Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations

Transcription

1 3036 J. Opt. Soc. Am. A/ Vol. 3, No. 1/ December 006 P. Réfrégier and J. Morio Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations Philippe Réfrégier Physics and Image Processing Group/Fresnel Institute UMR CNRS 6133, École Généraliste d Ingénieurs de Marseille, Domaine Universitaire de Saint-Jérôme, Marseille Cedex 0, France Jérôme Morio Physics and Image Processing Group/Fresnel Institute UMR CNRS 6133, École Généraliste d Ingénieurs de Marseille, Domaine Universitaire de Saint-Jérôme, Marseille Cedex 0, France, and Office National d Etude et Recherches Aérospatiales, Département Electromagnétisme et Radar, BA701, 13661, Salon Air Cedex, France Received April 13, 006; revised June 3, 006; accepted June 4, 006; posted July 13, 006 (Doc. ID 69909) We propose to analyze Shannon entropy properties of partially coherent and partially polarized light with Gaussian probability distributions. It is shown that the Shannon entropy is a sum of simple functions of the intensity, of the degrees of polarization, and of the intrinsic degrees of coherence that have been recently introduced. This analysis clearly demonstrates the contribution of partial polarization and of partial coherence to the characterization of disorder of the light provided by the Shannon entropy, which is a standard measure of randomness. We illustrate these results on two simple examples. 006 Optical Society of America OCIS codes: , INTRODUCTION The analysis of the coherence properties of partially polarized electromagnetic waves has been the subject of interesting new developments. 1 5 In particular, in Ref., Wolf proposed a definition of a degree of coherence that can be equal to one for partially polarized light. Another definition of degree of coherence has thus been proposed 3,4 to introduce invariance properties. With this definition, only totally polarized light can have a degree of coherence equal to one. In Ref. 6 it has been shown that one can introduce intrinsic degrees of coherence that are based on invariance properties to separate partial polarization and partial coherence. It has been shown 7 that these intrinsic degrees of coherence are related to the definition introduced in Ref.. In particular they can be equal to one for partially polarized light. It has also been shown in Ref. 8 that these intrinsic degrees of coherence appear as natural quantities if one analyzes the mutual information of the electric fields between two points in space and at two different times. Although different results have already been established, there are still open questions that can be of interest from both theoretical and practical points of views. In particular, the measure of statistical disorder, or randomness, of partially polarized light with the Shannon entropy has not been clearly analyzed until presently. However, this is an important question since, as mentioned in a fundamental paper by Glauber 9 in 1963, the coherence conditions restrict randomness of the fields rather their bandwidth. The analysis of the Shannon entropy will allow us to see that partial polarization, characterized by the degree of polarization, and partial coherence, measured with the intrinsic degrees of coherence, are two complementary measures of randomness that play different but symmetrical roles. The electric field of an electromagnetic radiation at point r (in the following, bold letters describe vectors or matrices) and at time t is classically represented by a complex random vector 10 E r,t. To simplify the notations, it will be written as E x =E r,t with x= r,t. Itis interesting to note that with these notations one could also consider different directions of propagation at different times instead of different locations r 1 and r. The electric field will also be assumed to correspond locally to a plane wave with a good approximation; it will thus be represented in dimension two E x = E X x,e Y x T where the superscript T stands for transpose. The coherence properties between spatiotemporal points x 1 and x of the complex random vectors E x 1 and E x can be represented by the mutual coherence matrix,4,10 14 x 1,x, defined by x 1,x = E x E x 1, where the symbol. denotes ensemble averaging (i.e., statistical averaging) and where the symbol denotes conjugate transpose. Equation (1) can be developed into 1 x 1,x = E X x E * X x 1 E X x E * Y x 1 E Y x E * X x 1 E Y x E * Y x 1, where the asterisk stands for conjugate. One can note that the standard coherency matrix 10 corresponds to the case x 1 =x or more precisely to r 1 =r and to t 1 =t. For the sake of clarity the standard coherency matrix will be called, the polarization matrix, which will thus correspond to /06/ /$ Optical Society of America

