1. INTRODUCTION

Physical image formation has a long and rich history due to its theoretical importance and practical interest [1–5]. It has been recognized that the polarization and coherence properties of light have significant influence on the image formation [5,6]. With the recent progress in device technology for high-resolution polarization cameras and liquid-crystal-based spatial polarization modulators, polarization imaging has attracted increasing interest for a wide range of applications such as biomedical diagnosis, remote sensing, and mineralogy [7–10].

In early studies on polarization imaging, the main efforts have been concentrated on the direct extension of the scalar imaging theory by taking into account the vector nature of the optical fields. McGuire and Chipman [9,11] have extended the concepts of the point spread function and optical transfer function (OTF) in scalar optics to corresponding matrix versions for coherent and incoherent polarization imaging. The nature of these early studies may be considered as a direct extension of conventional imaging because the image of interest was the distribution of the field (or intensity) at each single point on the image plane.

The effects of partial coherence and partial polarization in an imaging system were investigated and reviewed by Korotkova in her recent book [12], with a new focus being placed on the joint behavior of a pair of statistical electric fields at different locations. Her book laid a helpful basis for statistical polarization imaging, for example, by introducing the concept of the transmission cross-coefficient matrix (TCCM) as a natural extension of the transmission cross-coefficients (TCC) in scalar imaging theory. However, while an important theoretical basis was provided, no specific examples were given for practical optical engineers to analyze (and/or design) their real-world polarization imaging systems, such as a multistage imaging system in which polarization-dependent imaging optics (e.g., an objective lens with birefringence) and partially polarized/partially coherent illumination optics (e.g., a condenser lens plus an extended light source) are joined in tandem. On the other hand, Goodman, in his book [4], has given full discussion of these issues including practical multistage optical systems and specific test objects for image evaluation, but the analysis is based on scalar coherence theory.

With these observations in mind, the purpose of this paper is to fill some of the gaps left between the frameworks of these two renowned books [4,12] by extending and supplementing the theories in the relevant parts of the two books. To this end, we first develop simplified mathematical expressions (based on matrix-integral transforms and hypermatrices) for systematic analysis of polarization imaging systems with partially polarized and partially coherent illuminations. Next, using the simplified mathematical expressions, we present an analysis of Stokes imaging for a practical multistage imaging system composed of polarization-dependent imaging optics and partially polarized/partially coherent illumination optics, in which all the information about the image, object, and intermediate optical fields is represented by (generalized) Stokes vectors (rather than by a cross-spectral density matrix as in [12]), which would be preferred in practical applications. Finally, a generalized concept of the apparent transfer matrix is introduced to deal with the nonlinearity inherent in the polarization imaging system under partially coherent illumination.

In our attempt to fill the gaps between the two books, we will construct a bridge from the side of Goodman’s book [4] because most readers will be familiar with the scalar imaging theory well developed in this classic book. In order to facilitate reading by comparison with the scalar imaging theory, we will take a page out of Goodman’s book [4], follow its original descriptions and structure of logic, and reformulate them into a new matrix-based version in order to make it applicable to practical polarization imaging systems with partially polarized and partially coherent light.

For ease of reference, the primary vectors and matrices to be used for analysis of the polarization imaging in this paper are listed in Table 1.

Table 1. Vectors and Matrices Used in Polarization Imaging

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2. PRELIMINARIES

Before entering the main subjects of polarization imaging with partially polarized and partially coherent light, we first briefly review some of the matrix representations of stochastic electric fields propagating through linear optical systems.

A. Passage of Partially Polarized and Partially Coherent Light Through a Thin Transmitting Structure of Birefringent Media

For use in later analysis, we first consider the passage of light with arbitrary polarization and coherence through a thin transmitting structure of birefringence as shown in Fig. 1. The thin birefringent object is assumed to be imbedded in a uniform non-absorbing media, with the object itself taken to have variable thickness, two variable real indices of refraction, a variable orientation of fast/slow axis for birefringent media, and a variable component of absorption that accounts for the reduced intensity of the transmitted light.

 

Fig. 1. Passage of partially polarized and partially coherent light through a thin transmitting structure of birefringent media.

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When light passes through such a polarization-sensitive structure, the states of polarization of the electric field will change in general, and it is convenient to use Jones calculus to describe the relationship between the incident and transmitted electric fields [13], ${\textbf{E}^{\text{Obj}}}(\textbf{r},t) = \textbf{t}(\textbf{r}){\textbf{E}^{\text{Illm}}}(\textbf{r},t - \Delta t(\textbf{r}))$, where ${\textbf{E}^{\text{Illm}}}$ and ${\textbf{E}^{\text{Obj}}}$ are the analytic signal representations of the instantaneous electric field vectors under the narrowband assumption of the light; the superscripts “Illm” and “Obj” indicate the illumination and object light, respectively, $\Delta t$ is the time delay introduced into the electric field at location $\textbf{r}$, and $\textbf{t}$ is the amplitude transmittance matrix of the thin birefringent object, which is a Jones matrix of size 2 with its four elements fully describing the modulation effects of such a thin birefringent structure on the states of polarization of the wave. To find the effect of the thin birefringent object on the transmitted light with arbitrary polarization and coherence, we choose the generalized Stokes parameters [6,12,14] as a useful basis for representation of the joint polarization state of electric fields at two space–time points. In formal analogy to the column Stokes vector introduced by O’Neill [1], we define a column vector for the generalized Stokes parameters by the formula

It follows from Eqs. (1) and (2) that four components of the generalized Stokes vector representing the cross-correlation of the light at two spatio-temporal points can be expressed explicitly in terms of the components of the electric fields: ${S_0}({\textbf{r}_1},{\textbf{r}_2};\tau) = \langle {E_x}({\textbf{r}_1},t + \tau)E_x^*({\textbf{r}_2},t) \rangle + \langle {E_y}({\textbf{r}_1},t + \tau)E_y^*\def\LDeqbreak{}({\textbf{r}_2},t) \rangle $, ${S_1}({\textbf{r}_1},{\textbf{r}_2};\tau) = \langle {E_x}({\textbf{r}_1},t + \tau)E_x^*({\textbf{r}_2},t) \rangle - \langle {E_y}({\textbf{r}_1},t\def\LDeqbreak{} + \tau)E_y^*({\textbf{r}_2},t) \rangle $, ${S_2}({\textbf{r}_1},{\textbf{r}_2};\tau) = \langle {E_x}({\textbf{r}_1},t + \tau)E_y^*({\textbf{r}_2},t) \rangle + \langle {E_y}({\textbf{r}_1},t + \tau)E_x^*({\textbf{r}_2},t) \rangle $, and ${S_3}({\textbf{r}_1},{\textbf{r}_2};\tau) = i[ \langle {E_y}({\textbf{r}_1},t + \tau)E_x^*({\textbf{r}_2},t) \rangle - \langle {E_x}({\textbf{r}_1},t + \tau)E_y^*({\textbf{r}_2},t) \rangle ]$.

