ikx¯ ^′p) dp . (2.42) Puttingn = 0 in Eq. (2.41), we have

A¯⁰(x^′) = 2|x^′|, (2.43) which is shown in Fig. 2.5. We can conclude from Eq. (2.43) that there is no maximum limitation for the separation |x^′| between the points to recover J¯(x0;x^′) from Eq. (2.42). This method has also some fundamental constraints, for example, in this case it is problematic to recover the coherence function J¯(x0;x^′) for small values of|x^′| which, however, can be described through interpolation [30]. It is also difficult to extend this method to two dimensions [31].

0 2 4

-4 -2 x^′

−x^′

A¯⁰(x^′)

Figure 2.5: Function A¯⁰(x^′)for a binary phase mask represented by A(x) = sgn(x).

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2.2.3 Shadow method in one dimension

The main drawbacks of the previous methods are completely eliminated in the shadow method [29]. In this case, at thez = 0plane, rather than using an aperture or a binary phase mask, we use a transparent and uniform mask having a localized obscuration described by

A(x) = 1−a(x), (2.44)

where a(x) is a real distribution around x = 0 and bounded as 0 ≤ a(x) ≤ 1. A basic estimate of the mutual intensity J¯(x0;x^′) as a function of the centroid and difference coordinates is denoted by J¯0(¯x;x^′) and obtained by substituting x0 = ¯x into Eq. (2.42), leading to

J(¯¯x;x^′)≈J¯0(¯x;x^′) =

R∞

−∞∆I(p; ¯x) exp(i¯kx^′p) dp

A¯⁰(x^′) . (2.45)

In this equation,A¯⁰(x^′)can be found by substituting n= 0 into Eq. (2.39) resulting in

A¯⁰(x^′) =

Z ∞

−∞

"

1−A^∗ τ− x^′ 2

!

A τ +x^′ 2

!#

dτ. (2.46)

Inserting the mask function, defined in Eq. (2.44), into Eq. (2.46), we get A¯⁰(x^′) =

Z ∞

−∞

(

1−

"

1−a τ − x^′ 2

!# "

1−a τ+ x^′ 2

!#)

dτ,

=

Z ∞

−∞a τ −x^′ 2

!

dτ +

Z ∞

−∞a τ+ x^′ 2

!

dτ −

Z ∞

−∞a τ+ x^′ 2

!

a τ− x^′ 2

!

dτ.

(2.47) A circular obscuration with radiusdis represented by a(x) =circ(x/d)[28]. The first two terms on the right-hand side of Eq. (2.47) can be obtained from the area of the circle asπd². The last term of Eq. (2.47) is the obscuration’s autocorrelation function which can be calculated from the overlapping area of two symmetric circular functions as shown in Fig. 2.6. Finally, we get A¯⁰(x^′) as follows

A¯⁰(x^′) = 2πd²−2d²arccos |x^′| 2d

!

+ |x^′| 2

q

4d²− |x^′|²

, (2.48)

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where|x^′| ≤2d. It is evident thatA¯⁰(x^′)of Eq. (2.47), illustrated in Fig. 2.7, never becomes zero but goes to a constant for |x^′| larger than obscuration diameter and it is always larger than the half of this constant value. This is the main advantage of the shadow method. So in this approach we can get an estimate of the coherence for all pair of points which are symmetrically situated around the centre of the obscuration.

−x^′ 0 x^′

d

−x^′/2 x^′/2

Figure 2.6: Overlap of two circular functions with radiusd, centered at−x^′/2 and +x^′/2.

0 2

-2

3.14 6.28

-4 4

x^′/d

−x^′/d

A¯⁰(x^′)

Figure 2.7: A¯⁰(x^′) for a transparent mask with a localized obscuration of circular shapea(x) =circ(x/d) having a radius ofd.

