## Abstract

It is well known that vectorial analysis is essential to the study of high numerical aperture (NA) bright field scanning microscopes. We have constructed a high NA, vectorial model of a scanned Differential Interference Contrast (DIC) microscope which demonstrates that vectorial analysis is even more important to the study of this device. Our model is valid for coherent illumination and is able to model arbitrary scattering objects through the application of rigorous numerical methods for calculating electromagnetic scattering. We use our model to demonstrate how parameters such as sheer and bias affect imaging properties of both confocal and conventional scanning type DIC microscopes.

©2005 Optical Society of America

## 1. Introduction

Transmitted light differential interference contrast (DIC) microscopy is widely used in the study of phase objects which commonly occur in biological specimens. It is also used in reflection for studying surface profile. The principle behind DIC microscopy is to illuminate the sample by two orthogonally polarised and laterally adjacent light spots and then image the light scattered from the vicinity of the illuminated spots to a suitably placed detector. Before the scattered light reaches the detector it passes through a polariser which in turn permits interference of the two orthogonally polarised components. Interference transforms information about the relative phase of the two object points into amplitude information.

This paper describes a high numerical aperture (NA), vectorial model for a scanning type DIC microscope. Several models of the DIC microscope have already been proposed. Lessor *et al*. [1] presented a model of a DIC microscope based largely upon geometrical optics. This work provides many useful results particularly on modeling a Wollaston prism, however its scalar nature is ultimately limiting. Galbraith [2] developed a model for calculating the image of a point in a DIC microscope. This work was very useful for comparison with our model however it models a wide field DIC microscope and employs scalar analysis. Thus it is not suitable for modeling a high NA system. Holmes *et al*. [3] developed a more general model based upon (scalar) Fourier optics with the intention of understanding the signal processing characteristics of a DIC microscope. They considered only weak phase objects and also assumed that the objective lens does not truncate the spectrum of the illumination field. Cogswell *et al*. [4] derived a scalar model for scanning DIC microscope imaging using an optical transfer function approach to optical system modeling and was more rigorous than previous models. However, the analysis, employing a possibly partially coherent illumination, considered only weakly scattering objects. Preza *et al*. [5] and Preza [6] developed yet more advanced models not limited to coherent illumination. These models however are still by nature scalar and are thus unsuitable for the task at hand. It is thus the objective of this paper to develop a model of high NA DIC imaging capable of modeling arbitrary scattering objects assuming coherent illumination.

#### 1.1. Introduction to the scanning type DIC microscope

Figure 1 shows a schematic diagram of a scanning type DIC microscope. Note that although this diagram illustrates a reflection mode DIC microscope, transmission microscopes are at least as common. A polariser and possibly a *λ*/2 waveplate is used in the illumination path of the microscope to ensure that 45° polarised light is incident upon the Wollaston prism. The Wollaston prism, acting as a polarising beam splitter, generates two orthogonally polarised beams with angular displacement which are incident upon the objective lens. Two laterally displaced focused spots result. The reflected or transmitted light is then collected by the objective lens before propagating through the Wollaston prism again. This brings the two collimated beams into lateral overlap. An analyser placed before the detector lens has its axis crossed with that of the polariser to allow the two orthogonally polarised beams to interfere. The imaging properties of the microscope are determined principally by the shear (separation between focused spots) and the bias (phase shift between focused spots).

The remainder of the paper is organised as follows. First the theory for calculating the form of the focused illumination is described. It is then shown how the image of a point object is calculated. Following this it is shown how the image of an arbitrary scatterer may be calculated. This section includes brief section on the Finite Difference Time Domain (FDTD) method and the series of Mie. Finally, results investigating the form of the incident field and the image of a point, a sphere and a step are given.

## 2. Modeling Nomarski’s differential interference contrast microscope

#### 2.1. Coherent Illumination

The first step in modeling a scanning DIC microscope is to calculate the distribution of the focused illumination. The Richards-Wolf integral [7] is employed to do this. It however must be modified slightly to include the effect of the Wollaston prism. Consider the optical system shown in Fig. 2.

