## Abstract

Some different image formation models have been proposed for Nomarski’s differential interference contrast (DIC) microscope. However, the nature of coherence of illumination in DIC, of key importance in image formation, remains to be elucidated. We present a partially coherent image formation model for DIC and demonstrate that *DIC microscope images the coherent difference of shifted replicas of the specimen; but imaging of the each component is partially coherent.* Partially coherent transfer functions are presented for various DIC configurations. Plots of these transfer functions and experimental images provide quantitative comparison among various DIC configurations and elucidate their imaging properties. Approximations for weak or slowly varying specimens are also given. These improved models should be of great value in designing phase retrieval algorithms for DIC.

©2008 Optical Society of America

## 1. Introduction

Unstained biological specimens usually do not alter the amplitude of the incident light, i.e., they are phase specimens. Usually, detectors (eye, camera etc.) are sensitive only to the intensity. Hence, visualization of phase specimen requires some special method to convert phase information to intensity information. Zernike in 1930 realized that diffracted light from phase specimens is *λ*/4 out of phase with the direct light [1]. He noted that to observe the phase information contained in the diffracted light “all one has to do is to throw the diffraction image on a coherent background... ”, which he created using narrow annular illumination that was delayed and attenuated suitably in the back focal plane of the objective. Whereas phase contrast restricts the illumination and imaging apertures for producing an interference effect, Nomarski’s Differential Interference Contrast (DIC) is a wavefont shearing interferometer [2, Ch. 6] [3, Ch. 7] that allows imaging with large illumination aperture. Larger condenser apertures result in higher lateral resolution and better depth discrimination which have contributed much to the popularity of Nomarski’s DIC.

Currently, image formation models of DIC are of great interest as they are prerequisite to:

• accurate interpretation of DIC images in terms of specimen properties [4–8].

• quantitative retrieval of optical path length (i.e., phase) and absorption (i.e., amplitude) information from DIC images [9–13]

• improving DIC images by restoration [14, 15]

In Nomarski DIC transmission microscope, the polarization of the illumination is manipulated by a polarizer and a modifiedWollaston prism to produce two orthogonally polarized, but spatially coherent, beams which have a minute angular split between them [4]. These beams, after being focused by the condenser, illuminate the specimen a small (sub-resolution) distance apart from one another. After these beams traverse the specimen and the objective, they are brought in spatial registration by another modifiedWollaston prism, allowing them to interfere. Due to the spatial coherence, upon interference, a wave-field that depends on the phase difference between these two beams as well as their amplitudes is produced. The analyzer selects the interference term. Visibility of interference fringes (as defined by Young) is proportional to the spatial coherence between these two beams [16, Ch. 2, Ch. 7]. Effectively, the interference term represents subtraction of two laterally shifted and phase delayed replicas of the wavefront after the specimen [3, Sec. 7.2]. After being collected by the tube lens, the interference term results in an image that has phase-gradient contrast and hence a shadow-cast effect.

Although the requirement that the two beams need to be spatially coherent is well appreciated in geometric optics models of DIC, it is not always accounted for properly in diffraction based image formation models. For example, the most recent model for DIC that accounts for partially coherent illumination assumes that the coherence properties of Nomarski’s DIC are the same as conventional brightfield imaging [8, Fig 7]. In this paper, we aim to elucidate the effects of coherence on the image formation properties of DIC microscope.

Interestingly, various DIC configurations differ in the coherence of the illumination employed. We model following DIC configurations in detail using scalar partially coherent image formation theory, whose lightpaths are shown schematically in Fig. 1:

• Nomarski’s DIC with two Wollaston prisms (henceforth called Nomarski-DIC)

• A DIC microscope without a condenser-side Wollaston prism (henceforth called Köhler-DIC). Köhler-DIC configuration is not usually employed for visible light imaging, but a similar configuration has been developed as an X-ray DIC microscope [17].

• PlasDIC system developed by Zeiss [18, 19], which consists of a slit aperture in the condenser front focal plane (FFP) and a combination of polarizer, prism and analyzer inserted in objective back focal plane (BFP). As described in Sec. 3, PlasDIC can be modelled as a special design of the Köhler-DIC configuration.

In section 2, the coherence of various illumination methods and its effects on image formation are discussed. In the same section, the historical development of various DIC configurations is briefly reviewed to gain additional perspective of the relationship between various DIC configurations. In subsection 2.4, existing models of Nomarski-DIC are compared and their accuracy is considered. In section 3, we develop appropriate models for three DIC configurations in the unifying framework of image formation in the spatial frequency domain. Complete partially coherent transfer functions are presented, for the first time to our knowledge, in sections 4 and 5 for various in-focus DIC configurations without assuming that object is either slowly varying or weak. Approximations for weak and slowly varying specimens are also given. Relationship between symmetry of the transfer functions and the contrast observed in the image is clarified with experimental images. The last section summarizes the results.

## 2. DIC configurations and existing image formation models

Image formation in DIC has usually been described by considering the propagation of light from the condenser FFP to the image plane. However, it is instructive to segregate the objectiveside and the condenser-side light paths for various DIC configurations to illustrate effects of the coherence of illumination on image formation. Figure 1 depicts light paths for Köhler-DIC, Nomarski-DIC and PlasDIC configurations segmented in three key parts - illumination path, degree of coherence at the specimen plane, and imaging path. The figure also introduces the notation employed throughout the paper.