2.2.4 Shadow method in two dimensions

In order to extend the shadow method into two transverse dimensions, we begin with the propagation of monochromatic light from finite surfaces [14]. The wave propagates through an optical system and after being limited by an exit pupil it hits an open surface S as shown in Fig. 2.8. If we know the values of the second-order

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correlation functions (the cross-spectral density function or the mutual coherence function) for all pair of points S1(x1) and S2(x2) on the surface S, then the corresponding correlation functions at any pair of points P1(x^′1) and P2(x^′2) located distances R1 and R2, respectively, away from the surface S can be determined.

Optical system

Exit pupil

Light source

Surface

S1(x1)

S2(x2) R1

R2

P1(x^′1) P2(x^′2)

S

Figure 2.8: Illustration of the notation related to the propagation of the mutual intensity and the cross-spectral density.

Let us denote the complex light disturbances at points S1(x1) and S2(x2) by V1(x1, t)andV2(x2, t), respectively, while the related spectral amplitudes areV˜(x^′1, ν) and V˜(x^′2, ν), where ν is the frequency of field. From the Huygens-Fresnel princi- ple [2], the complex amplitudes at the pointsP1(x^′1)and P2(x^′2)can be expressed in terms of the complex amplitudes at all points located on surface S as follows

V˜(x^′_j, ν) =

Z

S

V˜(x_j, ν)exp(ikR_j) Rj

Λj(k)d²xj, (2.49) wherej ∈(1,2). Λ1(k),Λ2(k)are the inclination factors which can be approximated for small angles of diffraction at the limiting aperture as follows

Λ1(k)≈Λ2(k)≈ ik

2π. (2.50)

From Eq. (2.49) we can write V˜^∗(x^′1, ν) ˜V(x^′2, ν^′) =

Z

S

Z

S

V˜^∗(x1, ν) ˜V(x2, ν^′)

× exp[i(k^′R2−kR1] R1R2

Λ^∗1(k)Λ2(k^′) d²x1d²x2, (2.51)

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where the wave numbers at frequencies ν and ν^′ are k = 2πν/c and k^′ = 2πν^′/c, respectively. After taking an ensemble average over different realizations on both sides of Eq. (2.51), we get

DV˜^∗(x^′1, ν) ˜V(x^′2, ν^′)^E=

Z

S

Z

S

DV˜^∗(x1, ν) ˜V(x2, ν^′)^E

× exp[i(k^′R2−kR1] R1R2

Λ^∗1(k)Λ2(k^′) d²x1d²x2. (2.52) The cross-spectral density function, W(x1;x2, ν) which is a measure of the correlation between the spectral amplitudes at spatial pointsx1,x2 and at frequencyν, is defined by

DV˜^∗(x1, ν) ˜V(x2, ν^′)^E=W(x1;x2, ν)δ(ν−ν^′). (2.53) Substituting Eq. (2.53) into Eq. (2.52), we get the propagation law for the cross- spectral density function as follows

W(x^′1;x^′₂, ν) =

Z

S

Z

SW(x1;x2, ν)exp[ik(R2−R1)]

R1R2

Λ^∗1(k)Λ2(k) d²x1d²x2. (2.54) Assume next that the light is quasi-monochromatic whose effective bandwidth (∆ν) is small compared to its mean frequencyν, i.e.,¯ ∆ν/¯ν ≤1. In order to develop the propagation law in this case, we ignore the weak frequency dependence ofΛ1(k), Λ2(k)and replace them with the corresponding values Λ¯1, Λ¯2 at mean frequency ν.¯ Multiplying both sides of Eq. (2.54) with exp(−i2πντ2) and then integrating overν (0< ν <∞), we obtain

Z ∞

0 W(x^′1;x^′₂, ν) exp(−i2πντ2)dν=

Z ∞ 0

Z

S

Z

SW(x1;x2, ν)