Without loss of generality it will be assumed that the Wollaston prism is aligned as shown in Fig. 2. In particular, when relating the alignment of the prism to the polarisation of incident light, the *y*-polarisation direction lies parallel to the optical axis of the second wedge of the Wollaston and the *x* is parallel to the axis of the first.

If light, linearly polarised at 45°, is incident normally onto the Wollaston prism, the prism acts as a polarising beam splitter. It will be more instructive to consider the phase difference imparted upon the *x* and *y*-polarised beams rather than the angular splitting of the beams. In order to do this it is necessary to assume that normally incident rays deviate only after they have exited the prism and not while they are propagating through it. The validity of such an approximation is considered by Lessor *et al*. [1] and it will suffice to say that the approximation is reasonable since we are interested only in the salient features of the prism. Consider a ray propagating through a Wollaston prism as shown in Fig. 3.

At the prism the coordinates *η* and *ξ* are used instead of *x* and *y* to avoid confusion later. The position *η* = 0 represents the plane containing the optical axis of the system. The Wollaston prism is displaced in the *η* direction so that its center, where the thickness of both wedges is equal, is at position *η*_{c}
. Consider then a ray with field polarised in the *η* direction. In order to simplify notation, we now introduce *n*
_{+} = *n*_{o}
+ *n*_{e}
and *n*
_{-} = *n*_{o}
- *n*_{e}
, where *n*_{o}
and *n*_{e}
are the refractive index of ordinary and extraordinary directions in the prism wedges. The difference between the phase of the ray on entering the prism and on leaving the prism is given by:

where *k* = 2*π*/*λ* is the wavenumber with *λ* being the wavelength of light, 2*T* is the thickness of the prism, and *θ*_{w}
is the angle of the interface between the two wedges. Similarly, the phase difference for the orthogonally polarised ray is given by:

These expressions reveal the salient features of the prism. If it is assumed that the prism is infinitely thin and positioned at the back focal plane of a high NA objective lens, the form of the focused field may be found through application of the Richards-Wolf integral. The electric field in the vicinity of the focus may be found according to [7]:

where **ŝ** is a unit vector along a typical ray focused by the lens, described by spherical polar coordinates as **ŝ** = (cos*ϕ*sin*θ*,sin*θ*sin*ϕ*,cos*θ*), **ε**_{0} is the geometrical optics field vector on the Gaussian reference sphere, *f* is the focal length of the lens, and **r** = (*x*,*y*,*z*) is the location where we seek to calculate the field. Ω is governed by the NA of the objective and takes the form Ω = {**ŝ**|*ϕ* ∊ [0,2*π*),*θ* ∊ [*π*-*α*,*π*]} where *α* is the semi convergence angle of the objective.

With a Wollaston prism in place, *ε*_{0} takes the form:

where **ε**^{x}
_{0} is the geometrical optics field vector on the Gaussian reference sphere due to an *x*-polarised illumination and **ε**^{y}
_{0} is that for a *y*-polarised illumination.

For simplicity we consider **ε**^{x}
_{0} and **ε**^{y}
_{0} separately. Taking first **ε**^{x}
_{0} all we need do is include the additional phase term *ϕ*_{η}
in the Richards-Wolf integral to yield:

$$=-\frac{\mathit{\iota f}}{\lambda}\mathrm{exp}\left(-\iota k{\eta}_{c}n\_\mathrm{tan}{\theta}_{w}\right)\int {\int}_{\Omega}\frac{{\mathbf{\epsilon}}_{0}^{x}\left(\hat{\mathbf{s}}\right)}{{s}_{z}}\mathrm{exp}(\mathit{\iota k}\hat{\mathbf{s}}\bullet (x-f{n}_{-}\mathrm{tan}{\theta}_{w},y,z)d{s}_{x}d{s}_{y}$$