#### 2.1. Objective-side light path

All three configuration share a similar objective-side light path. In all configurations, the objective-side Wollaston prism (*W _{o}*), located in the BFP of the objective and sandwitched between crossed polarizers

*P*and

_{c}*P*, causes wavefront shear [20, Fig. 1.82]. Polarizer (

_{o}*P*) is employed before the FFP of the condenser in Köhler-DIC and Nomarski-DIC or just before objective BFP in PlasDIC. The prism

_{c}*W*introduces angular shear of 2

_{o}*e*in the wavefront, which corresponds to a lateral shear of 2ʔ in the specimen plane, which is by design smaller than the resolution limit.

In effect, the prism *W _{o}* combines (appropriately) polarized wavefronts that arrive at the objective BFP at an angle 2

*ε*with respect to one another, i.e., it causes coherent vector addition of amplitudes that have propagated from points separated by 2ʔ in the specimen plane. In Köhler-DIC and PlasDIC, these wavefronts (indicated by dashed and solid lines in Fig. 1(b) and 1(c)) are parallel polarized, whereas in Nomarski-DIC (Fig. 1(a)) such wavefronts are orthogonally polarized.

If the two wavefronts emerging from *W _{o}* (now polarized at 0° and 90° as shown in Fig. 1) have not experienced any relative phase shift, the resultant wavefront is linearly polarized. If the two component wavefronts have the same amplitude (which happens when

*P*and

_{c}*P*are set at extinction as is usual practice) the direction of polarization of the resultant wavefront is the same as the direction of the polarizer

_{o}*P*. The specimen induced space-varying phase difference (

_{c}*q*) between these two wavefronts affects the ellipticity of polarization of the resultant wavefront. A constant phase difference (2

*ϕ*) can be deliberately introduced as a bias to the phase difference induced by specimen. The analyzer (

*P*) which is crossed to

_{o}*P*rejects the common term polarized in the direction of

_{c}*P*and selects the interference term, increasing the contrast of the interference fringes. Note that the specimen induced phase difference

_{c}*θ*is proportional to the local phase gradient of the specimen because 2Δ is sub-resolution. In Nomarski-DIC, usually the bias is introduced by lateral movement of the prism

*W*, and the same holds true for Köhler-DIC and PlasDIC.

_{o}#### 2.2. Coherence of illumination

In full-field transmission systems, only the objective aperture performs imaging, whereas the condenser aperture controls the coherence of illumination and thus has an indirect effect on the imaging properties. The coherence of the imaging system varies with the coherence ratio, *S* defined as ratio of the numerical aperture of the condenser to the numerical aperture of the objective and affects the transfer function of the imaging system [21, 22].

In all three configurations mentioned above, producing sufficient interference contrast when two sheared wavefronts spatially overlap in the objective BFP requires that the points from which they originate on the specimen plane are illuminated with sufficient degree of coherence. Two possible means of increasing the coherence of illumination are: 1) restricting the illumination aperture with a side effect that lateral resolution and depth sectioning are compromised, and 2) employing wavefront shear with large illumination apertures which preserves resolution and depth discrimination.

In Fig. 1(b), a Köhler-DIC configuration with condenser aperture of the same size as objective aperture is shown. As shown in Fig. 1(c), PlasDIC employs a slit (*S _{c}*) to restrict the illumination aperture in the direction of shear. In both of these cases, the degree of coherence in the specimen plane is related to the condenser aperture geometry via the Van-Cittert Zernike theorem [23]. When shear is of the order of the resolution, the coherence between points separated by 2Δ is not sufficient for producing detectable phase gradient contrast in Köhler-DIC, whereas in PlasDIC the degree of coherence is expanded sufficiently in the direction of shear by reducing the aperture size [18, Fig. 3]. It is interesting to note that, in the area of X-ray microscopy, von Hofsten et al. [17] have proposed a DIC microscope that uses a diffractive optical element (instead of polarizing components) as a shearing objective and a condenser aperture having size of roughly 0.5 times the objective aperture. Therefore, the image formation model of Köhler-DIC should be helpful in studying the properties of the X-ray DIC microscopes.

In contrast to the other two configurations, Nomarski-DIC employs wavefront shear to coherently illuminate points separated by 2Δ* _{c}*. It employs (Fig. 1(a)) a modified Wollaston prism (

*W*) in the FFP of condenser to produce orthogonally polarized beams that have angular shear of 2

_{c}*ε*between them. Two orthogonally polarized beams can be considered to be independent of each other, which implies that the specimen is illuminated by individual beams in a partially coherent manner. However, the wavefronts of these two beams are mutually spatially coherent at distances 2Δ

_{c}*in the specimen plane. This nature of coherence is shown schematically in Fig. 1(a). The coherence of each wavefront at the specimen can be computed by the Van-Cittert Zernike theorem.*

_{c}Here, we have assumed that the light path from the condenser FFP to the objective BFP is free from spurious birefringence. The wavefront distortions in two orthogonally polarized beams induced by birefringent background (such as a plastic substrate on which specimen may be mounted) may mask the phase gradient information due to the specimen. In such situations, Nomarski-DIC fails to produce adequate interference contrast. Zeiss’s PlasDIC alleviates this problem by placing all polarization sensitive components responsible for wavefront shear after the objective, albeit with a compromise that the condenser aperture has to be narrowed down in the direction of shear to achieve sufficient degree of coherence between points separated by the shear distance.