2 P. Réfrégier and J. Morio Vol. 3, No. 1/ December 006/ J. Opt. Soc. Am. A 3037 x i = E x i E x i = E X x i E * X x i E X x i E * Y x i E Y x i E * X x i E Y x i E * Y x i. 3 It has recently been proposed 6 to analyze the spatiotemporal properties of partially polarized light so that one can separate partial polarization and partial coherence. For that purpose, useful invariance properties have been introduced that allow one to characterize intrinsic coherence properties of the optical light independently of the particular experimental conditions. This approach has lead to new intrinsic degrees of coherence that can be related to measurable quantities. These intrinsic degrees of coherence are the singular values of a normalized mutual coherence matrix that is simply M x 1,x = 1/ x x 1,x 1/ x 1. In this case, local modifications of the polarization state may not modify the intrinsic degrees of coherence. Their physical interpretation in interference experiments is analogous to the one introduced in Ref. and has also been analyzed in Refs. 6 and 7. Furthermore, it has recently been shown 8 that the spatiotemporal coherence properties of partially polarized light with Gaussian probability distributions can be analyzed using the mutual information. This last quantity is a standard measure of statistical dependence. It has been shown in Ref. 8 that this approach leads to the same intrinsic degrees of coherence as the ones introduced in Ref. 6. However, the precise relation of the degrees of polarization and the intrinsic degrees of coherence with the Shannon entropy, 15 which is a standard measure of randomness, 16 have not been studied in detail up to now. We propose here to analyze this relation in the case of partially polarized light with Gaussian fluctuations. Partially polarized light with Gaussian fluctuations can correspond to thermal light and is an interesting example of classical light. The generalization to quantum light with non-gaussian fluctuations is out of the scope of this paper since it needs higher-order statistics 9 and will be analyzed in future work. We furthermore note that the analysis of partial coherence with only the mutual coherence matrix, i.e., the covariance matrix, is complete if the fluctuations of the electric fields are fully characterized by this quantity, which is the case of Gaussian fluctuations. We thus demonstrate in this case that the entropy of partially polarized and partially coherent light is a sum of simple functions of the intensity, of the standard degree of polarization of the fields E x 1 and E x, and of the intrinsic degrees of coherence introduced in Refs. 6 and 8. Moreover, we show that the standard degrees of polarization and the intrinsic degrees of coherence have analogous and complementary influences on the entropy. These results show that one can obtain a unified approach and a clear physical meaning for the intrinsic degrees of coherence and the degrees of polarization. In the following, we analyze the Shannon entropy of partially polarized and partially coherent light at two different points in space and at two different times in Section. The relation of the Shannon entropy with the intrinsic degrees of coherence is analyzed in Section 3. In 4 Section 4, we illustrate these properties on simple examples. We conclude in Section 5.. ENTROPY OF PARTIALLY COHERENT AND PARTIALLY POLARIZED LIGHT WITH GAUSSIAN FLUCTUATIONS A. Background on Entropy of Partially Polarized Light The von Neumann entropy of partially polarized light has been studied for many years This approach is based on an analogy between the density matrix used in quantum mechanics and a normalized version of the polarization matrix. However, this approach provides only a partial entropy and cannot be easily used to describe large intensity fields (for which there are many quantum states), and its application to partial coherence is not straightforward. In the classical limit (i.e., large field value) the standard approach consists in using the Shannon entropy commonly defined in statistical thermodynamics. The Shannon entropy is well defined for random processes that take discrete values. 