Note that if we choose an identity matrix $\textbf{A} = \textbf{I}$, we have ${\langle}\textbf{E}({\textbf{r}_1},t + \tau) \otimes {\textbf{E}^*}({\textbf{r}_2},t) \rangle $, which is a vector representation of the mutual coherence matrix [6]. Next, under the condition that $| {\Delta t({\textbf{r}_1}) - \Delta t({\textbf{r}_2})} | \ll {1 / {\Delta v}} \approx {\tau _C}$ for all ${\textbf{r}_1}$ and ${\textbf{r}_2}$ (where $\Delta \nu$ is the bandwidth of a narrow band light with its coherence time ${\tau _C}$), the time average for the generalized Stokes parameters will be independent of time delay $\Delta t(\textbf{r})$. The advantage of the new expression in Eq. (1) is gained by making a suitable transformation in the four-dimensional space so that the generalized Stokes vector of the object light becomes

When Eq. (3) is derived, we have made use of the mixed-product property of Kronecker product, $(\textbf{A} \otimes \textbf{B})(\textbf{C} \otimes \textbf{D}) = (\textbf{AC}) \otimes (\textbf{BD})$ [15]. The matrix $\textbf{T}({\textbf{r}_1},{\textbf{r}_2})$ may be referred to as the mutual transmittance matrix due to the fact that all the elements of the matrix in Eq. (4) are functions of two positions ${\textbf{r}_1}$ and ${\textbf{r}_2}$. When ${\textbf{r}_1}$ and ${\textbf{r}_2}$ merge at a single point $\textbf{r}$, the transmittance matrix in Eq. (4) will degenerate to a Mueller matrix $^{\text{Mu}}\textbf{M}$. Under the condition of quasi-monochromatic illumination, Eq. (3) reduces to ${\textbf{S}^{\text{Obj}}}({\textbf{r}_1},{\textbf{r}_2}) = \textbf{T}({\textbf{r}_1},{\textbf{r}_2}){\textbf{S}^{\text{Illm}}}({\textbf{r}_1},{\textbf{r}_2})$, which agrees with the result given by Korotkova and Wolf [16].

B. Linear System for Polarization Optics

In modern optics, the linear system theory has played a fundamental role. The goal of this section is to specify the relationship between two sets of the generalized Stokes vectors at the input and output planes of a general linear system for polarization optics. Let us recall that, in the second-order coherence theory for the stochastic scalar fields, Hopkins formula provides a simple approach for calculation of the mutual coherence function at the output of a general linear optical system, and thereby serves as a building block for analyzing a more complex optical system [17]. Note also that the generalized Stokes vector for two different points in the space–time domain obeys a set of linear wave equations [18] just as the scalar coherence function obeys a pair of linear wave equations [4–6]. Therefore, we can introduce a concept of a generalized Stokes vector wave that behaves just like an optical field in the same manner as does the conventional scalar coherence function. Once we reach this understanding, it is no surprise to see that the generalized Stokes vector wave for two different points in the space–time domain obeys various physical laws, such as those of the linear system theory, which are already familiar to us with regard to scalar coherence theory. Here, we will extend and develop the Hopkins formula by taking the light polarization into account for the calculation of the generalized Stokes vector propagating through a linear system of polarization optics.

For the light (with an arbitrary state of polarization and coherence) incident on a linear optical system, we can express the electric field vector ${\textbf{E}^{\text{Out}}}$ at point $Q$ on the output plane in terms of the electric field vector ${\textbf{E}^{\text{In}}}$, leaving an elementary area (${d^2}P$) at point $P$ on the input plane: ${\textbf{E}^{\text{Out}}}(Q;P;t) = \textbf{h}(Q,P){\textbf{E}^{\text{In}}}(P;t - \Delta t){d^2}P$, where $\Delta t$ is the time delay incurred by the propagation of light from $P$ to $Q$, and the square matrix $\textbf{h}(Q,P)$ of size 2 may be referred to as the amplitude point-spread matrix. After substitution of ${\textbf{E}^{\text{out}}}$ into the definition for the generalized Stokes vector in Eq. (1), we have

When Eq. (5) is derived, we have made use of the quasi-monochromatic condition where the propagation delay from ${P_n}$ to ${Q_n}$ for $(n = 1,2)$ is much less than the coherence time, so that the influence of the time delay $\Delta {t_n}$ has been taken care of as a phase $\bar \omega \Delta {t_n}$ added to $\textbf{h}({Q_n},{P_n})$, with $\bar \omega$ being the central optical frequency. To find the generalized Stokes vector ${\textbf{S}^{\text{Out}}}({Q_1},{Q_2})$ averaged over all points $P$ at the input, we must integrate ${\textbf{S}^{\text{Out}}}({Q_1},{Q_2};{P_1},{P_2})$ over the input coordinates corresponding to ${P_1}$ and ${P_2}$,

As an additional special case of interest, consider the form of the general result when the light at the input plane is incoherent. In this case, the generalized Stokes vector of input light is given by ${\textbf{S}^{\text{In}}}({P_1},{P_2}) = \kappa \delta (| {{P_1} - {P_2}} |){\textbf{S}^{\text{In}}}({P_1})$, where $\delta (| {{P_1} - {P_2}} |)$ is a 2-D Dirac delta function and $\kappa$ has the dimensions of squared length. The generalized Stokes vector at the output plane becomes

All these relations, i.e., Eqs. (7)–(10), are shown diagrammatically in Fig. 2. They may also be regarded as a kind of polarization-coherence reciprocity relationship: if time is reversed, the distributions of the (generalized) Stokes vector across the scattering spot become the matrix-based integral transform of the (generalized) Stokes vector in the observation region.