#### 2.3. Evolution of DIC

Origins of DIC lie in the polarization based shearing interferometer designed by Lebedeff [4,24] [2, Ch. 6] in 1930, which has come to be known as Jamin-Lebedeff interferometer. Lebedeff’s design employed calcite prisms cemented on top of the objective and condenser to produce an image duplication system with large shear. The large shear was employed to interfere the wavefront that has passed through the specimen with the (almost flat) wavefront that has passed through the surroundings. Smith [25] proposed an image duplication microscope which employed Wollaston prisms placed in the BFP of the objective and the FFP of the condenser to produce shear. Nomarski [26, 27] improved upon Smith’s design by modifying the Wollaston prisms such that they can be placed beyond the focal planes of objective and condenser lenses. More importantly, he also employed sub-resolution shear providing contrast that is proportional to phase-gradient in the direction of shear rather than phase. All these methods employ matched birefringent prisms in illumination and imaging light paths so that rays separated by the shear distance in the specimen plane experience the same total optical path length (OPL) irrespective of the angle at which they illuminate the specimen (in the absence of intentionally introduced bias 2*ϕ*.) Such a matching of OPL is called compensation and is the same requirement as the requirement of coherence between the two beams.

Interestingly, Nomarski proposed a design in his paper [26, Fig 2] where slit illumination was employed to achieve sufficient coherence. Shearing was implemented by sandwiching a Wollaston prism between crossed polarizers placed in the BFP of objective as is done in PlasDIC. Nomarski’s original design with slit aperture was very similar to current PlasDIC configuration, except that PlasDIC employs a carefully computed width for the slit.

#### 2.4. Existing image formation models

Existing image formation models concern themselves with the problem of forward image calculation in Nomarski-DIC, as it is the almost exclusively used configuration in current biological research. We briefly review the most recent models and clarify the assumptions made by them that limit their utility to specific situations. We refer to earlier models only where necessary. Fuller discussion of earlier models can be found in the papers cited here.

Sheppard and Wilson [6] and Cogswell and Sheppard [7] provided the first detailed model based on transfer function theory for conventional and confocal Nomarski-DIC microscopes. In fact, Eq. (13) of Ref. [7] provides an accurate expression for the partially coherent transfer functions for conventional Nomarski DIC. In Ref. [7], the properties of the transfer function were considered along with effects of shear and bias settings for objects that are either weak or slowly varying. In sections 3–5, the transfer function theory reported in Ref. [7] is extended to objects that need not be weak or slowly varying.

Preza et al. [8] reported spatial and spatial frequency domain models for 2D and 3D imaging in Nomarski-DIC. The spatial domain models were employed for calculating images of known specimens, but frequency domain models were not discussed in detail. Equation (1) of Ref. [8] accounts for the effects of polarizer, analyzer and objective-side Wollaston prism by defining a point spread function (PSF) that is a linear combination of laterally-shifted and phase-shifted brightfield PSFs. However, the effect of the Wollaston prism on illumination side was not accounted for. As can be noticed from Eq. (3) and Fig. 7 in Ref. [8], the illumination was assumed to be standard Köhler illumination. In effect, Preza et al.’s model is for Köhler-DIC setup shown in Fig. 1(b). This difference between assumptions made in Ref. [7] and Ref. [8] led to an erroneous conclusion in Ref. [8] that Eq. (1)3 of Ref. [7] represents the transfer function of the Nomarski-DIC configuration *only in the coherent limit*. Preza et al.’s model has been successfully employed, albeit with assumption of small condenser aperture of up to 0.4 times the objective aperture, for computing images of known specimens [8] and designing phase-retrieval algorithms for uniformly absorbing [9, 12] and non-uniformly absorbing [14] specimens. As will be shown with the help of full partially coherent transfer functions and cell images in sections 3–5, Köhler-DIC and Nomarski-DIC have very similar imaging properties for condenser apertures smaller than around 0.4 times the objective aperture when the shear of around half the optical resolution is used. They behave identically when coherent illumination is employed. Thus, the imaging properties of Nomarski-DIC configuration with large condenser aperture have not been fully evaluated in [8], which we attempt to address in section 3.

Munro and Török have rigorously simulated imaging properties of Nomarski-DIC [28] under coherent illumination according to high NA, vectorial diffraction theory of Richards & Wolf. However, the spatial frequency domain approach has the distinct advantage of giving insight in how specimen’s transmission spectrum gets affected by the imaging system.

## 3. Accurate models for various DIC configurations

In this section, we describe spatial frequency domain models for in-focus imaging with three DIC configurations under quasi-monochromatic illumination. For the sake of continuity of argument and to clarify assumptions, some previously published equations have been re-derived in this section.

#### 3.1. Köhler-DIC and PlasDIC

PlasDIC is a special design of Köhler-DIC configuration in which a slit in the direction of shear is placed symmetrically with respect to the optical axis in the FFP of the condenser. Therefore, we derive the partially coherent transfer function only for Köhler-DIC.

Assume that the amplitude point spread function (PSF) of the objective is *h _{BF}*(

*x,y*) in the bright-field configuration. Where, (

*x,y*) are co-ordinates in the specimen plane (Fig. 1). Even without the condenser-side prism the polarizing components (

*W*, and

_{o}, P_{c}*P*) can introduce angular shear (2

_{o}*ε*) and phase bias (2

*ϕ*) to the wavefront in the objective BFP. Therefore, the PSF of the objective for Köhler-DIC can be given by,

where, *R* determines the relative amplitude of the two wavefronts. *R* is adjusted by rotating polarizers *P _{c}* and

*P*with respect to one another. 2Δ and 2

_{o}*ϕ*are the shear and bias, respectively, and the shear azimuth is assumed to be in

*X*direction.