16 In this case, if P n is the probability that the random variable x takes the value n, the entropy of the probability law is S P = P n log P n. 5 n The case of continuous processes is more complex. 16 One may simply use the differential entropy defined by the following equation: S D P x = P x E log P x E de, where...de stands for complex two-dimensional integration. However, in this case one has to consider dimensionless quantities. 1 Another approach consists in quantizing the electric field with a precision. If the probability density function (PDF) of E x at a spatiotemporal point x is P x E and if is sufficiently small, the probability of observing the value E with the precision on each component can be written as P x, E = 4 P x E. In this case, P x, E is a dimensionless quantity and has the dimension of the intensity. The entropy is thus 6 7 S P P x = E i P x, E i log P x, E i, 8 where the summation is performed over the different discrete values E i defined with a precision. When is small one can consider the continuous limit and thus S P P x = P x E log P x, E de. The advantage of this approach is that one can get a physical interpretation of the entropy with the asymptotic equipartition property. 16 Indeed, let us consider that one observes N measurements and that one analyzes the different series with a measurement precision on each component of the field. Then the number N of different 9

3 3038 J. Opt. Soc. Am. A/ Vol. 3, No. 1/ December 006 P. Réfrégier and J. Morio series that will appear when N is large is of the order of exp NS P x (i.e., N exp NS P x ). B. Shannon Entropy of Partially Polarized Light In the case of light with Gaussian fluctuations, the PDF of the complex random vector E x that describes the fluctuations of the electric field can be written as 1 P x E = det x exp E 1 x E, 10 where x is the polarization matrix at spatiotemporal point x and det x is its determinant. For values of sufficiently small, the probability of observing the value E with the precision on each component can be written as 4 P x, E = det x exp E 1 x E. The entropy is thus S P P x = P x E log P x, E de The problem is now analogous to Eq. (6) and one thus gets 1 S P P x = log e det x As long as is sufficiently small in comparison with the intensity of the light, its precise value is not relevant. Since the degree of polarization is defined as 10 P x = 1 4det x /tr x, 14 where tr x is the trace of x, the expression of the entropy of partially polarized light can also be written as S P P x = log e + log I x + log 1 P x 15 and where the intensity is I x = E X x E X x * + E Y x E Y x *. The contribution log 1 P x to the entropy is negative since it is related to the order created by the partial polarization of E x. If one considers totally incoherent light at two different points x 1 and x, the total entropy is simply the sum of the entropy at point x 1 and the entropy at point x. C. Entropy of Scalar Partially Coherent Light Let us now assume that the light is perfectly polarized. It means that the electric field can be written as E x = E x u x, 16 where E x is a scalar random field and where u x is a deterministic complex vectorial field in two dimensions. For the sake of simplicity one can consider that the light is perfectly linearly polarized along the X axis and thus that u x = 1,0 T. The randomness is due to E x, which is a random value for a given value of x. In this case the definition of the degree of coherence of the electric field of the beam at a pair of fields at points x 1 and x in the space time domain is given by 11 E x E * x 1 x 1,x = E x 1 E x. 17 The analysis of the Shannon entropy is simplified if one introduces the vector E x 1,x = E x 1,E x T. Indeed, in the case of light with Gaussian fluctuations, the PDF of the complex random vector E x 1,x can be written as 1 P x1,x E = det x 1,x exp E 1 x 1,x E, 18 with x 1,x = E x 1,x E x 1,x = E x 1 E * x 1 E x 1 E * x 1 E x E * x 1 E x E * 19 x. The analogy with the case of partially polarized light is clear. In Subsection.B we analyzed the influence of correlation in the space time domain between the components of the field on the entropy at the same space time point. Here, we describe the influence of correlation between the fields at two different space time points. However, since we introduced the vector E x 1,x we see that we have the same mathematical problem in both cases and thus S C P x1,x = log e det x 1,x, 0 where