 

Fig. 2. Matrix-based integral transform relations between the distributions of the (generalized) Stokes vectors on the input and output planes.

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3. SPACE-DOMAIN CALCULATION OF THE STOKES IMAGES FOR A POLARIZATION IMAGING SYSTEM

A. Relations of the Generalized Stokes Vectors in Space Domain

Consider the generic two-stage optical system shown in Fig. 3, which is a simple yet practically important example of a multistage optical system. The source with arbitrary polarization and coherence is assumed to be described by the generalized Stokes vector ${\textbf{S}^{\text{Sou}}}({P_1},{P_2})$ with its superscript “Sou” indicating the light source plane. The illumination optics is assumed to be described by a $4 \times 4$ impulse mutual response matrix $\textbf{F}({Q_1},{P_1};{Q_2},{P_2})$ given by Eq. (6), which in general is space variant. The thin object of birefringent media is described by its mutual transmittance matrix $\textbf{T}({Q_1},{Q_2})$. The polarization imaging optics is described by the (generally space-variant) $4 \times 4$ impulse response matrix $\textbf{H}({R_1},{Q_1};{R_2},{Q_2})$, where $R$ is an arbitrary point in the image plane. Note that all the elements in both $\textbf{F}$ and $\textbf{H}$ matrices have dimensions inverse length squared.

 

Fig. 3. Generic two-stage optical system for polarization optics.

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Consider first the mapping for the light leaving the source to that incident on the object. Applying Eq. (7), we have the generalized Stokes parameters ${\textbf{S}^{\text{Illm}}}({Q_1},{Q_2})$ of the light illuminating the thin object of birefringent medium:

After the light passes through the object of birefringent structure of $\textbf{T}({Q_1},{Q_2})$ in Eq. (4), we apply the input–output relation, Eq. (7), for a second time, this time to the region labeled “Polarization Imaging Optics” in Fig. 3, which is assumed to be characterized by an impulse mutual response matrix $\textbf{H}$. The expected relation is

This result with eight integrations looks formidable and is in need of simplification before it becomes useful. Now, we will simplify Eq. (13) step by step under several assumptions as follows.

(I) A first simplification arises if we assume that the source for illumination is fully incoherent, i.e., ${\textbf{S}^{\text{Sou}}}({P_1},{P_2}) = \kappa \delta (| {{P_1} - {P_2}} |){\textbf{S}^{\text{Sou}}}({P_1})$, with $\delta$ being a 2-D Dirac delta function. After using the shifting property of the delta function, we can simplify Eq. (13) as

(II) A small further simplification can be performed if we are interested in the Stokes images rather than the generalized Stokes vector at the image plane. We obtain

Note that the expression in the bracket in Eq. (15) simply represents the generalized Stokes vector ${\textbf{S}^{\text{Illm}}}({Q_1},{Q_2})$ of the light illuminating the object, for which the source is assumed to be spatially incoherent. Thus, we can rewrite Eq. (15) as

Note also that Eq. (16) can be obtained directly from Eq. (12) by putting ${R_1} = {R_2} = R$, without assuming incoherent source.

(III) Let us use a coordinate system $(u,v)$ in the image space and a reduced coordinate system $(\xi ,\eta)$ in the object space, where magnification and image inversion have been removed. If we assume the shift-invariance of the generalized Stokes vector ${\textbf{S}^{\text{Illm}}}$ incident on the object depends only on the difference of object coordinate $(\Delta \xi ,\Delta \eta)$, we can obtain the expressions for the Stokes images as

From the derivations in Eqs. (11)–(17), we can see that the proposed linear model for the Stokes imaging has been developed because the generalized Stokes vector obeys the vector wave equation [18]. Recently, another linear model of free propagation of polarized wave in inhomogeneous media has been reported where the evolution of the generalized Stokes parameters in the paraxial regime was studied [20].

B. Polarization-sensitive Illumination Optics

Based on the preceding discussion, we have realized that the polarization and coherence properties of light have a profound influence on the character of the recorded polarization images. Therefore, we should consider the generalized Stokes vector for the light incident on the object.

As illustrated in Fig. 4, an extended incoherent source is placed at an arbitrary distance ${z_1}$ in front of a positive condenser lens followed by a polarization-sensitive device for polarization selection and/or modulation, and the object lies at an arbitrary distance ${z_2}$ behind the lens. The source is assumed to subtend a sufficiently large angular subtense at the lens so that, as determined by the van Cittert–Zernike theorem, the coherence area of the light incident on the lens is much smaller than the area of the lens itself. As originally proposed by Zernike [19] and explained in detail by Goodman [4], the lens aperture itself may be regarded as an incoherent source, so that it is now expressed by a Stokes vector ${\textbf{S}^{\text{Con}}}(\tilde x,\tilde y) = {{\kappa ^\prime |{P^{\text{Con}}}(\tilde x,\tilde y){|^2}{{\boldsymbol{\cal S}}^{\text{Sou}}}(0,0)} / {{{(\bar \lambda {z_1})}^2}}}$, where the superscript “Con” indicates the condenser lens, $\kappa ^\prime $ has dimensions meters squared, ${P^{\text{Con}}}$ is the pupil function of the condenser lens, ${{\boldsymbol{\cal S}}^{\text{Sou}}}(0,0)$ is the zero spatial frequency component of the spectral Stokes vector of source, and $\bar \lambda$ is the nominal center wavelength of light. The transmitted Stokes vector $^{\prime}{\textbf{S}^{\text{Con}}}(\tilde x,\tilde y)$ just behind the polarization-sensitive device can be obtained from the Mueller calculus, that is $^{\prime}{\textbf{S}^{\text{Con}}}(\tilde x,\tilde y) \;=\; ^{\text{Mu}}{\textbf{M}^{\text{Con}}}\,{\textbf{S}^{\text{Con}}}(\tilde x,\tilde y)$, with $^{\text{Mu}}{\textbf{M}^{\text{Con}}}$ being the Mueller matrix for the polarization-sensitive element used in the condenser optics. Then, the generalized Stokes parameters illuminating the object may be calculated rather simply by applying the van Cittert–Zernike theorem again to $^{\prime}{\textbf{S}^{\text{Con}}}(\tilde x,\tilde y)$ as an incoherent polarization-selective source:

 

Fig. 4. Polarization-sensitive illumination system.

