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

Transcription

1 Measurement of spatial coherence of light beams with shadows and digital micromirror device AMAR NATH GHOSH Master of Science Thesis May 6 Department of Physics and Mathematics University of Eastern Finland

2 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 Setälä Dr. Henri Partanen Abstract In this thesis, a method for measuring the spatial coherence of a stationary, quasimonochromatic source is presented. The considered light fields are partially spatially coherent, almost fully spatially coherent, and almost incoherent. This method is implemented 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 coherent, almost fully spatially coherent, and almost incoherent optical fields.

3 Preface I am grateful to my supervisors Assoc. Prof. Tero Setälä 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. Iam thankful to Dr. Rahul Duttafor his valuablehelp andadvice 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 3, 6 Amar Nath Ghosh iii

4 Contents Introduction Optical coherence and its detection 4. Coherence functions Spatial coherence measurement methods Aperture method in one dimension Binary phase-mask method in one dimension Shadow method in one dimension Shadow method in two dimensions Experimental results with shadow method 6 3. Experimental setup Measurement and Results Comparison with Young s two-pinhole method Conclusions 4 Bibliography 43 iv

5 Chapter I Introduction Light has properties of both wave s and particle s. The nature of light was extensively studied around 7th century. In 69, Christian Huygens proposed light as a wave phenomenon [, ]. Later in 74 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 years, the particle theory dominated over the wave theory of light due to the superior status of Newton in the scientific community. Finally, in 8, 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

6 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 [], van Cittert [], Zernike [6], Pancharatnam [], Mandel, Agarwal, and Wolf [3, 4]. 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 conventional Michelson interferometer is a standard way to evaluate the temporal coherence of stationary light sources []. 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, 3]. 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 [5,6]. 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 allows us to measure the spatial coherence simultaneously for many pairs of points corresponding to any such pinholes [7]. 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 [8 ]. 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

7 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, 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. 3

8 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 [4]. 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.. 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 [4]. 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 [] as U(ρ, t) = where ν is the frequency and Ũ(ρ, ν) = Ũ(ρ, ν) exp( iπνt) dν, (.) U(ρ, t) exp(iπνt) dt. (.) 4

9 We consider the fields at two different spatial points ρ and ρ at two instants of time t and t. For stationary fields the coherence functions depend only on the time difference t = t t. 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 [4] Γ(ρ ; ρ, t) = U (ρ, t) U(ρ, t + t) t (.3) where the asterisk denotes the complex conjugate and the angular brackets with subscript t denote time averaging (or an ensemble average) of the function, generally defined as h(t) t = T T T h(t) dt, (.4) where T represents the time period. If we put ρ = ρ = ρ and t = t = t in Eq. (.3), the mean intensity of the field becomes I(ρ) = Γ(ρ; ρ, ) = U(ρ, t). (.5) The complex degree of coherence (normalized MCF) is defined as [4] µ(ρ ; ρ, t) = Γ(ρ ; ρ, t) I(ρ )I(ρ ). (.6) It can be shown that the absolute value of complex degree of spatial coherence satisfies the following inequalities µ(ρ ; ρ, t). (.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 =, these arguments hold for spatial coherence.. 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 5

10 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 x and 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 = can be described by the spatial domain correlation function named the mutual intensity which is defined by the ensemble average of the random field U(x, t) as J(x ; x ) = U (x, t) U(x, t), (.8) where the angular brackets are understood as ensemble averaging and x, x are any two spatial points at the z = plane. We take the spectral amplitudes at the points x and x in the z = plane to be Ũ(x, ν) and Ũ(x, ν), respectively. Then the spectral amplitudes at the points x and x at another plane distance z away (see Fig..) can be written in terms of the one-dimensional Fresnel integral [, 3] Ũ(x j, ν) = exp[i( kz π/4)] z λ [ i k(x j x j ) ] Ũ(x j, ν) exp dx j, (.9) z where k = π/λ represents the wavenumber, j (, ), while λ and ν are the wavelength and frequency of light, respectively. From Eq. (.9) we can write Ũ (x, ν) Ũ(x, ν ) = Ũ (x, ν) z λ Ũ(x, ν ) [ i k(x exp x ) ] z [ i k(x exp x ) ] dx dx. (.) z The cross-spectral density function, W(x ; x, ν) which is a measure of the correlation between the spectral amplitudes at different spatial points x, x and at frequency ν can be defined as U (x, ν) U(x, ν ) = W(x ; x, ν) δ(ν ν ), (.) where δ is the Dirac delta function [4]. Using Eq. (.) in Eq. (.) we can write 6

11 x j x x j θ z z z = Figure.: Illustration of notation related to the propagation of spectral amplitudes. The points x j and x j refer to the z = and z = z plane, respectively, j =,. the propagation law for the cross-spectral density function as W(x ; x, ν) = [ i k(x W(x ; x, ν) exp x ) ] z λ z [ i k(x exp x ) ] dx dx. (.) z Multiplying both sides of Eq. (.) with exp( iπντ ) and then integrating over ν ( < ν < ), we get W(x ; x, ν) exp( iπντ ) dν = k W(x ; x, ν) πz { [ exp iπν τ + (x x ) (x x ) ]} dx dx dν, (.3) z c z c where c is the vacuum speed of light and τ is a parameter representing a time difference(as wee see shortly). From the generalized Wiener-Khintchine theorem[4] 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 ; x, ν) = Γ(x ; x, τ ) = Γ(x ; x, τ ) exp(iπντ ) dτ, W(x ; x, ν) exp( iπντ ) dν. 7 (.4a) (.4b)