Usually, *P _{c}* and

*P*are crossed, leading to

_{o}*R*=0.5, i.e.,

We obtain the effective pupil of the objective, or coherent transfer function, as a 2D Fourier transform of the *h _{K}(x,y)*.

$$=i{P}_{\mathrm{BF}}(\xi ,\eta )\mathrm{sin}\left(2\pi \xi \Delta -\varphi \right).$$

ξ and *η* are transverse co-ordinates in the pupil-planes (condenser FFP and objective BFP as illustrated in Fig. 1) normalized by the numerical aperture of the objective, *NA _{obj}. P_{BF}*(ξ,

*η*) is the coherent transfer function of the brightfield microscope, which is simply a circle of radius equal to 1 [29]. Therefore, coherent transfer function for Köhler-DIC also has a normalized cutoff frequency of 1.

We emphasize that Eq. (2) and (3) describe imaging properties of the objective-side light path in the spatial and spatial frequency domains respectively. They provide a sufficient description of the imaging properties of Köhler-DIC if *coherent illumination, or point-illumination, is assumed*.

For partially coherent illumination, the partially coherent transfer function (also called transmission cross-coefficient) of the Köhler-DIC model can be written in terms of the objective pupil (*P _{K}*) and the condenser pupil(

*P*) as follows [21, 22]:

_{cond}Limits of all integrals in this paper are from -∞ to ∞. In all equations presented in this paper, (*m; p*) and (*n;q*) are spatial frequency pairs in *X* and *Y* directions, respectively, normalized by the frequency variable,

where *λ* is the wavelength of the quasi-monochromatic illumination.

The intensity image can be expressed in terms of the specimen spectrum, *T*(*m,n*)), and the partially coherent transfer function as,

The above equation illustrates that, in partially coherent imaging, the strength of the spatial frequencies produced in the image depends on pairs of spatial frequencies of the specimen, rather than individual frequencies.

Equations 4 and 6 describe the frequency transfer properties and the forward image calculation, respectively, for the Köhler-DIC configuration with a condenser aperture of any geometry.

#### 3.2. Nomarski-DIC

Assuming the absence of spurious birefringence, as described in subsection 2.2, Nomarski-DIC employs two independent beams each having large aperture. We can calculate wavefronts produced by the two beams individually until they interfere in the back-focal plane of the objective.

If the specimen transmission is given by *t*(*x,y*), two beams effectively image the transmission functions *t*(*x*+Δ,*y*) and *t*(*x*-Δ,*y*) because of the lateral shear. The effect of phase difference, i.e. bias of 2*ϕ*, between two beams can be modelled as equal and opposite phase offset added to the transmission functions. Therefore, the two beams can be considered to be imaging transmission functions *t*(*x*+Δ,*y*)exp(-i*ϕ*) and *t*(*x*-Δ,*y*)exp(*iϕ*), respectively.

The beams are spatially coherent to one another at distances 2Δ in the specimen plane, and *W _{o}* and

*P*optically compute their coherent difference. Therefore, Nomarski-DIC microscope effectively images a difference of transmission functions seen by the two beams. We can write the specimen transmission function that is effectively imaged as,

_{o}Therefore, the Fourier spectrum of the imaged transmission function is given by,

$$=2iT(m,n)\mathrm{sin}\left(2\pi m\Delta -\varphi \right).$$

We have accounted for the effects of all polarizing components in the above equation. Hence, we can assume that the imaging performed by the rest of the components is brightfield. As the prism *W _{o}* is placed in the BFP of the objective, angular-split of 2

*ε*has no effect on aperture size as seen by the specimen. Therefore, both beams illuminate the specimen from an aperture of the same size. Thus, the transmission spectrum

*T*is imaged in a partially coherent manner just as in a conventional bright-field microscope.

_{N}Therefore, the image in partially coherent Nomarski-DIC microscope is given by,

where,

is the partially coherent transfer function for brightfield imaging. By substituting *T _{N}* from Eq. (8) in Eq. (9), we can write an effective transfer function for Nomarski-DIC that images the original specimen spectrum

*T*(

*m,n*) as follows,

This is the same expression as Eq. (13) in Ref. [7].

In this paper, the frequency variables (*m,n; p,q*) are normalized by the frequency *m*
_{0}=*NA _{obj}*/

*λ*whereas in Ref. [7] they were normalized by the cutoff frequency 2

*NA*/

_{ob j}*λ*. This leads to cutoff of 2 for partially coherent transfer functions in this paper as opposed to cutoff of 1 as in Ref. [7].

#### 3.3. Equivalence of Köhler-DIC and Nomarski-DIC in coherent limit

When the illumination is coherent, i.e. when condenser aperture is closed to a point, the transmission cross coefficient for Köhler-DIC configuration *C _{K}* in Eq. (4) reduces to the frequency response function [23, pp.601–605] which can be written as,

where, *M _{BF}(m,n; p,q)=P_{BF}(m,n)P*_{BF}(p,q)*.

*M _{K}* relates the mutual intensity in the specimen plane to the mutual intensity in the image plane, whereas

*P*relates complex amplitudes.

_{K}*M*is sufficient description of imaging properties in coherent limit and is separable in

_{K}*P*(

_{K}*m,n*) and

*P*(

_{K}*p,q*). Thus, transfer function that affects complex amplitude can be written for Köhler-DIC as,

For Nomarski-DIC, the brightfield transmission cross coefficient *C _{BF}* in Eq. (11) similarly reduces to coherent transfer function

*P*can be considered to be imaging the complex amplitude spectrum

_{BF}. P_{BF}*T*(

_{N}*m,n*) (Eq. (8)). Therefore, the effective coherent transfer function (that images the original transmission spectrum

*T*(

*m,n*)) for Nomarski-DIC is given by,

Thus, both systems have the same transfer function in the coherent limit, and hence behave identically.