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Therefore, the calculation of the generalized Stokes vector incident on the object has been reduced to the problem of Fourier transform of the Stokes vector just behind the polarization-sensitive condenser lens. As an additional case of interest, consider a birefringence-free illumination system without any polarization-sensitive devices. In this case, the Mueller matrix is a unitary matrix $^{\text{Mu}}\textbf{M} = {\textbf{I}_4}$ with ${\textbf{I}_4}$ being the identity matrix of size 4, and therefore we have the constant Stokes vector ${{\boldsymbol{\cal S}}^{\text{Sou}}}(0,0)$ determined by the spatially averaged state of polarization of the light source itself.

In the illumination optics system, two most common forms of condenser systems are critical illumination and Köhler’s illumination. In critical illumination, the incoherent source is imaged by the condenser system with its polarization state selected by the polarization-sensitive device onto the object. In this case, the distances ${z_1}$ and ${z_2}$ in Fig. 4 should be chosen to satisfy the lens law $z_1^{- 1} + z_2^{- 1} = f_c^{- 1}$, with ${f_c}$ being the focal length of the condenser lens. The chief disadvantage of critical illumination is the fact that spatial variations of the Stokes parameters of the incoherent source are imaged on the object, leaving a nonuniform illumination pattern and nonuniform polarization states superimposed on the structure of the object. Therefore, the structures of the Stokes parameters from the source will be confused with the transmitting structure of the object. An alternative condenser geometry is the Köhler’s illumination, where both ${z_1}$ and ${z_2}$ in Fig. 4 are chosen to equal the focal length ${f_c}$ of the condenser lens. Thus, the incoherent source is in the front focal plane of the condenser lens and the object is in the rear focal plane. Because each point on the source is mapped into a plane-wave incident at a unique angle at the object, any spatial variations of the state of polarization of the source will not affect the uniformity of the illumination. In either case of the critical or Köhler illumination, if the size of the coherence area on the condenser lens, created by the extended incoherent source, is much smaller than the aperture size of the condenser lens, the approximation theory by Zernike [19] holds so that we can obtain the generalized and conventional Stokes images for the generic two-stage optical systems by substituting the result of Eq. (18) into Eqs. (12) and (16), respectively, with $({\xi _1},{\eta _1}) \Rightarrow {Q_1}$ and $({\xi _2},{\eta _2}) \Rightarrow {Q_2}$.

4. FREQUENCY-DOMAIN CALCULATION OF THE STOKES IMAGES’ SPECTRA

In the long and rich history of imaging optics, spatial frequency analysis has been so successfully applied that it now occupies a fundamental place in the theory of imaging systems [21–24]. In this section, we will develop a theory to understand how a polarization imaging system handles different spatial frequencies in the optical transmission chain for partially polarized and partially coherent light.

A. Relations of the Generalized Stokes Parameters in the Frequency Domain

We begin with Eq. (12), which relates the generalized Stokes vector in the image plane to that transmitted by the object. If we assume a coordinate system $(u,v)$ in the image space and a reduced coordinate system $(\xi ,\eta)$ in the object space, Eq. (12) can be rewritten as

This equation is in fact a four-dimensional superposition integral describing a four-dimensional linear system. The $4 \times 4$ matrix $\textbf{H}$ is the four-dimensional impulse mutual response of this polarization imaging system, that is, the generalized Stokes vector observed at the image coordinate pair $({u_1},{v_1})$ and $({u_2},{v_2})$ in response to the generalized Stokes vector transmitted from the object coordinate pair $({\xi _1},{\eta _1})$ and $({\xi _2},{\eta _2})$. In well-corrected imaging optics (or, at least, within an isoplanatic patch), the system is space invariant (or isoplanatic) so that the generic superposition integral above can be reduced to a simpler matrix convolution integral of the form

In a similar manner, we define the four-dimensional transfer matrix of the space-invariant linear system by

With a change of variables ${\xi _2} = {\xi _1} + \Delta \xi$ and ${\eta _2} = {\eta _1} + \Delta \eta$, this transform can be written as

The second double integral is recognized as the Fourier transform of a product of the matrix $\textbf{T}$ and the vector ${\textbf{S}^{\text{Illm}}}$, and as such can be evaluated as the matrix convolution of their individual Fourier transforms. With the help of shift property of Fourier transform, we find that the second double integral can be expressed as

After a combination of Eqs. (23) and (27), we obtain the overall polarization-imaging relationship in the frequency domain

This relation provides a path to understanding the frequency spectra of the generalized Stokes images obtained with a polarization imaging system illuminated by a partially polarized and partially coherent light, as we will see in the next section.

B. Transmission Cross-coefficient Matrix

When a polarization imaging system is operating in the partially polarized and partially coherent regime, the Stokes images ${\textbf{S}^{\text{Img}}}(u,v)$ can be obtained from the generalized Stokes vector ${\textbf{S}^{\text{Img}}}({u_1},{v_1};{u_2},{v_2})$ in the image plane by allowing the points $({u_1},{v_1})$ and $({u_2},{v_2})$ to merge at a common point $(u,v)$. Since the generalized Stokes vector in the image plane, ${\textbf{S}^{\text{Img}}}({u_1},{v_1};{u_2},{v_2})$, can be obtained from the inverse Fourier transform of ${{\boldsymbol{\cal S}}^{\text{Img}}}({f_1},{f_2},{f_3},{f_4})$, we can merge these two points at a single point such that $(u = {u_1} = {u_2},v = {v_1} = {v_2})$, and take the two-dimensional Fourier transform of the Stokes images to obtain

Because the two-dimensional Fourier transform in Eq. (29) yields a delta function, the sifting property of this delta function gives

Finally, we substitute Eq. (28) into the previous equation, yielding

With the change of variables ${f_1} = s - p$ and ${f_2} = t - q$, the relation in Eq. (31) becomes

With the aid of matrix index notation, we can rewrite Eq. (32) as

For comparison purpose, the conventional transmission cross-coefficient for the scalar imaging system [4] is given by $\text{TCC}(s,t;{f_u},{f_v}) = \iint_{- \infty}^\infty {{\mathfrak{h}}}(s - p,t - q){{{\mathfrak{h}}}^*}\def\LDeqbreak{}(s - p - {f_u},t - q - {f_v}) {{\cal J}^{\text{Illm}}}(p,q)\text{d}p\text{d}q$ with ${{\cal J}^{\text{Illm}}}$ being the two-dimensional Fourier transform of mutual intensity for illumination light.