is introduced for the same reason as in Subsection.A. One can check that det x 1,x = E x 1 E x 1 x 1,x, 1 which leads to S C P x1,x = log e + log I x 1 where I x i = E x i. 4 + log I x + log 1 x 1,x, D. Entropy of Partially Polarized and Partially Coherent Light To analyze the general case of partially polarized and partially coherent light, one needs to introduce the joint PDF P x1,x E 1,E of both vectorial random fields E x 1 and E x. Obviously one has P x1 E 1 = P x1,x E 1,E de and we have a symmetrical equation for P x E. The Shannon entropy of the couple E x 1, E x is defined by S CP P x1,x = P x1,x E 1,E log 8 P x1,x E 1,E de 1 de. 3 To analyze this problem, let us introduce the fourdimensional complex random vector:

4 P. Réfrégier and J. Morio Vol. 3, No. 1/ December 006/ J. Opt. Soc. Am. A 3039 E x 1,x = E X x 1,E Y x 1,E X x,e Y x T. 4 x 1,x = E x 1,x E x 1,x, 5 Its 4 4 complex global covariance matrix is or more explicitly XY x1,x1 XX x1,x XY x1,x YX x 1 1 YY x 1,x 1 YX x 1,x YY x 1,x x 1,x = XX x1,x1, 6 XX x,x 1 XY x,x 1 XX x,x XY x,x YX x,x 1 YY x,x 1 YX x,x YY x,x with UV x i,x j = E U x i E V x j *, 7 and U,V=X,Y and i,j=1,. This covariance matrix can also be written with the mutual coherence matrix and with the polarization matrices at point x 1 and at point x. Indeed, one has x 1,x = x 1 x 1,x. 8 x 1,x x The joint PDF of E x 1 and E x is the PDF of E x 1,x, and since we analyze the case of Gaussian light, one has 1 P x1,x E = 4 det x 1,x exp E 1 x 1,x E. 9 The expression of the entropy of complex Gaussian random fields in arbitrary dimension has been discussed in Ref. 1 and leads to S CP P x1,x = log 4 e 4det x 1,x. 30 To analyze this expression, it is of interest to introduce the joint polarization matrix T x 1,x = x x. 31 Assuming one can invert T x 1,x, one can also introduce the matrix x 1,x = T 1/ x 1,x x 1,x T 1/ x 1,x. 8 3 Using Eq. (8), and if one introduces the identity matrix I d in dimension two, one can see that (see Appendix A) I x 1,x = d M x 1,x, 33 M x 1,x I d where M x 1,x is the normalized mutual coherence matrix 6,8 defined by Eq. (4). With Eq. (3) one sees that det x 1,x =det x 1,x det T x 1,x. Furthermore, det T x 1,x =det x 1 det x and thus S CP P x1,x = log e det x 1 log + e det x 4 + log det x 1,x. 34 This relation shows that the entropy of the two electric fields can be decomposed as a sum of three contributions. The first one log e det x 1 is the entropy of the partially polarized light that is represented by the electric field E x 1. The second one log e det x is the entropy of the partially polarized light that is represented by the electric field E x. The last term log det x 1,x is the variation of the entropy related to partial coherence between E x 1 and E x. This result appears more clearly if one substitutes the expression of the first two terms, which also correspond to Eq. (13), with their expression provided by Eq. (15). S CP P x1,x = log e I x 1 + log 1 P x 1 + log e I x + log 1 P x + log det x 1,x. 35 The precise expression of the last term log det x 1,x is obtained if one considers the development of M x 1,x : M x 1,x = XX x,x 1 A direct calculation thus leads to det x 1,x 4 XY x,x 1 36 YX x,x 1 YY x,x 1. =1 XX x 1,x XY x 1,x YX x 1,x YY x 1,x + XX x 1,x + YY x 1,x + XY x 1,x YX x 1,x XX x 1,x XY x 1,x * YX x 1,x * YY x 1,x XX x 1,x * XY x 1,x YX x 1,x YY x 1,x *. 37

5 3040 J. Opt. Soc. Am. A/ Vol. 3, No. 1/ December 006 P. Réfrégier and J. Morio 3. ENTROPY AND RELATION WITH INTRINSIC DEGREES OF COHERENCE A more physical and meaningful expression of Eq. (35) can be obtained as a function of the intrinsic degrees of coherence. For that purpose it is interesting to first remark that det x 1,x = det I d M x 1,x M x 1,x. 38 The proof is simply obtained by a direct calculation of the right-hand side of Eq. (38). It has been shown 6,8 that the singular value decomposition of M x 1,x allows one to get intrinsic degrees of coherence. From a mathematical point of view, one uses the property that any matrix can be decomposed in singular values S x 1,x and I x 1,x such as where M x 1,x = N x 1,x D x 1,x N 1 x 1,x, 39 D x 1,x = S x 1,x 0 0 I x 1,x, 40 and where the matrices N 1 x 1,x and N x 1,x are unitary matrices and where S x 1,x I x 1,x 0. We show in Appendix B that the expression of the entropy given by Eq. (35) can thus be written as S CP P x1,x = log e I x 1 + log e I x + log 1 P x + log 1 P x 1 + log 1 S x 1,x + log 1 I x 1,x. 41 Since the entropy of