×exp{−i2πν[τ2−(R2−R1)/c]} R1R2

Λ¯^∗1Λ¯2d²x1d²x2dν, (2.55) where τ2 is a parameter representing a time difference (as wee see shortly). From the generalized Wiener-Khintchine theorem we know that the cross-spectral density function and the mutual coherence function are related to each other by the following Fourier transform pair:

W(x1;x2, ν) =

Z ∞

−∞Γ(x1;x2, τ2) exp(i2πντ2) dτ2, (2.56a) Γ(x1;x2, τ2) =

Z ∞

0 W(x1;x2, ν) exp(−i2πντ2) dν. (2.56b)

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Using Eq. (2.56b) in Eq. (2.55), we find the propagation law for the mutual coherence function as follows

Γ(x^′1;x^′₂, τ2) =

Z

S

Z

S

Γ[x1;x2, τ2−(R2−R1)/c] R1R2

Λ¯^∗1Λ¯2d²x1d²x2. (2.57) Since practically the coherence length (∼ c/∆ν) of the light is greater than the optical path difference|R2−R1|, it holds that

|R2−R1|

c ≪ 1

∆ν. (2.58)

In this situation using the properties of the envelope representation and Eq. (2.58), we can write the following

Γ[x1;x2, τ2−(R2−R1)/c]≈Γ(x1;x2, τ2) exp[i¯k(R2−R1)], (2.59) where k¯ = 2π¯ν/c is the wave number at mean frequency ν.¯ By substituting Eq. (2.59) into Eq. (2.57), the following form for the propagation law valid for quasi-monochromatic light

Γ(x^′1;x^′2, τ2) =

Z

S

Z

SΓ(x1;x2, τ2)exp[ik(R¯ 2−R1)]

R1R2

Λ¯^∗1Λ¯2d²x1d²x2. (2.60) When |τ2| ≪ 1/∆ν, the mutual intensity J(x1;x2) and the mutual coherence function are related by the expression

Γ(x1;x2, τ2)≈J(x1;x2) exp(−i2πντ¯ 2). (2.61) In order to extract the mutual intensity from the mutual coherence function, we consider equal-time correlations, i.e., we set τ2 = 0 in Eq. (2.61), to obtain the mutual intensity as follows

J(x1;x2) = Γ(x1;x2,0). (2.62) Furthermore, using Eqs. (2.62) and (2.50) in Eq. (2.60), we arrive at

J(x^′1;x^′₂) = k¯ 2π

!² Z

S

Z

SJ(x1;x2)exp[ik(R¯ 2 −R1)]

R1R2

d²x1d²x2, (2.63) which is also known as Zernike’s propagation law for the mutual intensity of quasi- monochromatic source.

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We next assume that the surface S coincides with a partially coherent, planar, quasi-monochromatic secondary source denoted byBillustrated in Fig. 2.9. Further, the pointsP1(x^′1)andP2(x^′2)are taken to be in the far-field of the source distancex1

andx2 away fromB. Letx^′₁andx^′₂be the position vectors whose originOis situated inside the source regionB. The unit vectorsp1 and p2 indicate the directions of the far-field points. Finally R1, R2 are the distances of the points P1(x^′1) and P2(x^′2) to the source points S1(x1) and S2(x2), respectively. The position vectors can be written as

x^′₁ =x1p₁, (2.64a)

x^′₂ =x2p₂. (2.64b)

When the distancesx1 and x2 are sufficiently large, we may approximate

R1 ∼x1 −p₁·x1, (2.65a)

R2 ∼x2 −p₂·x2. (2.65b)

Substituting Eqs. (2.65a) and (2.65b) into Eq. (2.63), the mutual intensity of the

o

Secondary source x1

x2

B S1(x1)

S2(x2)

R1

R2

x1p₁

x2p₂

P1(x^′1)

P2(x^′2)

Figure 2.9: Illustration of the notation related to the calculation of the mutual intensity of the far-field.