12 Using Eq. (.4b)in Eq. (.3), we get the propagationlaw for the mutual coherence function as follows Γ(x ; x, τ ) = k [ Γ x ; x, τ + (x x ) πz z c (x x ) ] dx dx. (.5) z c From the properties of the envelope representation we can write the following [ Γ x ; x, τ + (x x ) (x x ) ] Γ(x ; x, τ ) z c z c { [ (x exp i k x ) (x x ) ]}. z z (.6) Inserting Eq. (.6) into Eq. (.5) leads to Γ(x ; x, τ ) = k { Γ(x ; x, τ ) exp i k πz [ (x x ) (x x ) ]} dx dx. z z (.7) Assuming that τ / ν, where ν is the frequency band, the mutual intensity J(x ; x ) and the mutual coherence function are related by the expression [4] Γ(x ; x, τ ) J(x ; x ) exp( iπ ντ ). (.8) In order to extract the mutual intensity from the mutual coherence function, we assume equal time correlation, i.e., we set τ = in Eq. (.8), to obtain the mutual intensity as follows Furthermore using Eq. (.9), Eq. (.7) takes the form J(x ; x ) = Γ(x ; x, ). (.9) J(x ; x ) = k J(x ; x ) πz [ ( x exp i k x x + x x + x x x )] dx dx. (.) z Next we consider the case that the points x, x are located far from the z = plane. If we assume x = x = x and λz x, x, the mutual intensity, expressed in Eq. (.), can then be written as J(x ; x ) = k { [ (x x )x ]} J(x ; x ) exp i k dx dx. (.) πz z 8

13 In the exponential term of Eq. (.), x /z = tan θ = sin θ = p, where p is named as directional variable and θ is the observation angle (see Fig..). The mutual intensity of the far field using Fraunhofer diffraction formula [5] can be written as J(p) = k J(x ; x ) exp[ i k(x x )p ] dx dx. (.) πz So the radiant intensity, defined as I(p) = z J(p), becomes I(p) = k J(x ; x ) exp[ i k(x x )p ] dx dx. (.3) π Next we perform a coordinate transformation from x and x to the centroid coordinate ( x) and difference coordinate (x ) which are given by x = x + x, (.4) x = x x. (.5) The Jacobian related to this transformation is equal to one. We can thus express the coordinates x, x in terms of the centroid and difference coordinates such that x = ( x x /) and x = ( x + x /). Then the radiant intensity can be written as where I(p) = k π J( x; x ) exp( i kx p) dx d x, (.6) J( x; x ) = J( x x x ; x + ). (.7) 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 x ) is next introduced at the z = plane where x is a lateral displacement as shown in Fig... Denoting the field after the mask by U A (x), the mutual intensity can be written as J A (x ; x ) = U A(x ) U A (x ), = A (x x ) U (x ) U(x ) A(x x ), = A ( x x / x ) U (x ) U(x ) A( x + x / x ), ( ) ( ) = A τ x U (x ) U(x ) A τ + x, (.8) 9

14 x I A (p; x ) x mask θ z z = far field Figure.: Depiction of notation related to the propagation of the mutual intensity to the far zone after introducing a mask A(x x ) at the z = plane. where τ = x x and the subscript A refers to the mask. Inserting Eq. (.8) into Eq. (.7) and developing Eq. (.6) leads to the radiant intensity in the presence of the mask I A (p ; x ) = k ( ) ( ) A τ x A τ + x J(x + τ; x ) exp( i kx p) dx dτ, π (.9) where x on the left-hand side emphasizes that the mask is centered at x. 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... Aperture method in one dimension Like some other coherence measurement techniques [6], 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 [7]. The coherence function J(x + τ; x ) can be extracted from the radiant intensity I A (p ; x ), but it is difficult for those points whose separations are greater than the windows extent. In order to explain this we

15 expand J(x +τ; x ) as a Taylor series around the aperture centroid x as follows [4] J(x + τ; x ) = J(x ; x ) + J x (x ; x ) (x + τ x ) + J! x (x ; x ) (x + τ x )! + 3 J x (x ; x ) (x + τ x ) , 3 3! n J = x (x ; x ) τ n n n!. (.3) n= Substituting Eq. (.3) into Eq. (.9), we get I A (p ; x ) = k π = k π n= n= ( ) ( ) A τ x A τ + x n J x (x ; x ) τ n n n! exp( i kx p) dx dτ, n J x n (x ; x ) A n (x ) exp( i kx p) dx, (.3) where A n (x ) = τ n ( ) ( ) n! A τ x A τ + x dτ. (.3) For a real and symmetric aperture function, we get A n (x ) = when n is odd. If we neglect the higher order even terms in Eq. (.3) and approximate this equation to its leading term n =, results in I A (p ; x ) = k π J(x, x )A (x ) exp( i kx p) dx. (.33) Therefore a basic estimate of J(x ; x ) can be retrieved by performing the inverse Fourier transform [8] of Eq. (.33) as J(x ; x ) A (x ) I A (p ; x ) exp(i kx p) dp, (.34) where A (x ) = ( ) ( ) A τ x A τ + x dτ, (.35) is the autocorrelation of the aperture function. For a rectangular opening of width w, the aperture function is A(x) = rect(x/w) [8]. The autocorrelation A (x ) is