We note that Eq. (2), (3), (4), and (6) correspond to Eq. (1), (2), (14), and (15) in Ref. [8], respectively. Ref. [8] (which effectively models Köhler-DIC configuration) notes that the partially coherent transfer function of Eq. (4) reduces to frequency response function of Eq. (12) in the coherent limit [8, Eq.17]. The forms of Eq. (11) (which describes partially coherent image formation for Nomarski-DIC) and Eq. (12) (which describes coherent image formation for Köhler-DIC) are similar, but they describe two different quantities. Perhaps, this led to an erroneous conclusion in Ref. [8] that Eq. (13) of Ref. [7] (in fact, valid for partially coherent imaging in Nomarski-DIC) is applicable only to the coherent illumination.

## 4. Partially coherent transfer functions

#### 4.1. Computation of transfer functions

As can be seen from Eq. (4) and (10), complete partially coherent transfer function can be computed as the area of overlap of the following three pupils: the squared magnitude of the condenser pupil, the objective pupil shifted by *m* and *n*, and the conjugate of the objective pupil shifted by *p* and *q* [22]. We have implemented an algorithm in MATLAB to compute the complete partial coherent transfer functions on the discrete 4D grid of (*m,n; p,q*) using this approach. The value of *C*(*m,n; p,q*) computed as an area of overlap of three pupils is normalized with area of the objective pupil making computed values independent of fineness of the grid. Note that for a brightfield system employing matched illumination, the partially coherent transfer function for the DC term of the specimen spectrum (*C _{BF}*(0, 0;0,0)) is the same as area of the objective pupil. Therefore, when computed with our algorithm,

*C*(0, 0;0,0)=1.

_{BF}Pupil radii are normalized by the numerical aperture of the objective pupil (*NA _{obj}*). Therefore, the objective pupil has radius of 1 and the condenser pupil has the radius equal to coherence ratio,

*S*. As an example, pupils used for computing transfer functions of the three configurations with coherence ratio

*S*=0.7, shear 2Δ=1/4

*m*

_{0}and bias 2

*ϕ*=

*π*/2 are shown in Fig. 2. The frequency variables, (

*m,n; p,q*), are normalized by characteristic frequency,

*m*. Correspondingly, the spatial variables such as shear are expressed as multiples of 1/

_{o}=NA_{obj}/l*m*=

_{o}*λ*/

*NA*. We Note that in Ref. [7], the normalizing frequency variable (

_{obj}*m*

_{0}) is 2

*NA*. Therefore, normalized values of shear used in this paper correspond to half those reported in Ref. [7].

_{obj}/λTo compute transfer functions for Köhler-DIC and PlasDIC, we first compute the objective pupil as per Eq. (3). The condenser pupil for Köhler-DIC is a circle with radius of *S*. For PlasDIC, the condenser pupil is a slit and in our simulation we take coherence ratio, *S* to be the width of the slit. Then the partially coherent transfer function, *C _{K}*, is computed from Eq. (4).

To compute the partially coherent transfer function for Nomarski-DIC, we first compute *C _{BF}* from Eq. (10) with appropriate objective and condenser pupils.

*C*is modulated as shown in Eq. (11) to compute

_{BF}*C*.

_{N}Although it is possible to compute the image of a specimen having arbitrary 2D transmission using the four dimensional partially coherent transfer function, we assume a one dimensional specimen allowing a reduction to two dimensions. From Eq. (6) and (9), we can see that the image intensity along the *X*-direction depends only on the spatial frequency variable pairs (*m; p*). Assuming a 1D specimen allows us to plot partially coherent transfer functions that describe imaging properties of a general one dimensional object and correlate their structure with contrast expected in the image.

As an example, Fig. 3 shows partially coherent transfer functions for one dimensional objects which have variations either parallel to the direction of shear or perpendicular to the direction of shear. Without loss of generality, shear azimuth is assumed to be in *X* direction.

#### 4.2. Symmetry of transfer function and contrast

Symmetry of the partially coherent transfer function determines the specimen information that gets transferred to intensity of the image [30, 31]. If the transfer function has even symmetry around *m*=-*p* axis, the resultant image has pure amplitude contrast, whereas odd symmetry of the transfer function around *m*=-*p* axis results in differential phase contrast.

As can be seen from Fig. 3(b), in the direction perpendicular to the shear, the partially coherent transfer function has even symmetry around *m*=-*p* axis. This observation holds true for all DIC configurations. Moreover, Fig. 3(b) has the same structure as for the brightfield microscope as reported in Ref. [32]. Thus, in all DIC configurations, imaging in the direction perpendicular to shear is similar to that in a brightfield microscope. Therefore, only amplitude information is expected to be imaged within the passband of the system. However, strong phase information that lies outside the passband of the system will affect the intensity of the image and in that sense a brightfield microscope does image phase information. This observations agrees with observation in Ref. [33, Appendix-1] that DIC microscope images specimens which have constant phase in the direction of shear but have ‘step-like’ phase in the perpendicular direction.

Having determined the structure of the transfer function in the direction perpendicular to shear in all DIC configurations, hereafter we only consider imaging in the direction of shear. Therefore, the transfer functions that we are concerned with are as follows:

For Nomarski-DIC,

where,

and for Köhler-DIC,

where *P _{K}* is defined by Eq. (3).