where the constant phase term of exp(*ιkTn*
_{+}) has been omitted for convenience and the substitution *η* = - *f*sin*θ*cos*ϕ* has been made. Then, if ${\mathbf{E}}_{0}^{x}$(**r**) is the field obtained when an *x*-polarised beam is focused *in the absence* of a Wollaston prism, the field obtained with the prism present may be written as:

where *ϕ*_{b}
is the bias defined by *ϕ*_{b}
= -2*kη*_{c}*n*
_{-}tan*θ*_{w}
and the shear *x*_{s}
is defined by *x*_{s}
= 2*fn*
_{-}tan*θ*_{w}
. Similarly, if **E**^{y}
_{0} is the field due to a focused *y*-polarised beam in the absence of a Wollaston prism, the field in the presence of the prism is given by:

The meanings of *ϕ*_{b}
and *x*_{s}
are evident from these expressions: *ϕ*_{b}
represents the phase difference between the two focused beams and *x*_{s}
represents the lateral shift between the two focused beams. Thus the focused field is formed by coherently adding the two focused spots originating from orthogonally polarised beams and mutually displaced. Thus the electric field in the focus of a DIC microscope is given by the coherent sum of Eqs. 5 and 6 with **E**^{x}
_{0} (*x,y,z*) and **E**^{y}
_{0}(*x,y,z*) given by [7]:

where *ϕ* = arctan(*y*/*x*) and the *I*_{n}
integrals are defined as:

$${I}_{1}={\int}_{0}^{\alpha}\sqrt{\mathrm{cos}\theta}{\mathrm{sin}}^{2}\theta {J}_{1}\left(\rho \right)\mathrm{exp}(\iota \Theta )\mathit{d\theta}$$

$${I}_{2}={\int}_{0}^{\alpha}\sqrt{\mathrm{cos}\theta}\mathrm{sin}\theta \left(1-\mathrm{cos}\theta \right){J}_{2}\left(\rho \right)\mathrm{exp}(\iota \Theta )\mathit{d\theta}$$

with *ρ* = *k*$\sqrt{{x}^{2}+{y}^{2}}$, *α* is the semi-convergence angle of the objective and Θ = *kz*cos*θ*.

#### 2.2. Image of a point object

We now describe how to calculate the field at the detector for the optical system shown in Fig. 4(a) for spherical scatterers with radius *r* satisfying the constraint (2*πr*/*λ*)^{2} ≪ 1. In this case the object is regarded as a point scatterer and the scattered field is the same as that of a harmonically oscillating dipole with moment proportional to the field incident upon it [8]. Consider first the optical system shown in Fig. 4(b) which is the same as that in Fig. 4(a) except that it does not have a Wollaston prism. For such a system, the field due to an on-axis dipole of moment **p** is given by [9, 10]:

$${E}_{y,\mathbf{p}}^{\mathit{det}}={p}_{y}\left({I}_{0}^{A}+{I}_{2}^{A}\mathrm{cos}2\varphi \right)+{p}_{x}{I}_{2}^{A}\mathrm{sin}2\varphi -2\iota {p}_{z}{I}_{1}^{A}\mathrm{sin}\varphi \text{}$$

$${E}_{z,\mathbf{p}}^{\mathit{det}}=2\iota \left({p}_{x}\mathrm{cos}\varphi +{p}_{y}\mathrm{sin}\varphi \right){I}_{1}^{B}-2{p}_{z}{I}_{0}^{B}$$

where *ϕ* = arctan(*y*/*x*) with *x* and *y* co-ordinates in the detector plane now and the *I*_{n}
functions are defined as:

$${I}_{0}^{B}={\int}_{0}^{{\alpha}_{2}}\sqrt{\frac{\mathrm{cos}{\theta}_{2}}{\mathrm{cos}{\theta}_{1}}}{\mathrm{sin}}^{2}{\theta}_{2}\mathrm{sin}{\theta}_{1}{J}_{0}\left(\rho \right)\mathrm{exp}(-\iota \Theta ){\mathit{d\theta}}_{2}$$