16 obtained from the overlapping area of two rectangular functions (see Fig..3) as follows A (x ) = w x. (.36) The function A (x ) is non-zero within the limit w x +w and otherwise zero as illustrated in Fig..4. The same treatment for a Gaussian aperture function is shown in Ref. [9]. Therefore it is obvious from Eq. (.35) that for the pair of points having a separation x larger than the width of aperture function, we can not recover the coherence function J(x ; x ). This is a fundamental limitation of the aperture method. w x x / x / x Figure.3: Overlap of two rectangular functions with width w and height centered at x / and +x /. A (x ) x /w - - x /w Figure.4: Autocorrelation function A (x ) for a rectangular aperture represented by A(x) = rect(x/w)... 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 [3]. In this method, instead

17 of an aperture, a binary transparent phase mask is employed at z = plane and the difference of two radiant intensities measured with and without the mask is calculated. This is implemented by subtracting Eq. (.9) from Eq. (.6) as follows: I(p ; x ) = I(p) I A (p ; x ), = k J( x; x ) exp( i kx p) dx d x k π π ( ) ( ) A τ x A τ + x J(x + τ; x ) exp( i kx p) dx dτ. (.37) Substituting x = x + τ and Eq. (.3) into Eq. (.37), we get I(p ; x ) = k π n= k A = k π = k π π ( n= ) τ n n J n! x (x ; x ) exp( i kx p) dx dτ n τ n n ( ) J n! x (x ; x )A τ x n τ + x exp( i kx p) dx dτ, n J n= x (x ; x ) exp( i kx p) n τ n ( ) ( [ A τ x A τ + x n! n= )] dτdx, n J x n (x ; x ) Ān(x ) exp( i kx p) dx, (.38) where Ā n (x ) = τ n ( ) ( )] [ A τ x A τ + x dτ. (.39) n! After performing the inverse Fourier transform of Eq. (.38), we obtain n= n J x (x ; x ) Ān(x ) = I(p ; x n ) exp(i kx p) dp. (.4) As the binary phase mask is transparent with a single discontinuity [], we consider it to be a signum function A(x) = sgn(x). In this case, the value of the term A ( ) ( ) τ x A τ + x in Eq. (.39) is either or -. In order to obtain a non-zero value of Ā n (x ), the mask function product in Eq. (.39) has to be equal to -, which 3

18 is possible if τ is within the limit x / τ x /. Under these conditions, we get Ā n (x ) = n! [( = x x τ n [ ( )] dτ, ) n + ( ) n ] x n+ (n + )!. (.4) From Eq. (.4) it is clear that Ān(x ) = for odd n. If we neglect the higher order even terms and approximate the series in Eq. (.39) by its leading term n =, we get J(x ; x ) Ā(x ) Putting n = in Eq. (.4), we have I(p ; x ) exp(i kx p) dp. (.4) Ā (x ) = x, (.43) which is shown in Fig..5. We can conclude from Eq. (.43) that there is no maximum limitation for the separation x between the points to recover J(x ; x ) from Eq. (.4). This method has also some fundamental constraints, for example, in this case it is problematic to recover the coherence function J(x ; x ) for small values of x which, however, can be described through interpolation [3]. It is also difficult to extend this method to two dimensions [3]. Ā (x ) x -4-4 x Figure.5: Function Ā(x ) for a binary phase mask represented by A(x) = sgn(x). 4

19 ..3 Shadow method in one dimension The main drawbacks of the previous methods are completely eliminated in the shadow method [9]. In this case, at the z = plane, 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) = a(x), (.44) where a(x) is a real distribution around x = and bounded as a(x). A basic estimate of the mutual intensity J(x ; x ) as a function of the centroid and difference coordinates is denoted by J ( x; x ) and obtained by substituting x = x into Eq. (.4), leading to J( x; x ) J ( x; x ) = I(p ; x) exp(i kx p) dp. (.45) Ā (x ) In this equation, Ā (x ) can be found by substituting n = into Eq. (.39) resulting in Ā (x ) = [ A ( τ x ) A ( τ + x )] dτ. (.46) Inserting the mask function, defined in Eq. (.44), into Eq. (.46), we get Ā (x ) = = { [ ( )] [ ( )]} a τ x a τ + x dτ, ( ) a τ x ( ) dτ + a τ + x ( dτ a τ + x ) a ( ) τ x dτ. (.47) A circular obscuration with radius d is represented by a(x) = circ(x/d) [8]. The first two terms on the right-hand side of Eq. (.47) can be obtained from the area of the circle as πd. The last term of Eq. (.47) is the obscuration s autocorrelation function which can be calculated from the overlapping area of two symmetric circular functions as shown in Fig..6. Finally, we get Ā(x ) as follows Ā (x ) = πd d arccos ( x ) + ) x ( 4d d x, (.48) 5