statistically independent subsystems is additive, Eq. (41) shows that we obtained a relevant decomposition of the different sources of randomness measured with the Shannon entropy. This relation shows that the entropy of the two electric fields can be decomposed as a sum of three physical contributions. The first one i=1 log ei x i / is the sum of the entropies of independent and totally unpolarized electric fields with Gaussian PDF and of intensities I x 1 and I x. The second contribution i=1 log 1 P x i is the decrease of entropy due to the order created by the partial polarization of E x 1 and E x. The third one K S,I log 1 K x 1,x is the decrease of entropy due to the order created by the partial statistical dependence, or partial coherence, of E x 1 and E x. Furthermore, one can see that the intrinsic degrees of coherence and the degrees of polarization have analogous and symmetrical expressions in their contribution to the Shannon entropy. Since they appear as additive terms in the entropy, their influence is analogous to one of the contributions of the independent subsystems. This is a new argument in favor of considering the intrinsic degrees of coherence to characterize coherence properties of partially polarized light. This result is of course related to the invariant decomposition discussed in Ref. 6 that has been introduced to separate partial polarization and partial coherence. It is important to note that the entropy goes to minus infinity either if the intensity goes to 0 or if one of the values of the degree of polarization or the intrinsic degrees of coherence goes to 1. This is a consequence of the fact that one considers PDFs instead of probability laws. This behavior is standard in statistical physics and means that these expressions are not applicable if the intensity goes to 0 or if the light becomes perfectly polarized since in those cases one should not neglect the quantum fluctuations. The divergence that appears when one of the intrinsic degrees of coherence goes to 1 is also a consequence of the fact that we neglect quantum sources of fluctuations. One can note that the entropy can also be written as S CP P x1,x = S P P x1 + S P P x + log 1 S x 1,x + log 1 I x 1,x. 4 This result shows that partial coherence introduces negative contributions to the entropy of incoherent partially polarized light. This a consequence of the fact that partial coherence decreases the randomness in the sense that it decreases the number N of different series that will appear when one observes N (with N large) measurements with a precision on each component of the field. 4. ILLUSTRATION WITH SIMPLE EXAMPLES A. Partially Temporally Coherent Light Mixing Let us first consider a simple case for which one mixes two independent optical beams with orthogonal linear polarizations. The first one is assumed linearly polarized along the X axis while the second one is polarized along the Y axis. Let us assume that one considers stationary plane waves propagating in the Oz direction. One can thus write x i = z i,t i. Furthermore, if one assumes propagation in the vacuum, one can write E x i =E z i,t i =E 0,t i z i /c where c is the celerity of light in vacuum. In this case one has E X x E * X x =I X and one has E Y x E * Y x =I Y. One can thus write E X x E X * x 1 = I X f X, E Y x E * Y x 1 = I Y f Y, 43 with = z z 1 /c+ t 1 t and where f X, f Y are two complex functions. The degrees of polarization are equals: P x 1 =P x =P = I X I Y / I X +I Y where a is the modulus of a. If, for example, f Y f X, one gets S x 1,x = f X and I x 1,x = f Y. In this case one has x 1,x = E x E x 1 = I Xf X I Y f Y. Since for i=1,, one gets x i = I X 0 0 I Y 45

6 P. Réfrégier and J. Morio Vol. 3, No. 1/December 006/J. Opt. Soc. Am. A 3041 M x 1,x = f X 0 0 f Y, 46 D x 1,x = f X f Y. The situation is thus simple and clear in this case. The entropy is simply S P x1,x = S I,P + log 1 f X + log 1 f Y, 48 where S I,P = 4 log e I X + I Y + log 1 P. Let us now analyze a standard example of a Gaussian shape spectral density that leads to f U = exp exp i 0, 49 U where U=X,Y, i = 1, and where 0 is the central optical frequency. The coherence times can be chosen equal to U for the X or Y polarizations and are positive. In this case, the evolution of the entropy evolves accordingly to Eq. (48): S P x1,x = S I,P + log 1 exp + log 1 exp Y X. 50 We show in Fig. 1 the evolution of the entropy difference S P x1,x S I,P for that example as a function