The transmission cross-coefficient matrix depends on both the polarization-dependent illumination and polarization imaging systems, but not on the object itself. From Eqs. (35) and (36), it is seen that the spatial frequency spectra of the Stokes images are a sum of contributions from all the modified spatial frequency spectra of the object, where the modifications represent a combined effects of the polarization and coherence of the illumination, plus the diffraction and aberrations of the polarization-dependent imaging system. Equation (35) is one of the main results in this paper which provides the frequency-domain calculation of the Stokes image spectra.

To apply the hypermatrix TCCM to a practical optical system, we should first specify the amplitude transfer matrix ${\boldsymbol {\mathfrak{h}}}$ for a polarization imaging system. Here, we take a system-theoretical approach based on a view that all imaging elements and polarization-sensitive devices may be lumped into a single “black box” representing the integrated system, and that the polarization imaging properties of the system can be completely described by specifying the relation between the optical fields at an appropriately chosen set of input and output terminals. As shown in Fig. 5, the input and output “terminals” of this black box are chosen as the planes of the entrance and exit pupils, respectively. As with the conventional scalar imaging theory [26], the limited spatial extent of the wavefront gives rise to the diffraction of the electric field, which takes place when the converging light from the exit pupil propagates over the distance ${z^{\text{Img}}}$ to the image plane. As a consequence, a certain portion of the diffracted electric field generated by a complicated polarization-sensitive object is intercepted by this finite pupil to form the Stokes images. Specifically, the high-frequency components of the electric field vector from the object amplitude transmittance will be intercepted by the pupil and make no contribution to polarization imaging.

 

Fig. 5. Generalized model of a polarization imaging system.

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In analogy to the amplitude spread function in scalar imaging theory [26], the sought-after amplitude point-spread matrix $\textbf{h}$ for the polarization imaging system is equal to a scaled inverse Fourier transform of the generalized pupil matrix $\textbf{P}$ of the exit pupil of the polarization imaging system [9,11]

The amplitude transfer matrix ${\boldsymbol {\mathfrak{h}}}$ is itself equal to the generalized pupil matrix $\textbf{P}$, which is a product of the scalar pupil function (aperture function) $P$ of the conventional scalar imaging system and the Jones matrix $\textbf{J}$ representing the polarizing modulation (or polarization aberration) introduced by the imaging optics into the electric field at the exit pupil plane. The relation in Eq. (38) is of importance since it gives the basic information about the complex-amplitude behavior of diffraction-limited polarization imaging systems in the spatial frequency domain.

With Eqs. (33), (36), and (38) in hand, we are ready to give physical interpretation of the newly introduced TCCM. Note that the Jones matrices are operators that act on the electric field vector, and these matrices are associated with (or physically implemented by) the corresponding polarization-sensitive elements. Since each element of ${\boldsymbol {\mathfrak{h}}}$ is set to zero by the pupil function $P$ when the point lies outside the area of the exit pupil, the amplitude transfer matrix for the polarization imaging system can be specified by the polarization element that modulates the polarization states on the exit pupil aperture. Similarly to the Zernike approximation in the scalar case [4,19], when illuminated by a sufficiently large incoherent source, the illumination on the polarization-sensitive object can be considered to be that made by an extended spatially incoherent source placed on the exit pupil of the condenser lens. In this case, the van Cittert–Zernike theorem in Eq. (18) implies that ${{\boldsymbol{\cal S}}^{\text{Illm}}}(p,q) = \kappa {| {{P^{\text{Con}}}(- \bar \lambda {z^{\text{Con}}}p, - \bar \lambda {z^{\text{Con}}}q)} |^2}^{\text{Mu}}{\textbf{M}^{\text{Illm}}}\,{{\boldsymbol{\cal S}}^{\text{Sou}}}(0,0)$, where ${z^{\text{Con}}}$ is the distance from the exit pupil of the condenser lens to the object, $\kappa$ is a constant with dimensions in meters squared, $^{\text{Mu}}{\textbf{M}^{\text{Illm}}}$ is the Mueller matrix of the illumination optics, and ${{\boldsymbol{\cal S}}^{\text{Sou}}}(0,0)$ is a constant DC Stokes vector obtained by integrating (i.e., averaging) the Stokes vector over the incoherent source. Ignoring multiplicative constants and assuming a circular exit pupil of radius ${r^{\text{Illm}}}$ for the polarization-selective illumination optics and a circular exit pupil of radius ${r^{\text{Img}}}$ for the polarization imaging system, the calculation of each element in TCCM can be performed by finding the overlapping areas of the one circle embedded with polarization information from the illumination system and the two circular plates with the polarization modulation/selection associated with the imaging system shown in Fig. 6 as functions of $(s,t)$ and $({f_u},{f_v})$.

 

Fig. 6. Overlapping areas of one circle (corresponding to the pupil aperture of the condenser lens in which the polarization information of the illumination system is embedded) and two circular plates (each corresponding to the pupil aperture of the polarization imaging system) contribute to the frequency-domain calculation of all the elements in the transmission cross-coefficient matrix.

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Therefore, the proposed TCCM can be understood as the generalization of the conventional transmission cross-coefficient for the scalar imaging system in order to deal with the stochastic vectorial optical fields by taking into account the modulation of the polarization states [4,5]. It is important to note that the calculation of the Stokes images in the partially polarized and partially coherent case is not a trivial task since we must find all the overlapping areas (i.e., TCCM) for all arguments in frequency domain that affect the final outcome.