20 where x d. It is evident that Ā(x ) of Eq. (.47), illustrated in Fig..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. d x x / x / x Figure.6: Overlapof twocircularfunctionswithradius d, centeredat x / and +x /. Ā (x ) x /d -4-4 x /d Figure.7: Ā (x ) for a transparent mask with a localized obscuration of circular shape a(x) = circ(x/d) having a radius of d...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 [4]. 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..8. If we know the values of the second-order 6

21 correlation functions (the cross-spectral density function or the mutual coherence function) for all pair of points S (x ) and S (x ) on the surface S, then the corresponding correlation functions at any pair of points P (x ) and P (x ) located distances R and R, respectively, away from the surface S can be determined. Exit pupil R P (x ) Light source S (x ) S (x ) R P (x ) Optical system Surface S Figure.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 S (x ) and S (x ) by V (x, t)and V (x, t),respectively, whiletherelatedspectralamplitudesare Ṽ (x, ν) and Ṽ (x, ν), where ν is the frequency of field. From the Huygens-Fresnel principle [], the complex amplitudes at the points P (x ) and P (x ) can be expressed in terms of the complex amplitudes at all points located on surface S as follows Ṽ (x j, ν) = S Ṽ (x j, ν) exp(ikr j) R j Λ j (k)d x j, (.49) where j (, ). Λ (k), Λ (k) are the inclination factors which can be approximated for small angles of diffraction at the limiting aperture as follows From Eq. (.49) we can write Ṽ (x, ν)ṽ (x, ν ) = S Λ (k) Λ (k) ik π. (.5) S Ṽ (x, ν)ṽ (x, ν ) exp[ i(k R kr ] R R Λ (k)λ (k ) d x d x, (.5) 7

22 where the wave numbers at frequencies ν and ν are k = πν/c and k = πν /c, respectively. After taking an ensemble average over different realizations on both sides of Eq. (.5), we get Ṽ (x, ν)ṽ (x, ν ) Ṽ = (x, ν)ṽ (x, ν ) S S exp[ i(k R kr ] Λ R R (k)λ (k ) d x d x. (.5) The cross-spectral density function, W(x ;x, ν) which is a measure of the correlation between the spectral amplitudes at spatial points x, x and at frequency ν, is defined by Ṽ (x, ν)ṽ (x, ν ) = W(x ;x, ν) δ(ν ν ). (.53) Substituting Eq. (.53) into Eq. (.5), we get the propagation law for the crossspectral density function as follows W(x ;x, ν) = S S W(x ;x, ν) exp[ik(r R )] R R Λ (k)λ (k) d x d x. (.54) Assume next that the light is quasi-monochromatic whose effective bandwidth ( ν) is small compared to its mean frequency ν, i.e., ν/ ν. In order to develop the propagation law in this case, we ignore the weak frequency dependence of Λ (k), Λ (k) and replace them with the corresponding values Λ, Λ at mean frequency ν. Multiplying both sides of Eq. (.54) with exp( iπντ ) and then integrating over ν ( < ν < ), we obtain W(x ;x, ν) exp( iπντ )dν = S S W(x ;x, ν) exp{ iπν[τ (R R )/c]} R R Λ Λ d x d x dν, (.55) where τ 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(x ;x, ν) = Γ(x ;x, τ ) = Γ(x ;x, τ ) exp(iπντ ) dτ, W(x ;x, ν) exp( iπντ ) dν. 8 (.56a) (.56b)

23 UsingEq.(.56b)inEq.(.55), wefindthepropagationlawforthemutualcoherence function as follows Γ(x ;x, τ ) = S S Γ[x ;x, τ (R R )/c ] R R Λ Λ d x d x. (.57) Since practically the coherence length ( c/ ν) of the light is greater than the optical path difference R R, it holds that R R c ν. (.58) In this situation using the properties of the envelope representation and Eq. (.58), we can write the following Γ[x ;x, τ (R R )/c] Γ(x ;x, τ ) exp[ i k(r R )], (.59) where k = π ν/c is the wave number at mean frequency ν. By substituting Eq. (.59) into Eq. (.57), the following form for the propagation law valid for quasi-monochromatic light Γ(x ;x, τ ) = S S Γ(x ;x, τ ) exp[i k(r R )] R R Λ Λ d x d x. (.6) When τ / ν, the mutual intensity J(x ;x ) and the mutual coherence function are related by the expression Γ(x ;x, τ ) J(x ;x ) exp( iπ ντ ). (.6) In order to extract the mutual intensity from the mutual coherence function, we consider equal-time correlations, i.e., we set τ = in Eq. (.6), to obtain the mutual intensity as follows J(x ;x ) = Γ(x ;x, ). (.6) Furthermore, using Eqs. (.6) and (.5) in Eq. (.6), we arrive at ( ) k J(x ;x ) = J(x ;x ) exp[ i k(r R )] d x d x, (.63) π S S R R which is also known as Zernike s propagation law for the mutual intensity of quasimonochromatic source. 9

24 We next assume that the surface S coincides with a partially coherent, planar, quasi-monochromatic secondary source denoted by B illustrated in Fig..9. Further, the points P (x ) and P (x ) are taken to be in the far-fieldof the source distance x and x away from B. Let x andx bethepositionvectors whose origin O issituated inside the source region B. The unit vectors p and p indicate the directions of the far-field points. Finally R, R are the distances of the points P (x ) and P (x ) to the source points S (x ) and S (x ), respectively. The position vectors can be written as x = x p, x = x p. (.64a) (.64b) When the distances x and x are sufficiently large, we may approximate R x p x, R x p x. (.65a) (.65b) Substituting Eqs. (.65a) and (.65b) into Eq. (.63), the mutual intensity of the R P (x ) S (x ) x x S (x ) x p R P (x ) o x p Secondary source B Figure.9: Illustration of the notation related to the calculation of the mutual intensity of the far-field.