Although, the symmetry of the transfer function can provide information about the type of contrast present in the image, variations of the transfer function govern the strength with which different frequencies of the object are imaged. Frequency support of the transfer function determines resolution of the imaging system.

If the object is weak, a line *C*(*m*;0) through the 2D transfer function *C*(*m; p*) is sufficient description of imaging properties of the system and is called the weak object transfer function (WOTF). The even component of WOTF determines the strength with which amplitude is imaged and the odd component determines the strength with which phase gradient is imaged. If the object is slowly varying with respect to the resolution element, a line *C*(*m;m*) provides sufficient description of imaging properties and is called phase gradient transfer function (PGTF). The symmetry of PGTF shows how the imaging system behaves for increasing or decreasing slopes in optical path length of the object.

In the following section, we examine the structure of the partially coherent transfer functions in the direction of shear with respect to different values of coherence ratio and bias to investigate the resultant contrast and resolution. Contrast and resolution predicted by the structure of transfer function are evaluated with experimental images.

## 5. Imaging properties of three configurations

We first consider effects of the condenser aperture size, i.e., degree of coherence, on the imaging properties of Nomarski-DIC, Köhler-DIC, and PlasDIC in the direction of shear. A fixed shear of 2Δ=0,25*λ/NA _{obj}* (i.e., Δ=1/8

*m*) and bias of 2

_{0}*ϕ*=

*π*/2 have been used in simulations as they are suitable for linear imaging of phase information in conventional DIC configuration with good contrast [7, pp.88–94].

Figure 4 (Media 3) is the image sequence showing transfer functions in the direction of shear computed for *S*=1,…,0.1. Comparing different features of transfer functions of three configurations as the coherence ratio (*S*) decreases, one can quantitatively predict imaging properties as the illumination becomes more coherent. We note that it is meaningful to compare minimum and maximum values of the transfer function as *S* changes only between PlasDIC and Köhler-DIC as they are described by the same imaging model. For interpretation of absolute values of plots, please see Sec. 4. Frequency support and shape of the transfer function provides information about resolution and fidelity, respectively, with which information is imaged. Relative strength with which amplitude and phase gradient information is imaged is governed by the relative strength of even and odd components of the transfer function.

Figure 5 shows the even and odd parts of the weak object transfer function and the phase gradient transfer functions for three configurations for *S*=0,2,0.4,0.6,0.8 and 1.

From Fig. 4 (Media 3) and Fig. 5 we can make the following observations. In the following, we refer to the partially coherent transfer functions as *C*(*m; p*), the weak object transfer function as *C*(*m*;0) and the phase gradient transfer function as *C*(*m;m*).

For all three configurations, the even part of *C*(*m; p*) is stronger than odd part of *C*(*m; p*) for all values of *S*. Thus, the configurations considered here will not produce good phase contrast when the specimen contains strong amplitude information. However, as *S* reduces the odd part of *C*(*m;p*) becomes more similar in strength to the even part. Thus, smaller values of *S* lead to better phase gradient contrast.

Consider the case of matched illumination, i.e., *S*=1. The odd part of *C*(*m; p*) for Nomarski-DIC has broadest frequency support without zero crossings. Presence of zero crossings could lead to imaging of certain frequencies with reversed contrast and hence artifacts. For Köhler-DIC and PlasDIC, the odd part of *C*(*m; p*) does not transmit certain frequency pairs (*m; p*). Therefore, when matched illumination is employed, the Nomarski-DIC configuration images phase gradient information with highest resolution and fidelity.

At *S*=1, Köhler-DIC and PlasDIC are the same configuration. They cannot image weak phase information at all because the odd part of *C*(*m*;0) is zero. However, *C*(*m;m*) for both is nonzero and therefore expected to image slowly varying phase information. *C*(*m;m*) for Nomarski-DIC has sharper changes with value of phase gradient. Therefore, a specimen which has both positive and negative gradients will have more pronounced shadow-casting and highlighting in Nomarski-DIC rather than in Köhler-DIC. These effects can be observed in Fig. 8. We note that the curves for Nomarski-DIC for the even part of *C*(*m*;0), the odd part of *C*(*m*;0) and *C*(*m;m*) are identical to those presented in Ref. [7] for the same shear and bias.

As condenser is stopped down to around *S*=0.7, structures of *C*(*m*;0) and *C*(*m;m*) for Nomarski-DIC and Köhler-DIC become slightly similar. Köhler-DIC and PlasDIC can now image weak phase information as *C*(*m*;0) in both configurations have non-zero odd component. However, *C*(*m; p*) for Köhler-DIC and PlasDIC still do not transmit as wide a range of frequencies as for Nomarski-DIC. *C*(*m*;0) for PlasDIC has a stronger even component than Köhler-DIC, which will lead to stronger imaging of amplitude of the weak object. PlasDIC also has stronger *C*(*m;m*) than Köhler-DIC, which leads to stronger imaging of slowly varying phase information.

As the value of *S* is reduced further, the transfer functions for all three configurations become rather similar. At *S*=0.4, three configurations are characterized by similar *C*(*m; p*),*C*(*m*;0) and *C*(*m;m*). Therefore, when the condenser aperture size is around 40% of the objective aperture size, the use of condenser-sideWollaston prism is redundant. In such a situation, PlasDIC leads to a brighter image background than Köhler-DIC as its transfer functions are stronger at zero spatial frequency. We note that the value of *S*=0.4 is dependent on the assumption that shear is 2Δ=1/4*m*0. For bigger shear, the condenser will need to be stopped down further to expand the distance over which illumination is coherent. Conversely, systems employing smaller shear can work with a more open condenser. *This is the key conclusion about dependence of phase gradient contrast on degree of coherence of illumination*.