5. INCOHERENT AND COHERENT LIMITS FOR POLARIZATION IMAGING

In this section, we explore the properties of the Stokes images in the limiting cases of perfectly incoherent and perfectly coherent object illuminations, which can be predicted by the theorems presented in previous sections. For the sake of mathematical simplicity, the polarization vector ${\textbf{E}^{\text{Illm}}}$ and the corresponding Stokes vector ${\textbf{S}^{\text{Illm}}}$ of illumination are assumed, without loss of generality, to be uniform constant vectors independent of the space coordinates.

A. Incoherent Case

The case of perfect incoherence of the object, illumination can be represented by the generalized Stokes parameters of the object illumination of the form ${\textbf{S}^{\text{Illm}}}(\Delta \xi ,\Delta \eta) = \kappa \delta (\Delta \xi ,\Delta \eta){\textbf{S}^{\text{Illm}}}$, where $\kappa$ is again a constant with dimensions meters squared. Substituting into Eq. (17) and making use of an assumption of a space-invariant system, we have

When Eq. (39) is derived, we have made use of the fact that the mutual transmittance matrix $\textbf{T}$ in Eq. (17) degenerates into the Mueller matrix $^{\text{Mu}}\textbf{M}$, and therefore ${\textbf{S}^{\text{Obj}}}(\xi ,\eta) \;=\; ^{\text{Mu}}\textbf{M}(\xi ,\eta){\textbf{S}^{\text{Illm}}}$ is the transmitted Stokes vector from the object. Thus, up to a constant multiplier, the Stokes images are found to be the matrix convolution of the impulse point-spread matrix $\textbf{H}$ of the polarization imaging system and the Stokes vector ${\textbf{S}^{\text{Obj}}}$ transmitted from the object, and therefore the polarization imaging system is linear with respect to the Stokes vector. For comparison purpose, the image intensity for the corresponding scalar imaging system under perfectly incoherent illumination has been expressed in a shorthand notation [4]: ${I^{\text{Img}}}(u,v) = \kappa [|h(u,v){|^2} * |t(u,v){|^2}]{I^{\text{Illm}}}$.

Turning next to the spectra of the generalized Stokes parameters, we have ${{\boldsymbol{\cal S}}^{\text{Illm}}}(p,q) = \kappa {{\boldsymbol{\cal S}}^{\text{Illm}}}$ for incoherent illumination. After substitution into Eqs. (32)–(34) (with change of variables $p + s \Rightarrow p$ and $q + t \Rightarrow q$), the spectra of the Stokes images become

B. Coherent Case

As a coherent case, we take the generalized Stokes parameters of the object illumination to be ${\textbf{S}^{\text{Illm}}}(\Delta \xi ,\Delta \eta) = {\textbf{S}^{\text{Illm}}}$, implicitly assuming a uniform plane-wave illumination (from the direction normal to the plane of the object) with definite polarization state. For such a fully coherent and completely polarized illumination, we have the constant polarization vector ${\textbf{E}^{\text{Illm}}}$. When light passing through a thin structure of birefringent medium with its amplitude transmittance matrix $\textbf{t}$, the exiting electric field is given by Jones calculus as ${\textbf{E}^{\text{Obj}}} = \textbf{t}\,{\textbf{E}^{\text{Illm}}}$. With an assumption of a space-invariant system, we obtain

We see that the polarization vector at the image plane ${\textbf{E}^{\text{Img}}}$ is proportional to a matrix convolution of the amplitude point-spread matrix $\textbf{h}$ with the polarization vector ${\textbf{E}^{\text{Obj}}}$ transmitted from the object. Clearly, the fully coherent polarization imaging system is linear with the complex electric field. To find the corresponding Stokes images, we follow the definition of $\textbf{S}$ given by O’Neill [1] by substituting the polarization vector in the image plane:

By way of comparison, we list the image intensity for conventional scalar imaging system under fully coherent illumination [4]: ${I^{\text{Img}}}(u,v) = {| {h(u,v) * t(u,v)} |^2}{I^{\text{Illm}}}$.

Having assumed perfectly coherent object illumination, we take the Fourier transform of the generalized Stokes paraments to be ${{\boldsymbol{\cal S}}^{\text{Illm}}}(p,q) = \Im \{{{\textbf{S}^{\text{Illm}}}} \} = \delta (p,q){{\boldsymbol{\cal S}}^{\text{Illm}}}$, which represents illumination with a fully polarized and coherent plane-wave incident normal to the plane of the object. Substitution of this into Eqs. (32)–(34) and making use of the shifting property of the delta function and the mixed-product property of Kronecker product yields

Equation (45) is recognized as a matrix direct correlation. In a similar way for comparison [4], the image spectrum for conventional scalar imaging system under fully coherent illumination is given by $\mathcal{I}^{\text{Img}}(f_u, f_v)=\{[\mathfrak{h}(f_u, f_v) \mathfrak{t}(f_u, f_v)] \star\def\LDeqbreak{}[\mathfrak{h}(f_u, f_v) \mathfrak{t}(f_u, f_v)\} \mathcal{I}^{\text {Illm }}$.

From these results, we see that the behaviors of coherent and incoherent polarization imaging systems are fundamentally different; the incoherent polarization imaging system is linear for the spectral Stokes vectors, while the coherent polarization imaging system is linear for complex electric field vectors. Whether one of the two extreme conditions holds, or the system must be treated as a partially polarized and partially coherent polarization imaging system, depends on the polarization and coherence of the object illumination determined by the design of the source and illumination optics.

6. FREQUENCY RESPONSE FOR DIFFRACTION LIMITED POLARIZATION IMAGING

As shown in the Section 4, the relation between the amplitude transfer matrix ${\boldsymbol {\mathfrak{h}}}$ and the generalized pupil matrix $\textbf{P}$ is of utmost importance because it provides helpful information about the characteristics of diffraction-limited polarization imaging systems in the spatial frequency domain. In this section, we will continue our investigation of the spatial frequency response of a diffraction-limited polarization imaging system under incoherent and partially coherent illumination.

A. Optical Transfer Matrix

We are now ready to deal with a wider class of polarization-related imaging problems involving partially coherent illumination, but they are generally more complicated than those for incoherent imaging. Since objects are illuminated with incoherent light in most imaging systems used in our daily life (e.g., cameras, telescopes), we first start our discussion on a diffraction-limited polarization imaging system with the special but most familiar case of incoherent illumination.