$${I}_{1}^{A}={\int}_{0}^{{\alpha}_{2}}\sqrt{\frac{\mathrm{cos}{\theta}_{2}}{\mathrm{cos}{\theta}_{1}}}\mathrm{sin}{\theta}_{2}\mathrm{cos}{\theta}_{2}{\mathrm{sin}{\theta}_{1}J}_{1}\left(\rho \right)\mathrm{exp}(-\iota \Theta )d{\theta}_{2}$$

$${I}_{1}^{B}={\int}_{0}^{{\alpha}_{2}}\sqrt{\frac{\mathrm{cos}{\theta}_{2}}{\mathrm{cos}{\theta}_{1}}}{\mathrm{sin}}^{2}{\theta}_{2}\mathrm{cos}{\theta}_{1}{J}_{1}\left(\rho \right)\mathrm{exp}(-\iota \Theta )d{\theta}_{2}$$

$${I}_{2}^{A}={\int}_{0}^{{\alpha}_{2}}\sqrt{\frac{\mathrm{cos}{\theta}_{2}}{\mathrm{cos}{\theta}_{1}}}\mathrm{sin}{\theta}_{2}(1-\mathrm{cos}{\theta}_{1}\mathrm{cos}{\theta}_{2}){J}_{2}\left(\rho \right)\mathrm{exp}(-\iota \Theta )d{\theta}_{2}$$

where Θ = *kz*_{d}
cos*θ*
_{2}, *ρ* = *k*$\sqrt{{x}^{2}+{y}^{2}}$, *z*_{d}
is the axial position of the dipole and *α*
_{2} is the semi-convergence angle of lens L_{2}.

We must now make two modifications to these expressions to include the effect of the Wollaston prism. One could re-derive the detector field including the Wollaston prism as a phase function in a manner similar to that for the illumination field. It is however possible to make some simplifying approximations. Since lens L_{2} is generally of low NA, the longitudinal component of the detector field will be weak compared to the transverse components. Furthermore, it may be assumed that the *x* component of the detector field may be found via an application of a Fourier transform on the *x* component of the field at the detector back focal plane and likewise for the *y* component [11]. Then using the shifting property of Fourier transforms it can be shown that the detector field for the optical system shown in Fig. 4(a), for a dipole of moment **p** at position (*x*_{d}
,*y*_{d}
,*z*_{d}
) is given by:

$${E}_{y}^{\mathit{det}}={E}_{y,\mathbf{p}}^{\mathit{det}}\left(x+\beta \left({x}_{d}+\frac{{x}_{s}}{2}\right),y+\beta {y}_{d},z\right)\mathrm{exp}\left(\frac{-\iota {\varphi}_{b}}{2}\right)$$

$${E}_{p}^{\mathit{det}}=0$$

where *β* = *f*
_{2}/*f*
_{1} = sin*θ*
_{1}/ sin*θ*
_{2} is the lateral magnification of the optical system. Because of the linearity of Fourier transforms, the analyser may be modeled as if it were applied directly to
the field at the detector. Once the detector field is found, the detector signal may be calculated from:

where *D* is the sensitivity function of the detector of surface *S*. The arrangement analysed here is frequently referred to as a “confocal” set-up even though strictly speaking only the application of a point-like detector would make it such. Conversely, when a large detector is employed the same arrangement is often called a “conventional” set-up.

## 2.3. Image of a general object

Point objects are of limited interest to the study of DIC microscopy. Furthermore, DIC microscopy relies upon interference between scattered light and directly transmitted or reflected light. This is difficult to model without knowing the relative intensities of scattered and directly transmitted or reflected light. General objects that may be encountered practically are much more complex than point objects. Solving for the field scattered by such objects is in general difficult to do analytically. This is because analytic solutions to Maxwell’s equations seldom exist for inhomogeneous regions with even the simplest of boundary conditions. The only option is thus to employ rigorous numerical techniques to solve for the light scattered by arbitrary objects. We have implemented the FDTD method [12] for this purpose. We have also implemented a method for calculating scattering of a focused beam by spheres of arbitrary radius. These two methods are discussed in the following sections.