To verify whether Köhler-DIC and Nomarski-DIC behave as predicted above with change in S, we imaged 16*µm* thick mouse intestine section (Invitrogen Fluocells prepared slide #4) with 20X 0.75 NA objective and 0.9 NA condenser. The value of *S* was changed with motorized condenser aperture. The microscope control software allowed calibrated change in the size of the motorized condenser aperture which was visually verified by looking at the objective back focal plane with a Bertrand lens. Using a condenser top lens of NA 0.9 allowed the maximum value of *S* to be 1.2. The smallest value of *S* was limited by the amount by which the aperture diaphragm could be closed. The bias was approximately set to *π*/4 with a translatable modified Wollaston prism on objective side. The amount of shear employed by our microscope is not calibrated.

Figure 6 shows images for Nomarski-DIC and Köhler-DIC configurations for matched illumination and *S*=0.4. It is evident that around *S*=0.4, both Nomarski-DIC and Köhler-DIC produce similar phase gradient contrast. However, with matched illumination Köhler-DIC loses phase gradient contrast especially for weak phase changes, while contrast in Nomarski-DIC is retained. Associated media Fig. 6 (Media 4) shows the variation in the image contrast as the value of *S* is increased from 0.3 to 1.2. It can be seen that as condenser aperture is opened, the role of the Wollaston prism in providing coherent illumination for two points separated by the shear becomes more important.

Next, we consider the effect of changing the bias in Köhler-DIC and Nomarski-DIC for two important condenser aperture sizes discussed above.

Figures 7 and 8 correlate the partially coherent transfer functions with the image of an optical fiber of diameter 50*µm* under Nomarski-DIC and Köhler-DIC at bias values of 2*ϕ*=0 (left column), 2*ϕ*=*π*/2 (center column), and 2*ϕ*=*π* (right column).

The optical fiber is a good approximation of a one dimensional slowly varying phase specimen except near the edges. Since the fiber was aligned with its axis perpendicular to the direction of shear, one can estimate the relative strengths with which phase gradients (along the direction of shear) are imaged by observing relative strength of the transfer function along line (*m,m*). Therefore, this specimens’ imaging can be described to good accuracy with a *C*(*m;m*) line through the transfer functions. As we move across fiber along the direction of shear, we first encounter a high positive phase gradient at one edge of the fiber which slowly reduces to zero at the center of the fiber, and then increases in opposite direction until we reach opposite edge of the fiber. It can be seen from Fig. 7 and 8 that for all cases, the variation in intensity in the image of fiber for different phase gradients follows closely the variation in *C*(*m,m*) for corresponding values of *m*.

It can be seen from Fig. 7 that for condenser aperture size of *S*=0.4, both systems have almost the same transfer functions for all values of bias. The contrast observed in the images for the two systems is almost identical and correlates well with the structure of the transfer function. Transfer functions for 2*ϕ*=0 do not transmit low spatial frequencies and hence, both systems behave as dark-field imaging system. For bias of 2*ϕ*=*π*/2, both systems are seen to produce a typical shadow-cast effect and for 2*ϕ*=*π* both systems are seen to behave like brightfield systems with low cutoff.

However, when the condenser is opened to give matched illumination, we observe from Fig. 8 that Köhler-DIC does not image phase information very well. In particular, at 2*ϕ*=0 Köhler- DIC no longer behaves like a dark-field system whereas Nomarski-DIC does. Nomarski-DIC retains good shadow-cast imaging at 2*ϕ*=*π*/2, whereas Köhler-DIC does not. For 2*ϕ*=*π*, both systems behave almost like brightfield system with matched condenser.

## 6. Summary and conclusions

We have presented and verified an accurate model based on scalar diffraction theory for Nomarski’s DIC. The model is summarized by Eqs. 11 and 9. This model accounts for effects of coherence of illumination due to the condenser-side Wollaston prism. Another means of achieving sufficient degree of coherence is to restrict the condenser pupil in the direction of shear. Zeiss has developed a DIC system, termed PlasDIC, based on this principle that allows imaging of phase gradient information for birefringent specimens. Properties of Nomarski-DIC and PlasDIC are examined alongwith a configuration, termed Köhler-DIC, that is similar to Nomarski-DIC but without the condenser-side prism. Imaging properties of PlasDIC and Köhler-DIC are described by Eqs. 4 and 6. It has been shown that when typical settings for shear and bias are employed, closing the condenser to approximately 0.4 times the objective aperture results in similar contrast for all three configurations. For wider condenser apertures, the second Wollaston prism is necessary to produce sufficient degree of coherence. Complete partially coherent transfer functions for one dimensional objects have been computed without assuming either a weak or a slowly varying object. The symmetry of the computed transfer functions has been discussed in light of the contrast seen in the image. Experimental images are in good agreement with predictions made from transfer functions (Fig. 6,7, and 8).

These accurate models for DIC configurations will enable accurate calculation of images of known specimen using the frequency domain approach. The models should be valuable in developing accurate phase retrieval algorithms from DIC images.

## Acknowledgment

C.J.R. Sheppard acknowledges support from the Singapore Ministry of Education Tier-1 funding (grants R397000022112 and R397000033112). S.B. Mehta acknowledges graduate scholarship from Graduate Programme in Bioengineering-NGS, National University of Singapore.

## References and links

**1. **F. Zernike, “How I discovered phase contrast,” Science **121**, 345–349 (1955). [CrossRef] [PubMed]

**2. **M. Françon and S. Mallick, *Polarization Interferometers: Applications in Microscopy and Macroscopy* (Wiley-Interscience, 1971).