of = / X and of = Y / X. It is interesting to see that, depending on the values of, one can expect different shapes for the entropy increase as a function of. It is important to remark that the entropy diverges to when 0. As mentioned above, this is the consequence of using continuous entropy that diverges when a deterministic relation exists between the two electric fields. The divergence disappears when one takes into account noise sources such as photon noise or electronic noise. Let us now assume that the independent lights that are mixed do not have orthogonal linear polarizations. If they do not have parallel polarization one can write them as the action of a nonsingular Jones matrix J on the electric field E x considered above such as the actual electric field is A x =JE x. Since the intrinsic degrees of coherence are invariant by the multiplication of the electric field by the action of a deterministic Jones matrix, only the term S I,P of the total entropy can be modified when J varies. A modification of the entropy can appear also if there is no modification of the total intensity since one can expect in the general case to have a modification of the degree of polarization with J. However, the only modification in the entropy behavior will be the addition of a term that will not affect the general shape of evolution of the entropy with the time delay that allows one to analyze temporal coherence. B. Partially Spatially Coherence Introduced by a Moving Ground Glass Plate We propose to analyze in this subsection the evolution of the entropy when the spatial coherence is modified with a moving birefringent random phase screen. For that purpose, we consider an example analogous to the one considered by Shirai and Wolf. 5 Let us introduce the notation r,z,t where r is now a two-dimensional position vector in the plane perpendicular to the z axis. Let us thus assume that a phase screen with a transmission matrix T r has been put at location z=0, which action is represented by E T r,t = T r E I r,t, 51 where E I r,t =E I r,0,t and E T r,t =E T r,0,t are, respectively, the electric fields before and after the phase screen. Before the phase screen, the mutual coherence matrix is written as I r 1,r,z 1,z,t 1,t while after the phase screen it is written as T r 1,r,z 1,z,t 1,t. Since we are interested in spatial coherence in this subsection, we consider wide-sense stationary fields, and the mutual coherence matrices will be evaluated at times t 1 =t. In this case, l r 1,r,0,0,t,t, with l=i,t, are independent of time t and one will simply write l r 1,r = l r 1,r,0,0,t,t. After the phase screen in the z=0 plane, one gets the mutual coherence matrix T r 1,r = T r I r 1,r T r 1, 5 where the angle brackets represent statistical averages that can correspond to temporal averages in the case of a moving phase screen. The polarization matrix becomes T r = T r I r T r, 53 where l r = l r,0,t. The normalized mutual coherence matrix is simply M l r 1,r = l r 1/ l r 1,r l r 1 1/, 54 Fig. 1. Evolution of the part of the entropy due to partial temporal coherence (i.e. of the entropy difference S P x1,x S I,P as a function of = / X and of = Y / X =1 (solid curve) and for = Y / X =100 (dashed curve). where l=i (respectively, l=t) for the field before (respectively, after) the phase screen. One considers the general case of a birefringent random phase screen for which the transmission matrix can be written as

7 304 J. Opt. Soc. Am. A/ Vol. 3, No. 1/ December 006 P. Réfrégier and J. Morio Fig.. Evolution of the part of the entropy due to partial spatial coherence (i.e., of the entropy difference S P x1,x S I,P ) as a function of r 1 r / 1 for different values of 1 / and of 1 and.(a) 1 / =1; from top curve (solid curve) to bottom curve (dashed curve), 1 = =10, 1, 0.1. (b) 1 = =1; from top curve (solid curve) to bottom curve (dashed curve); 1 / =100, 1, Fig. 3. Analogous curves to Fig. but (a) 1 / 1 =1, 1 =1; and from top curve (solid curve) to bottom curve (dashed curve), =10, 1, 0.1. (b) 1 =1; and from top curve (solid curve) to bottom curve (dashed curve), =10, 1, 0.1 and 1 / =10, 1, 0.1. T r = exp i 1 r 0 0 exp i r, 55 where n r = n r + n r and where the phases n r are deterministic and n r are assumed random with a Gaussian distribution with zero mean, so that n r 1 n r = n exp r 1 r n, 56 1 r 1 r =0. It can be shown (see Ref. 5, for example) that exp i n r = exp n /, exp i n r 1 i n r = F n r 1 r, 59 where for the sake