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far-field becomes J(x^′1;x^′₂) = ¯k

2π

!² Z

B

Z

BJ(x1;x2)exp[i¯k(x2−p2·x2−x1+p1·x1)]

(x1−p₁ ·x1)(x2−p₂·x2) d²x1d²x2. (2.66) Since x1 and x2 are large, we can neglect the terms p₁ · x1 and p₂ · x2 in the denominator implying that

J(x^′1;x^′₂) = ¯k 2π

!² Z

B

Z

BJ(x1;x2)exp[i¯k(x2−p₂·x2−x1+p₁·x1)]

x1x2

d²x1d²x2. (2.67) We assume that for the distancesx1 =x2 =xand for the unit vectors p1 =p2 =p wherep = (p, q), the mutual intensity at a single point P(x^′) in the far-field due to the pair of source pointsS1(x1)and S2(x2) in plane B can be written as

J(p) = ¯k 2π

!²

1 x²

Z

B

Z

BJ(x1;x2) exp[−i¯k(x2−x1)·p] d²x1d²x2. (2.68) So the radiant intensity, defined by I(p) = x²J(p) [32], becomes

I(p) = k¯ 2π

!² Z

B

Z

BJ(x1;x2) exp[−i¯k(x2−x1)·p] d²x1d²x2. (2.69) Next we repeat most of the steps done in one-dimensional case in section 2.2.

First we perform the coordinate transformation from x1 and x2 into the centroid coordinate (¯x) and difference coordinate (x^′), which are frequently used in the thesis.

The transformation is

¯

x= x1+x2

2 , (2.70)

x^′ =x2−x1, (2.71)

where x1 = (x1, y1), x2 = (x2, y2), x¯ = (¯x,y), and¯ x^′ = (x^′, y^′). After expressing the coordinates x1, x2 in terms of the centroid and difference coordinates as x1 = (¯x−x^′/2)and x2 = (¯x+x^′/2), the radiant intensity becomes

I(p) = ¯k 2π

!² Z Z ∞

−∞

J(¯¯x;x^′) exp(−ikx¯ ^′·p) d²x^′d²x,¯ (2.72)

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where

J(¯¯x;x^′) =J(¯x− x^′

2; ¯x+x^′

2), (2.73)

in the finite surface B and J¯(¯x;x^′) is zero outside of it. When a mask with an amplitude function A(x− x0) is introduced at the source plane B, the resultant radiant intensity is obtained as

IA(p;x0) = k¯ 2π

!² ZZ ∞

−∞A^∗ τ − x^′ 2

!

A τ +x^′ 2

!

J(x¯ 0+τ;x^′)

×exp(−ikx¯ ^′·p) d²x^′d²τ, (2.74) where x0 = (x0, y0), τ = (τ, η) = ¯x−x0. Expanding J¯(x0 +τ) in a Taylor series around the aperture centroid x0, we get

J(x¯ 0+τ;x^′) = ¯J(x0;x^′, y0;y^′) + ∂²J¯

∂²x¯(x0;x^′, y0;y^′)(x0+τ −x0) 1!

+ ∂J¯

∂y¯(x0;x^′, y0;y^′)(y0+η−y0) 1!

+ ∂²J¯

∂x¯²(x0;x^′, y0;y^′)(x0+τ −x0)² 2!

+ ∂²J¯

∂y¯²(x0;x^′, y0;y^′)(y0+η−y0)² 2! +...,

=

∞

X

n,m=0

∂ⁿ⁺^mJ¯

∂x¯ⁿ∂y¯^m(x0;x^′)τⁿη^m

n!m!. (2.75)

We consider a transparent and uniform mask having a localized obscuration given by

A(x) = 1−a(x), (2.76)

where a(x) is a real distribution within the limit 0 ≤ a(x) ≤ 1 confined around x=0. So the difference of two radiant intensities, measured with and without the

Measurement of spatial coherence of light beams with shadows and digital micromirror device