**3. **M. Pluta, *Advanced Light Microscopy*, vol. 2 Specialized Methods (PWN-Polish Scientific Publishers, Warszawa, 1989).

**4. **R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy.” Z Wiss Mikrosk **69**, 193–221 (1969). [PubMed]

**5. **R. D. Allen, J. L. Travis, N. S. Allen, and H. Yilmaz, “Video-enhanced contrast polarization (AVEC-POL) microscopy: a new method applied to the detection of birefringence in the motile reticulopodial network of Allogromia laticollaris.” Cell Motil **1**, 275–289 (1981). [CrossRef] [PubMed]

**6. **C. J. R. Sheppard and T. Wilson, “Fourier imaging of phase information in scanning and conventional optical microscopes,” Phil. Trans. Roy. Soc. London, Series A **295**, 513–536 (1980). [CrossRef]

**7. **C. Cogswell and C. Sheppard, “Confocal differential interference contrast(DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. **165**, 81–101 (1992). [CrossRef]

**8. **C. Preza, D. L. Snyder, and J.-A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference-contrast microscopy,” J. Opt. Soc. Am. A **16**, 2185–2199 (1999). [CrossRef]

**9. **C. Preza, “Rotational-diversity phase estimation from differential-interference-contrast microscopy images,” J. Opt. Soc. Am. A **17**, 415–424 (2000). [CrossRef]

**10. **M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc. **199**, 79–84 (2000). [CrossRef] [PubMed]

**11. **M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. **214**, 7–12 (2004). [CrossRef] [PubMed]

**12. **S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. **13**, 024020 (2008). [CrossRef] [PubMed]

**13. **M. Shribak, J. LaFountain, D. Biggs, and S. Inoué, “Quantitative orientation-independent differential interference contrast (DIC) microscopy,” Proceedings of SPIE **6441**, 64411L (2007). [CrossRef]

**14. **J. A. O’Sullivan and C. Preza, “Alternating minimization algorithm for quantitative differential-interference contrast (DIC) microscopy,” Proceedings of SPIE **6814**, 68140Y (2008). [CrossRef]

**15. **Z. Kam, “Microscopic differential interference contrast image processing by line integration (LID) and deconvolution,” Bioimaging **6**, 166–176 (1998). [CrossRef]

**16. **M. Françon, *Optical Interferometry* (Academic Press, 1966).

**17. **O. von Hofsten, M. Bertilson, and U. Vogt, “Theoretical development of a high-resolutiondifferentialinterference-contrast optic for x-raymicroscopy,” Opt. Express **16**, 1132–1141 (2008). http://www.opticsexpress.org/abstract.cfm?URI=oe-16-2-1132. [CrossRef] [PubMed]

**18. **R. Danz, A. Vogelgsang, and R. Kathner, “PlasDIC - a useful modification of the differential interference contrast according to Smith/Nomarski in transmitted light arrangement,” Photonik (2004). www.zeiss.com/C1256F8500454979/0/366354E1E8BA8703C1256F8E003BBCB9/$file/plasdic photonik 2004march e.pdf.

**19. **R. Danz, P. Dietrich, A. Soell, C. Hoyer, and M. Wagener, “Arrangement and method for polarization-optical interference contrast,” (2006). US Patent No. 7046436.

**20. **M. Pluta, *Advanced Light Microscopy*, vol. 1 Principles and Basic Properties (PWN-Polish Scientific Publishers, Warszawa, 1988).

**21. **H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A, Math. and Phys. Sci. **217**, 408–432 (1953). [CrossRef]

**22. **C. J. R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” J. Mod. Opt. **24**, 1051–1073 (1977).

**23. **M. Born and E. Wolf, *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light*, 7th ed. (Cambridge University Press, Cambridge, 1999). [PubMed]

**24. **A. A. Lebedeff, “L’interféromètre à polarisation et ses applications,” Rev. d’Opt9, 385–413 (1930). (“Polarization interferometer and its applications”).

**25. **F. H. Smith, “Interference Microscope,” (1952). US patent no. 2601175.

**26. **G. Nomarski, “Microinterféromètre différentiel à ondes polarisées,” J. Phys. Radium16, 9–13 (1955). (“Differential microinterferometer with polarized waves”).

**27. **M. Françon, “Polarization interference microscopes,” Appl. Opt. **3**, 1033–1036 (1964). [CrossRef]

**28. **P. Munro and P. Török, “Vectorial, high numerical aperture study of Nomarski’s differential interference contrast microscope,” Opt. Express **13**, 6833–6847 (2005). http://www.opticsexpress.org/abstract.cfm?URI=oe-13-18-6833. [CrossRef] [PubMed]

**29. **J. W. Goodman, *Introduction to Fourier Optics*, 2nd ed. (McGraw-Hill, New York, 1996).

**30. **T. Wilson and C. J. R. Sheppard, “Coded apertures and detectors for optical differentiation,” in *Int. Optical Computing Conference*, vol. 232, pp. 203–209 (Washington DC, 1980).

**31. **T. Wilson and C. J. R. Sheppard, *Theory and Practice of Scanning Optical Microscope* (Academic Press, London, 1984).

**32. **B. Möller, “Imaging of a straight edge in partially coherent illumination in the presence of spherical aberrations,” J. Mod. Opt. **15**, 223–236 (1968).

**33. **C. Preza, “Phase estimation using rotational diversity for differential interference contrast microscopy,” Ph.D. thesis, Washington University (1998).