of simplicity we have introduced F n r 1 r = exp n exp r 1 r n Let us assume that the incident light is totally unpolarized with the following spatial mutual coherence matrix: I r 1,r = I 0 I, 61 exp r 1 + r 4 S where I is the two-dimensional identity matrix and where I 0 and S as are constant parameters. In this example, the input field is thus assumed temporally coherent and spatially coherent. A direct calculus 5 leads to with and where T r 1,r = I 0 exp r 1 + r 4 S L T r 1,r = T 0 r L r 1,r T 0 r 1, L T r 1,r, 6 63 L r 1,r = F 1 r 1 r 0 0 F r 1 r, 64 T 0 r = exp i 1 r 0 0 exp i r. 65 The two intrinsic degrees of coherence are thus F 1 r 1 r and F r 1 r. Since F 1 0 =1, the degree of polarization of the output light is still equal 5 to 0, and the phase fluctuations do not modify polarization fluctuations as would have been the case with polarized input light. The entropy is thus S P x1,x = S I,P + log 1 F 1 r 1 r + log 1 F r 1 r, 66 where one simply has S I,P =4 log e. One has F n 0 =1 and F n r 1 r exp n when r 1 r +. The evolution of the part of the entropy due to partial coherence is shown in Figs. and 3 as a function of r 1 r / 1 for different optical configurations. More precisely, we have drawn the entropy difference S P x1,x S I,P as a function of r 1 r / 1 for different values of 1 / and of 1 and. I 0

8 P. Réfrégier and J. Morio Vol. 3, No. 1/ December 006/ J. Opt. Soc. Am. A 3043 A pure phase screen without a birefringent effect corresponds to 1 = and to 1 = and the behavior of the entropy is simpler in that case. The experiment for which the phase mask is fixed is described by considering n =0 for n=1,. In this case the transmission matrix is a deterministic matrix, and it has been shown in Ref. 6 that the action of linear deterministic transformations does not modify the intrinsic degrees of coherence. The intrinsic degrees of coherence are thus the same before and after the phase mask in the plane perpendicular to the z axis and located at z=0. This is also the case of the degrees of polarization at different locations r. The entropy of the light is thus not increased by the action of the fixed birefringent phase mask, which shows that the deterministic birefringent phase mask does not introduce randomness. It is interesting to observe that after the phase screen and in the case of completely unpolarized light, the degree of coherence defined in Ref. is equal to 1 exp i 1 r i 1 r 1 +exp i r i r 1 and thus has a modulus that can vary between 0 and 1. The physical meaning of intrinsic degrees of coherence equal to one is simple to get. Indeed, it has been shown 6,7 that the largest intrinsic degree of coherence is the maximal value of the modulus of the fringe visibility one can observe in interference experiments. This largest value is obtained using a linear polarizer aligned along the horizontal or the vertical directions. After such a polarizer, the degree of coherence defined in Ref. is equal to exp i 1 r i 1 r 1 for a horizontal alignment, or to exp i r i r 1 for a vertical alignment and is thus in both cases of unit modulus. Obviously, with a moving random phase screen, using a polarizer does not allow one to recover fringe visibility equal to one in interference experiments, which indeed corresponds to intrinsic degrees of coherence smaller than one. These different experiments explain the different values of the intrinsic degrees of coherence when the phase mask is moving or not. The increase of entropy is thus clearly due to averages over different possible random phase screens. In other words, using a moving random phase screen introduces randomness that is characterized by an increase of entropy. A corollary of that property is the decrease of the intrinsic degrees of coherence. 5. CONCLUSION AND PERSPECTIVES In this paper we have analyzed the Shannon entropy of partially polarized and partially coherent light. Since the Shannon entropy is a standard measure of disorder, it is a useful quantity to characterize the randomness between the electric fields at two points in space and at two different times, which is, as mentioned by Glauber, 9 a characteristic of partial coherence. It has been demonstrated that the Shannon entropy is the sum of simple functions of the intensity, of the degrees of polarization, and of the intrinsic degree of coherence introduced in Refs. 6 and 8. We have analyzed this result on simple but illustrative examples. It clearly appears that the intrinsic degrees of coherence are simply related to the Shannon entropy that measures the degree of randomness. There exist different perspectives to this work. We have considered the case of two-dimensional electric fields; however, it will be interesting to generalize this approach to three-dimensional electric fields. 