From Eq. (41), we note that the transfer of frequency components of Stokes vector is governed by a $4 \times 4$ matrix ${\boldsymbol{\cal M}}({f_u},{f_v})$,

As our answer to the question, we propose normalization with the Frobenius norm [27] of the frequency transfer matrix at zero spatial frequency

Such way of normalization is justifiable from Haagerup’s inequality, $\left\| {\int {\textbf{F}(x) \otimes {\textbf{G}^*}(x)\text{d}x}} \right\| \le {\left\| {\int {\textbf{F}(x) \otimes {\textbf{F}^*}(x)\text{d}x}} \right\|^{{1 / 2}}}\def\LDeqbreak{}{\left\| {\int {\textbf{G}(x) \otimes {\textbf{G}^*}(x)\text{d}x}} \right\|^{{1 / 2}}},$ which is a generalized version of the Cauchy–Schwarz inequality [28]. Due to the fact that $\left\| {\textbf{A}(\textbf{F} \otimes {\textbf{G}^*}){\textbf{A}^{- 1}}} \right\| = \left\| {\textbf{F} \otimes {\textbf{G}^*}} \right\|$ and letting $\textbf{F} = {\boldsymbol {\mathfrak{h}}}(p + {{{f_x}} / 2},q + {{{f_y}} / 2})$ and $\textbf{G} = {\boldsymbol {\mathfrak{h}}}(p - {{{f_x}} / 2},q - {{{f_y}} / 2})$, we have

The left side of this inequality above is $\left\| {\boldsymbol{\mathcal{M}}({f_u},{f_v})} \right\|$ from its definition, and the right side of the inequality can be written as $\left\| {\boldsymbol{\mathcal{M}}(0,0)} \right\|$. Therefore, it follows from the definition that the Frobenius norm of the OTM $\left\|\pmb{\mathscr{M}}\!\left(f_x, f_y\right)\right\|=\left\|\boldsymbol{\mathcal { M }}\!\left(f_x, f_y\right)\right\| /\|\boldsymbol{\mathcal { M }}(0,0)\|$ is never greater than unity. It should be pointed out that a different normalization with the matrix trace has been adopted by McGuire and Chipman [11], leading to the corresponding matrix norm of the OTM larger than unity. It is the general property $\|\pmb{\mathscr{M}}\!\left(f_x, f_v\right) \| \leq 1$ that justifies the definition given in Eq. (47) to represent the appropriate linear mapping of the spectral distributions of the Stokes vectors in the polarization imaging.

Similar to the modulation transfer function (MTF) and the phase transfer function (PTF) in conventional imaging systems to represent the amplitude and phase information of OTM, respectively, we can also introduce the Modulation Transfer Matrix (MTM) and the Phase Transfer Matrix (PTM) based on the proposed OTM for polarization imaging system. They are

Instead of a $4 \times 4$ matrix for OTM, the MTF here is a function of $({f_u},{f_v})$ and will provide a convenient means to represent a certain kind of measure on the overall performance of polarization imaging as compared with the corresponding imaging for scalar optics without polarization modulation. Since the MTF of a polarization imaging system is the Frobenius norm of OTM, we can easily find its property: $\text{MTF}({f_u},{f_v}) \le 1$.

Similar to the optical transfer function for scalar imaging, the OTM is not only useful for the design of polarization imaging system; it is also valuable to characterize a manufactured system for polarization optics, and all the elements of the OTM can be determined from simple experiments. Let an extended test object with its birefringence-free amplitude transmittance ${t_0}(\xi ,\eta)$ be placed in the specified object plane. When the test object such as a sinusoidal grating or sharp edge [29] is uniformly illuminated by an incoherent beam with six different states of polarization achieved by applying six different optical elements for polarization selection, the corresponding Stokes vectors at the object plane become

B. Apparent Transfer Matrix

Similar to the case of partially coherent imaging in scalar optics [26], the polarization imaging system is in general nonlinear system under partially coherent illumination, and therefore does not possess a transfer matrix in the usual sense. Nonetheless, it is still possible to apply a sinusoidal amplitude grating as an example of the extended test object at the input and to measure a periodic output. However, unlike the linear case for incoherent polarization imaging, knowledge of the frequency response of the polarization imaging system to a sinusoidal input does not allow one to predict what the output will be for other types of inputs. As an example to demonstrate the theory of MTCC developed in previous section, we now find the frequency response of a partially coherent polarization imaging system to a sinusoidal amplitude grating with its amplitude transmittance matrix of the form $\textbf{t}(\xi ,\eta) = {t_0}(\xi ,\eta){\textbf{I}_2}$ with ${t_0}(\xi ,\eta) = 0.5[1 + \cos (2\pi {f_0}\xi)]$ and ${\textbf{I}_2}$ being the identity matrix of size 2. Here, we have neglected any spatial bound that may limit the extent of the object. The intensity transmittance of such a grating is given by

Note the fact that incoherent illumination is a mathematical idealization and all optical fields which occur in nature possess a certain level of coherence. Due to the finite degree of polarization coherence, fundamental difficulties arise in the attempt to apply the linear theory to the evaluation of the Stokes imagery. To understand the nonlinearity associated with the partially coherent imaging system, we wish to compare the distributions of the Stokes images that appear in the image plane with the Stokes parameters of the transmitted light, taking into account the partial polarization and partial coherence of the object illumination.