#### 2.3.1. The FDTD method

It is beyond the scope of this paper to give a detailed explanation of the FDTD method however a brief introduction is given here. Maxwell’s equations result in a set of coupled partial differential equations, one example from this set, for a source free region, is:

where *σ* is the conductivity and ∊ is the permittivity of the region.

All numerical methods require discretisation of field values throughout space. The FDTD method employs discretisation reported by Yee [13]. The field quantities on the Yee cell are described by an indexing system of the form (*i*,*j*,*k*) which corresponds to a position (*i*Δ*x*,*j*Δ*y*, *k*Δ*z*) where Δ*x*, Δ*y* and Δ*z* are the physical dimensions of the Yee cell.

The field values must also be discretised in time and they are calculated at intervals of a specifically chosen time step Δ*t*. The electric and magnetic field values are however known half a time step apart. This allows an indexing scheme for time such that time of index *n* refers to real time *n*Δ*t*. Using this system, Yee showed how each partial differential equation of the form of Eq. 12 can be approximated accurately by a difference equation of the form [12]:

$$-\frac{{H}_{y}{\mid}_{i,j+1/2,k+1}^{n}-{H}_{y}{\mid}_{i,j+1/2,k}^{n}}{\Delta z})$$

and similarly for other components of **E** and **H**. Note that *α*
_{i,j+1/2,k+1/2} and *β*
_{i,j+1/2,k+1/2} are functions of Δ*t* and material properties at location (*i*,*j* + 1/2,*k* + 1/2) and that the superscripts *n*+1/2,*n* and *n*-1/2 are time indices. The set of difference equations allows an incident field to be introduced to the computational grid and the fields leap frogged in time. If the incident field is introduced as a Gaussian pulse and the resulting field given time to decay to be negligibly small, the scattered field at the center wavelength of the pulse may be found from the time domain data through an application of a discrete Fourier transform.

The FDTD method may be used to calculate light from focused beams scattered by arbitrary objects. The primary limitation on it is the type of objects which can be accurately modeled. For example, the Yee cell size must be no larger than the smallest feature to be modeled. This can lead to enormous memory requirements when modeling large objects with fine details. Curved and angled surfaces are also poorly represented due to the stair casing method of modeling objects.

#### 2.3.2. Calculation of scattering by a sphere using Mie series

Due to the difficulty of modeling curved surfaces in the FDTD method, an additional method is used for calculating the field scattered by an arbitrary sized, conducting or dielectric sphere. The method, first described by Török *et. al* [14], combines the Richards-Wolf integral with the Mie series [15] solution to scattering of a plane wave by a sphere. The Richards-Wolf integral expresses the focused field as a truncated angular spectrum of plane waves and so it is possible to find the total scattered field for a focused wave by summing the scattered field corresponding to each plane wave component.

In the case of linearly polarised light it is possible to obtain a simple expression for the polarisation of each plane wave using the generalised Jones matrix formalism [9]. The end result is that each plane wave, propagating in direction **ŝ**, takes the form:

where each term above retains the same meaning as in Eq. 3. Our numerical implementation of the Mie series, based upon that of Bohren *et al*. [15] provides the solution to the field scattered for a plane wave incident along the *z*-axis polarised in the *x*-direction. Thus, for each plane wave component, it is necessary to transform all observation points into a coordinate system with its *z*-axis parallel to the direction of propagation **ŝ** and *x*-axis parallel to the polarisation of the plane wave. Field values must be transformed back to the original coordinate system. A different transformation is required for every plane wave component. Application of the generalised Jones matrix formalism [9] shows that the transformation matrices required for calculating scattering by *x* and *y* polarised beams, T_{x} and T_{y}, may be conveniently expressed through Euler matrices [16] as:

$${T}_{y}={R}_{z}\left(\frac{\pi}{2}-\varphi \right){R}_{y}\left(\theta \right){R}_{z}\left(\varphi \right)$$

where

shear and bias are modeled by shifting and biasing each plane wave component. The scattered fields from the two polarisation cases are summed to give the total scattered field.