25 far-field becomes ( ) k J(x ;x ) = π B B J(x ;x ) exp[ i k(x p x x +p x )] (x p x )(x p x ) d x d x. (.66) Since x and x are large, we can neglect the terms p x and p x in the denominator implying that J(x ;x ) = ( ) k J(x ;x ) exp[ i k(x p x x +p x )] d x d x. π B B x x (.67) We assume that for the distances x = x = x and for the unit vectors p = p = p where p = (p, q), the mutual intensity at a single point P (x ) in the far-field due to the pair of source points S (x ) and S (x ) in plane B can be written as J(p) = ( ) k J(x π x ;x ) exp[ i k(x x ) p] d x d x. (.68) B B So the radiant intensity, defined by I(p) = x J(p) [3], becomes I(p) = ( ) k J(x ;x ) exp[ i k(x x ) p] d x d x. (.69) π B B Next we repeat most of the steps done in one-dimensional case in section.. First we perform the coordinate transformation from x and x into the centroid coordinate( x)anddifferencecoordinate(x ), which arefrequently usedinthethesis. The transformation is x = x +x, (.7) x = x x, (.7) where x = (x, y ), x = (x, y ), x = ( x, ȳ), and x = (x, y ). After expressing the coordinates x, x in terms of the centroid and difference coordinates as x = ( x x /) and x = ( x +x /), the radiant intensity becomes I(p) = ( ) k π J( x;x ) exp( i kx p) d x d x, (.7)

26 where J( x;x ) = J( x x x ; x + ), (.73) in the finite surface B and J( x;x ) is zero outside of it. When a mask with an amplitude function A(x x ) is introduced at the source plane B, the resultant radiant intensity is obtained as I A (p;x ) = ( ) ( ) ( ) k A τ x A τ + x π J(x + τ;x ) exp( i kx p) d x d τ, (.74) where x = (x, y ), τ = (τ, η) = x x. Expanding J(x + τ) in a Taylor series around the aperture centroid x, we get J(x + τ;x ) = J(x ; x, y ; y ) + J x (x ; x, y ; y ) (x + τ x )! + J ȳ (x ; x, y ; y ) (y + η y )! + J x (x ; x, y ; y ) (x + τ x )! + J ȳ (x ; x, y ; y ) (y + η y ) +...,! n+m J = x n ȳ (x ;x ) τ n η m m n!m!. (.75) n,m= by We consider a transparent and uniform mask having a localized obscuration given A(x) = a(x), (.76) where a(x) is a real distribution within the limit a(x) confined around x=. So the difference of two radiant intensities, measured with and without the

27 obscuration, can be obtained using Eqs. (.69) and (.74) as I(p;x ) = I(p) I A (p;x ), ( ) k = J( x;x ) exp( i kx p)d x d x π ( ) ( ) ( ) k A τ x A τ + x π J(x + τ;x ) exp( i kx p) d x d τ. (.77) Substituting x = x + τ and Eq. (.76) into Eq. (.78), implies ( ) k I(p;x ) = π n,m= ( ) k π n,m= τ n η m n!m! n+m J x n ȳ m(x ;x ) exp( i kx p) dx dy dτdη τ n η m n+m J n!m! x n ȳ (x ;x )A m ( τ x exp( i kx p) x dy τ η, ( ) k d n+m J = π n,m= d x n dȳ m(x ;x ) exp( i kx p) ( ) ( )] [ A τ x A τ + x dx dy dτdη, ( ) k = π n,m= ) A ( τ n η m n!m! τ + x n+m J x n ȳ m (x ;x ) Ān,m(x ) exp( i kx p) dx dy, ) (.78) where Ā n,m (x ) = τ n η m n!m! [ A ( τ x ) A ( τ + x )] dτ dη. (.79) After performing the inverse Fourier transform of Eq. (.78), we can write n,m= n+m J x n ȳ m(x ;x ) Ān,m(x ) = I(p;x ) exp(i kx p) dp dq. (.8) If we approximate the series in Eq. (.8) by its first term n, m =, we get J(x ;x ) Ā,(x ) I(p;x ) exp(i kx p) dp dq. (.8) 3