When the object is illuminated by partially coherent light, Eq. (35) will be used to determine the Stokes images’ spectra ${{\boldsymbol{\cal S}}^{\text{Img}}}$ for evaluation of $\boldsymbol{\mathcal{M}}$. The matrix evaluated in accordance with Eq. (54) will not, of course, be interpretable as a transfer matrix, but it may be referred to as an apparent transfer matrix (ATM) denoted by ${\boldsymbol{\mathcal{M}}^{\text{ATM}}}$ in a similar way to the customary definition of the apparent transfer function used in scalar imaging systems. For computational simplicity, we assume a square exit pupil and a square effective source (condenser lens), allowing us to solve a one-dimensional problem for a variable separable system. Consider a polarization imaging system consisting of an imaging lens and quarter wave plate. The magnification of the lens is assumed to be unity and we ignore image inversion. For a square aperture with its width $2w$, the amplitude point-spread matrix $\textbf{h}$ is given by

For such a polarization-sensitive illumination optics system consisting of a square condenser lens with width $2L$ on a side and polarization selective devices, the generalized Stokes vector incident on the object can be obtained from the van Cittert–Zernike theorem taking the form

When Eq. (57) is written so, we have made use of the assumption that the aperture of the condenser lens itself has been regarded as the incoherence source with uniform distributions of the Stokes parameters when six different polarization states have been used. After Fourier-transforming the amplitude point-spread matrix of Eq. (56) and the generalized Stokes vector of Eq. (57), we obtain the corresponding amplitude transfer matrix ${\boldsymbol {\mathfrak{h}}}$ and the spectral Stokes vector for illumination $^K{{\boldsymbol{\cal S}}^{\text{Illm}}}$, with their elements taking the forms of the rectangular functions. Substitution of these expressions into the definition of ${\boldsymbol{\mathcal{M}}^{\text{TCC}}}$ in Eq. (36), we note that most of the elements are equal to zero except for the 16 elements, i.e., $^K{\mathcal{M}}_{0 0 n}^{\text{TCC}},^K{\mathcal{M}}_{1 1 n}^{\text{TCC}},^K{\mathcal{M}}_{2 3 n}^{\text{TCC}}$ and $^K{\mathcal{M}}_{3 2 n}^{\text{TCC}}$ for $(n = 0 \sim 3)$. Just for mathematical simplification without loss of generality, we can rewrite the one-dimensional equivalents of Eqs. (34) and (35) as

After substitution of ${\boldsymbol {\mathfrak{t}}}(s) = [{1 / 2}\delta (s) + {1 / 4}\delta (s - {f_0}) + {1 / 4}\delta (s + {f_0})]{\textbf{I}_2}$ to obtain ${\boldsymbol{\cal T}}(s;s - f)$, the spectra of the Stokes images become

When these simplified expressions above are derived, we have made use of the properties ${{\cal T}_{\textit{mn}}} = 0$ when $m \ne n$. After inverse Fourier transforms of $^K{\cal S}_l^{\text{Img}}$ in Eq. (60), we have the Stokes images taking the forms of the fringes referred to as the Stokes fringes

Note from Eq. (54) that the numerator of each element in $\boldsymbol{\mathcal{M}}$ is a sum or difference of two Stokes images’ spectra, and all these elements in $\boldsymbol{\mathcal{M}}$ share the common denominator, i.e., $2\kappa {I_0}{{\cal T}_0} = 2\kappa {I_0}\Im \{|{t_0}{|^2}\}$. Although the partially coherent polarization imaging system is nonlinear, it is still possible to define the apparent transfer matrix based on the cosinusoidal components of the Stokes fringes at frequencies ${f_0}$ and $2{f_0}$ at input and output, respectively. The definition used for each element of the unnormalized ATM is

Figure 7 shows plots of the normalized apparent transfer matrices $\pmb{\mathscr{M}}^{\text{ATM}}(f)=\boldsymbol{\mathcal{M}}^{\text{ATM}}(f) /\left\|\boldsymbol{\mathcal{M}}^{\text{ATM}}(0)\right\|$ for both cosinusoidal components of the Stokes parameters with $\sigma = {{L{z^{\text{Img}}}} / {(w{z^{\text{Con}}})}}$ as a parameter. As expected for fully incoherent illumination (i.e., $\sigma = \infty$), the apparent transfer matrix approaches the desired optical transfer matrix: $\lim _{\sigma \rightarrow \infty} \pmb{\mathscr{M}}^{\text{ATM}}\!\left(f_0\right)=\pmb{\mathscr{M}}\!\left(f_0\right)$. Note also that for the component at frequency ${f_{\,0}}$, fully coherent illumination (i.e., $\sigma = 0$) yields higher contrast than fully incoherent illumination when ${f_0} \le {f_c}$ with, ${f_c} = {w / {(\bar \lambda {z^{\text{Img}}})}}$ being coherent cutoff frequency, while the opposite is true for ${f_0} \gt {f_c}$. Furthermore, as expected for the polarization optics, the DC components of the ATM and/or OTM at the zero-frequency construct a constant $4 \times 4$ matrix, which agrees with the well-known Mueller matrix corresponding to the particular polarizing elements used in the polarization imaging system. For the component at $2{f_0}$, fully coherent illumination always yields higher contrast than fully incoherent illumination. It should be clarified that these conclusions are true only for a sinusoidal amplitude grating. For other types of objects, different conclusions about which type of illumination is superior will in general hold.

 

Fig. 7. Apparent transfer matrices for the terms $\cos (2\pi {f_0}u)$ (upper) and $\cos (4\pi {f_0}u)$ (bottom) in the Stokes fringes as functions of the parameter $\sigma$.

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7. CONCLUSIONS

Due to its unique ability to detect optical polarization information modulated by the object, polarization imaging is an advanced imaging technique with great importance. To study the effects of partial polarization and partial coherence in polarization imaging systems, we developed a new matrix-based formalism for quantifying and calculating Stokes imaging with compact and simplified expressions. We first reviewed some matrix representations on the joint behaviors of the stochastic electric fields at two points propagating through a generic polarization imaging system. Before discussing the polarization-selective illumination optics, we provided a formula for the space-domain calculation of the Stokes images. To explore the frequency response of a polarization imaging system, we also presented frequency-domain calculation of the Fourier spectra of the Stokes images and gave a detailed expression for MTCC to represent the combined effects of the polarization imaging system and the illumination optics in tandem. With the help of matrix convolution and matrix direct correlation, the coherent and incoherent limits in polarization imaging were discussed to reveal the underlying mathematical principles of nature philosophy for the Stokes imaging process. With the aid of the matrix’s Frobenius norm, the optical transfer matrix (OTM) was reformulated as a normalized version of matrix direct correlation of the generalized pupil matrix, along with its general properties and its example for a diffraction-limited polarization imaging system with incoherent illumination. Finally, a new matrix referred to as the apparent transfer matrix (ATM) was introduced to address the nonlinearity of a polarization imaging system under partially coherent illumination.