1,3,4 The analysis of the relation between intrinsic degrees of coherence and the condition of complete electromagnetic coherence introduced in Ref. 9 is a fundamental subject that requires further investigation. APPENDIX A Our purpose is to prove Eq. (33). With Eq. (3) one has x 1,x = T 1/ x 1,x x 1,x T 1/ x 1,x. Equation (8) is and Eq. (31) is One has and thus A1 x 1,x = x 1 x 1,x, A x 1,x x T x 1,x = x x. A3 1/ T x 1,x = 1/ x / A4 x, x 1,x = 1/ x 1 x 1 1/ x 1 1/ x 1 x 1,x 1/ x 1/ x x 1,x 1/ x 1 1/ x x 1/, A5 x which proves Eq. (33). APPENDIX B Using Eq. (39) one gets M x 1,x M x 1,x = N x 1,x D x 1,x N x 1,x. B1 other words, S x 1,x and I x 1,x are solutions of the equation det M x 1,x M x 1,x I d =0. B This result means that S x 1,x and I x 1,x are the square roots of the eigenvalues of M x 1,x M x 1,x.In In particular, one has

9 3044 J. Opt. Soc. Am. A/ Vol. 3, No. 1/ December 006 P. Réfrégier and J. Morio det I d M x 1,x M x 1,x which clearly shows that = 1 S x 1,x 1 I x 1,x, B3 log det x 1,x = log 1 S x 1,x + log 1 I x 1,x, B4 which ends the proof. ACKNOWLEDGMENTS The authors thank the members of the Phyti team, François Goudail, and Christian Brosseau for fruitful discussions and region Provence-Alpes-Côte d Azur for partial financial support. The corresponding author is Philippe Réfrégier: phone, (33) ; fax, (33) ; , philippe.refregier@fresnel.fr. REFERENCES 1. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, Coherent-mode decomposition of partially polarized, partially coherent sources, J. Opt. Soc. Am. A 0, (003).. E. Wolf, Unified theory of coherence and polarization of random electromagnetic beams, Phys. Lett. A 31, (003). 3. J. Tervo, T. Setälä, and A. T. Friberg, Degree of coherence of electromagnetic fields, Opt. Express 11, (003). 4. J. Tervo, T. Setälä, and A. T. Friberg, Theory of partially coherent electromagnetic fields in the space frequency domain, J. Opt. Soc. Am. A 1, (004). 5. T. Shirai and E. Wolf, Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space, J. Opt. Soc. Am. A 1, (004). 6. Ph. Réfrégier and F. Goudail, Invariant degrees of coherence of partially polarized light, Opt. Express 13, (005). 7. Ph. Réfrégier and A. Roueff, Coherence polarization filtering and relation with intrinsic degrees of coherence, Opt. Lett. 31, (006). 8. Ph. Réfrégier, Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations, Opt. Lett. 30, (005). 9. R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, (1963). 10. J. W. Goodman, Some first-order properties of light waves, in Statistical Optics (Wiley, 1985), pp L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995). 1. Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 004). 13. T. Setälä, J. Tervo, and A. T. Friberg, Complete electromagnetic coherence in the space frequency domain, Opt. Lett. 9, (004). 14. P. Vamihaa and J. Tervo, Unified measures for optical fields: degree of polarization and effective degree of coherence, Pure Appl. Opt. 6, (004). 15. C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 7, , (1948). 16. T. M. Cover and J. A. Thomas, The asymptotic equipartition theory, in Elements of Information Theory (Wiley, 1991), pp R. Barakat and C. Brosseau, Von Neumann entropy of n interacting pencils of radiation, J. Opt. Soc. Am. A 10, (1993). 18. R. Barakat, Polarization entropy transfer and relative polarization entropy, Opt. Commun. 13, (1996). 19. R. Barakat, Some entropic aspects of optical imagery, Opt. Commun. 156, (1998). 0. C. Brosseau, Entropy of the radiation field, in Fundamentals of Polarized Light A Statistical Approach (Wiley, 1998), pp Ph. Réfrégier, F. Goudail, P. Chavel, and A. Friberg, Entropy of partially polarized light and application to statistical processing techniques, J. Opt. Soc. Am. A 1, (004).. J. W. Goodman, Coherence of optical waves, in Statistical Optics (Wiley, 1985) pp T. Setälä, M. Kaivola, and A. T. Friberg, Degree of polarization in near fields of thermal sources: effects of surface waves, Phys. Rev. Lett. 88, 1390 (00). 4. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, Degree of polarization of statistically stationary electromagnetic fields, Opt. Commun. 48, (005).