28 A basic estimate of the mutual intensity J(x ;x ) as a function of the centroid and difference coordinates is denoted by J ( x;x ). By inserting x = x into Eq. (.8) indicates that J( x;x ) J ( x;x ) = J ( x;x ) I(p; x) exp(i kx p) dp dq. (.8) Ā, (x ) The function Ā,(x ) can be found by substituting n, m = into Eq. (.79), i.e., ( ) ( )] Ā, (x ) = [ A τ x A τ + x d τ. (.83) Inserting the mask function, defined in Eq. (.76) into Eq. (.83), we get { [ ( )] [ ( )]} Ā, (x ) = a τ x a τ + x d τ, ( ) ( ) = a τ x d τ + a τ + x d τ ( ) ( ) a τ + x a τ x d τ. (.84) For the obscuration of a circular shape and radius d we have a(x) = circ(x/d)[8]. Thefirsttwotermsontheright-handsideofEq.(.84)canbeobtainedfromthearea ofthecircleas πd. Thelasttermistheobscuration sautocorrelationfunctionwhich can be calculated from the overlapping area of two symmetric circular functions as shown in Fig... Finally, we get Ā,(x ) as follows ( x Ā, (x ) = πd d ) arccos + x d ( 4d x ), (.85) where x d. It is obvious that Ā,(x ) of Eq. (.85), illustrated in Fig.., 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. From the mutual intensity given in Eq. (.8), we can also get the normalized degree of spatial coherence by using Eqs. (.6), (.7), and (.7) as follows µ( x;x ) = J ( x;x ) J ( x x /, ) J ( x +x /, ). (.86) In order to measure coherence between many pair of points we have to scan the obscuration across the light field, i.e., by varying the centroid x of the obscuration. 4

29 η x / τ d x / Figure.: Overlap of two circular functions with radius d, centered at x / and +x / in the (τ, η) plane. Ā, (x ) x /d -4-4 x /d Figure.: Ā, (x ) for a transparent mask with a localized obscuration of circular shape a(x) = circ(x/d) of radius d. 5

30 Chapter III Experimental results with shadow method In this chapter, the measurement of the spatial coherence properties of stationary, quasi-monochromatic sources is demonstrated. Here we study three different cases in which the optical fields are considered to be partially spatially coherent, almost fully spatially coherent, and almost incoherent. An almost fully coherent light source is used to derive other two kind of light fields using some optical elements for both the shadow method and Young s two-pinhole method. 3. Experimental setup The experimental setup for measuring the partially spatially coherent wave fields is shown in Fig. 3.. At first a partially spatially coherent beam is generated. In order to do so we have focused a beam emanating from a helium-neon (HeNe) laser (central wavelength λ = 633 nm) onto the edge of a rotating diffuser by a focusing lens L (focal length f = 4 mm). Then the beam coming out from the diffuser is collimated using a lens L (f = 5 mm). After that the collimated beam is limited by an iris I and is incident on a digital micromirror device (DMD) chip. We assume the laser beam to be close to Gaussian Schell-model (GSM), i.e., both the intensity profile and the complex degree of spatial coherence profile of the beam follow Gaussian distribution. The intensity profile of a GSM beam depends on the absolute coordinate and the complex degree of spatial coherence profile of the GSM beam depends on the difference of two spatial coordinates [33 36]. The diffuser is made of plastic whose surface has some irregularities created by sand paper. The measuringpartofthesetupconsistsofthedmddevice, aniris I,aninterchangeable 6

31 Camera sensor ND filter L 4 L 3 I HeNe laser L Rotating L diffuser I DMD Camera sensor Figure 3.: Illustration of the DMD-based shadow-method setup. imaging lens L 3 (f = 75 mm) and a converging lens L 4 (f = 5 mm), a neutral density (ND) filter, and a CMOS camera. Here the DMD device is a modified Texas Instruments DLP Light Crafter video projector module where the DMD chip is revealed by removing the LED light source and the projector lens from the original device [37] as shown in Fig. 3.. The chip consists of square mirrors with a diagonal length of.8 ñm. These digital micromirrors are micromechanical devices arranged in 45 rotated diamond orientated array as shown in Fig The mirrors are always tilted diagonally by ± when the DMD is switched on otherwise they are flat. The device is controlled with a computer by digital HDMI video signal and we have used the mirrors to draw or create the obscuration or pinholes. The DMD array works as a reflective grating due to the periodicity of the micromirrors. The DMDs are also used in many other applications like measurement and modification of light [38, 39]. When the collimated beam hits the DMD chip, it gets reflected and thereby produces several diffraction orders in different directions. We use the iris I to pass only one of the strongest diffraction orders. This beam is then detected by the camera sensor through a converging lens. The converging lens is placed in such a way that the camera sensor is at the focal plane of the lens. So the converging lens is working as a Fourier-transforming lens. Alternatively, another lens (imaging lens) is used to image the DMD chip plane onto the camera. The imaging lens is located 7

32 between the Fourier-transforming lens and the DMD chip. When the imaging lens is in use, the Fourier transforming lens has been removed and vice versa. The CMOS camera sensor has 8 4pixels with the pixel size of 5. µm[4]. The ND filter is used on the camera sensor to drop the intensity of the beam so that the camera does not burn in case of high intense beam. Further, it is also used to increase the exposure time of the camera so that it becomes longer than the rotation time of the diffuser. The camera sensor in Fig. 3. is used for Young s two-pinhole method which is discussed in section 3.3. Figure 3.: Modified Texas instruments digital light crafter. x x x z y Figure 3.3: Arrangement of the micromirrors of the DMD device. For an almost fully spatially coherent light field, we remove all the intermediate 8

33 optical elements between the laser and the DMD, so that the beam directly hits the DMD chip. The coherence after the diffuser depends on the size of the secondary source, i.e., laser spot on the diffuser. Without the focusing lens between the laser source and the rotating diffuser, the spot is larger. Therefore, the produced light possesses low spatial coherence. In both cases the rest part of the setup is the same as for partially spatially coherent field. 3. Measurement and Results Here the DMD device and the camera is connected and controlled by a computer. The DMD screen is acting as a secondary display which is set to the resolution in the computer using Matlab and Java function tool. The mirrors in the DMD and the camera settings are operated through a Matlab programme. We can also manually adjust the camera settings through the camera software. When the laser and the DMD are switched on, we use the imaging lens by removing the Fourier-transforming lens to image the DMD chip plane. A circular obscuration with a diameter of µm is drawn on the DMD chip plane in a way such that it remains at the middle portion of the beam by adjusting and compensating the mirrors through Matlab script. Then after removing the imaging lens, we use the Fourier transforming lens through which the beam hits the camera. Here we adjust the exposure time, frame rate and gain of the camera so that any portion of the beam does not saturate and write them into the setting file of the Matlab script. In order to save the measurement data, we create a folder and copy the path into the Matlab script. By compiling the Matlab script, two images of the beam after the Fourier transforming lens, i.e., the radiant intensities, are then captured, one with the circular obscuration at the DMD plane and another without the circular obscuration. We also capturedtwo images of the DMDplane, one with thecircular obscurationat the DMD plane and another without the circular obscuration, using the imaging lens and those two images are shown in Fig. 3.4 for an almost fully spatially coherent light field. Next we calculate the mutual intensity J ( x;x ) and the complex degree of spatial coherence µ( x;x ) by analyzing the captured images (radiant intensities) with the Fourier transforming lens for all three different types of light fields (partially spatially coherent, almost fully spatially coherent, and almost incoherent). 9

34 The entire numerical analysis is performed in Matlab environment. For this purpose those two images, with the circular obscuration [I(p; x)] and without the circular obscuration [I(p)], are loaded in Matlab as a function of p and q which are shown in Fig. 3.5(a) for a partially spatially coherent light field, in Fig. 3.5(b) for an almost fully spatially coherent light field, and in Fig. 3.5(c) for an almost incoherent light field without obscuration with obscuration Figure 3.4: Example of measured two images, with the circular obscuration and without the circular obscuration, using imaging lens for almost fully spatially coherent light fields. Color scale is in arbitrary 8 bit camera values. After that the two images are subtracted from each other and the resultant image I(p; x) is plotted as a function of p and q as shown in Fig. 3.6 for all three cases in the similar manner as in Fig Here the elements p, q are calculated from the camera pixels 8 4 and the focal length of the Fourier transforming lens (f = 5 mm) using the following equations: p = X f, q = Y f, (3.a) (3.b) respectively, where X and Y are the horizontal and vertical axis on the camera plane. Parameters X and Y are determined using the camera pixels and their size. We perform the Fourier transform of the resultant image I(p; x). During the Fourier transform, using p, q and Fast Fourier Transform [4, 4], we calculate the new coordinates which give the value of the elements x and y, respectively, where x = (x, y ). Now using Eq. (.8), the mutual intensity J ( x;x ) is obtained 3

35 q [µm] q [nm] q [nm] (a) I(without obscuration) p [nm] (b) I(without obscuration) p [nm] (c) I(without obscuration).... p [µm] q [nm] q [nm] q [µm].5.5 I A (with obscuration) p [nm] I A (with obscuration) p [nm] I A (with obscuration).. p [µm] Figure 3.5: Example of measured two images, with the circular obscuration [I(p; x)] and without the circular obscuration [I(p)] for(a) a partially spatially coherent light field, (b) an almost fully spatially coherent light field, and (c) an almost incoherent light field. by dividing the Fourier transformed image with the circular obscuration function Ā, (x ) given in Eq. (.85). Since we can not get the mutual intensity of the whole beam from just one position of the obscuration, we scanned the whole beam by changing the position of the obscuration s centroid ( x) along the vertical axis x and 3

36 q [nm] (a) I p [nm] q [nm] (b) I p [nm] 3 (c) I 3 q [µm] p [µm] Figure 3.6: Resultant image I(p; x) which is the difference of measured two images, with the circular obscuration [I(p; x)] and without the circular obscuration [I(p)] for(a) a partially spatially coherent light field,(b) an almost fully spatially coherent light field, and (c) an almost incoherent light field. for each position the image pairs are captured with and without the obscuration. For each image pair, we calculate the mutual intensity J ( x;x ) by following the same procedure from the beginning. We take the central column of the mutual intensity matrices for each position of the obscuration s centroid and create a new matrix which is the mutual intensity of the whole beam. The absolute value of this new mutual intensity, plotted as a function of centroid coordinate x and difference coordinate x (ȳ and y are zero as the obscuration is scanned along vertical x axis), is depicted in Fig. 3.7 for all three cases. Here we use an interpolation method to get sufficient number of points in order to calculate the complex degree of spatial 3

37 (a) J( x, x ) x [µm] x [µm] (b) J( x, x ) x [µm] x [µm] x [µm] (c) J( x, x ) x [µm] Figure 3.7: Absolute value of the mutual intensity for (a) a partially spatially coherent light field, (b) an almost fully spatially coherent light field, and (c) an almost incoherent light field. The black areas around the figures indicate lack of data. coherence. Using Eq. (.86), the normalized degree of spatial coherence µ( x;x ) is evaluated and plotted as a function of the centroid coordinate x and the difference coordinate x as shown in Fig. 3.8 for all three cases. 33

38 (a) µ( x, x ) 5.8 x [µm] (b) µ( x, x ) 5 5 x [µm] 8 (c) µ( x, x ) x [µm].6.4 x [µm] x [µm] x [µm] Figure 3.8: Absolute value of the normalized degree of spatial coherence for (a) a partially spatially coherent light field, (b) an almost fully spatially coherent light field, and (c) an almost incoherent light field. 3.3 Comparison with Young s two-pinhole method In this section we discuss about the Young s two-pinhole method to measure spatial coherence of stationary light fields as in Refs. [43,44]. This method is based on the classical Young s double-pinhole experiment. We consider the light fields to be quasi-monochromatic. If the two pinholes are located at coordinates x and x, then 34

39 using Eq. (.6) the normalized degree of spatial coherence can be written as µ(x ; x ) = J(x ; x ) I(x )I(x ), (3.) where J(x ; x ) is the mutual intensity and I(x) = J(x; x) is the field intensity at coordinate x. From the intensity I(x) and using the visibility of interference fringes, we can obtain the absolute value of degree of spatial coherence. The phase of the degree of spatial coherence φ(x ; x ) can also be measured from the lateral position of the fringes on the detector plane. So the mutual intensity becomes J(x ; x ) = µ(x ; x ) I(x )I(x ) exp[ iφ(x ; x )]. (3.3) Now the entire mutual intensity of the light fields can be obtained by scanning the pinholes for each possible combinations of x, x and by measuring the intensity profile I(x) at each step. Using paraxial approximation the intensity profile on the detector plane can be written as [ I( x) = I ( x) + I ( x) + [I ( x)i ( x)] / µ(x ; x ) cos φ(x ; x ) + 4πa ] x, (3.4) dλ where a represents the distance between the two pinholes and d is the separation between the detector plane and the pinhole plane. Further, I j ( x) represents the intensity when jth pinhole is open. Adjusting the origin of x coordinate in terms of the centre of the pinhole coordinate x, x we get x = x (x x )/. In order to obtain the complex degree of coherence we measure the intensities I( x), I ( x), and I ( x). Then the normalized fringes are expressed as F( x) = I( x) I [ ( x) I ( x) = µ(x [I ( x)i ( x)] / ; x ) cos φ(x ; x ) + 4πa ] x. (3.5) dλ In Young s method we have used the same laser source and the same experimental setup of shadow method as shown in Fig. 3.. Here using the digital micromirrors, the pinholes are created on the DMD plane. The resultant fringe pattern after the beam reflected from the DMD chip plane are detected in the camera sensor. From those fringes we have calculated the coherence properties of a partially spatially coherent light field, an almost fully spatially coherent light field, and an almost incoherent light field. The absolutevalueof themutual intensity, J(x ; x ), andthe absolute value of the normalized degree of spatial coherence, µ(x ; x ), are plotted 35

40 as a function of general coordinates x and x in Fig. 3.9 for all three cases. We compare the results of Young s two-pinhole method with the results of the shadow method. At first the absolute value of the mutual intensity and the absolute value of the normalized degree of spatial coherence are plotted as a function of the centroid coordinate x and the difference coordinate x. The comparison of the absolute value of mutual intensity between the shadow method and Young s two-pinhole method is shown in Fig. 3. for all three cases. Figure 3. shows the comparison of the absolute value of the normalized degree of spatial coherence between the shadow method and Young s two-pinhole method. The coherence width of the degree of spatial coherence is also compared between the shadow method and Young s two-pinhole method for a partially spatially coherent light field, an almost fully spatially coherent light field, and an almost incoherent light field as shown in Fig

41 (a) J(x, x ) µ(x, x ) x [µm].6.4 x [µm] x [µm] 5 5 x [µm] 4 (b) J(x, x ) 4 µ(x, x ) x [µm] x [µm] x [µm] x [µm] x [µm] 5 5 (c) J(x, x ) x [µm] x [µm] 5 5 µ(x, x ) x [µm] Figure 3.9: Absolute value of the mutual intensity J(x ; x ) and the absolute value of the normalized degree of spatial coherence µ(x ; x ) for (a) a partially spatially coherent light field, (b) an almost fully spatially coherent light field, and (c) an almost incoherent light field. 37

42 x [µm] 5 5 (a)shadow J( x, x ) x [µm] 5 5 Young s J( x, x ) x x [µm] x [µm] (b)shadow J( x, x ) Young s J( x, x ) x [µm].6.4 x [µm] x [µm] x [µm]. x [µm] 8 (c)shadow J( x, x ) x [µm] x [µm] Young s J( x, x ) x [µm] x Figure 3.: Comparison of the absolute value of the mutual intensity between the shadow method and Young s two-pinhole method for (a) a partially spatially coherent light field, (b) an almost fully spatially coherent light field, and (c) an almost incoherent light field. 38