#### 2.3.3. Detection of scattered light

Having calculated the scattered light it must be imaged onto the detector. This is done by first constructing a plane of equivalent dipoles, normal to the optical axis, in the vicinity of the focus of the objective lens. Each dipole has moment proportional to the value of the field incident upon it. The detector field due to each equivalent dipole is calculated and the total detector field is found by summing the contribution of each dipole.

The plane of dipoles must be sufficiently wide in order that most of the energy of the scattered field propagates through it. It must also be in the far field region of the scattered field. This is not a problem for the case of a spherical scatterer as the field may be calculated at any observation point when the Mie series method is employed. When the FDTD method is employed the scattered field is calculated on a cuboid surface very close to the scatterer with faces usually insufficiently wide to be used as the equivalent dipole surface. In this case we employ a near-to far-field transformation similar to the Stratton-Chu integral [17].

## 3. Numerical Results

#### 3.1. Illumination and point spread function

The complex nature of the polarisation of the field in the vicinity of the focus a high NA lens is now well understood (see for example Richards *et al*. [7]). This has important implications for high NA scanning DIC microscopes. Previous models of DIC microscopes [1, 2, 3, 4, 5, 6] assume scalar models for the illumination field. The following example demonstrates that at high NA it is essential to use vectorial analysis for the illumination field.

The plots in this paper employ a right handed Cartesian co-ordinate system with origin in the Gaussian focus of the lens with no Wollaston prism employed. When a scanned image or line scan is presented, the center of the coordinate system coincides with the scatterer. Figure 5 shows various field components associated with the incident field of a DIC microscope with NA=0.95 (dry), *λ*=632.8nm, *ϕ*_{b}
=0 and *x*_{s}
= 0.5.

Note that the shear is quoted in normalised units of *x*_{s}
= 2*r*NA/*λ*. A shear of 0.5 is considered to provide a good trade-off between image quality and image luminance [2]. Recalling Eqs. 5 and 6 now for zero bias, the top row of images in Fig. 5 show the components of ${\mathbf{E}}_{0}^{x}$ (*x* - *x*_{s}
/2,*y*,*z*) and the middle row shows the components of ${\mathbf{E}}_{0}^{y}$(*x* + *x*_{s}
/2,*y*,*z*). The bottom row shows the coherent sum of these two. These images reveal two important points. The first point is that each of the focused beams possesses cross-polar terms. This leads to interference between the two focused beams as, for example, the *y* component of ${\mathbf{E}}_{0}^{y}$(*x* + *x*_{s}
/2,*y*,*z*) may interfere with the *y* component of ${\mathbf{E}}_{0}^{x}$(*x* - *x*_{s}
/2,*y*,*z*) and vice versa, resulting in a focal distribution significantly different from that predicated by scalar analysis. The second and perhaps more important point is the presence of a strong longitudinal field component yields profound differences in imaging when compared to scalar description.

The intensity of the electric field incident upon the sample is shown in Fig. 6. The distributions are quite different from the intensity pattern of two superimposed spots that may have been expected for a scalar description. An interesting feature to note is that the intensity of the magnetic field has a distribution complementary to that of the electric field leading to an electromagnetic energy density with the symmetry one would expect as is shown in Fig. 6.

In order to further probe the imaging properties of the confocal DIC, the image of a point scatterer for a point-like detector, i.e. the point spread function (PSF) was calculated for the reflection case and the result is shown in Fig. 7.

The same parameters as were used in the illumination example are used again here with the exception that the PSF was calculated for *ϕ*_{b}
= *π*/2 as well as *ϕ*_{b}
= 0. The detection system had transverse magnification of 100×. The PSF for *ϕ*_{b}
= 0 has significant asymmetries that one would not expect and is not predicted by scalar theory [2]. This is due to the nature of the illumination field as previously discussed. This example thus underlines that even for the simplest of objects vectorial analysis is required when modeling high NA scanning DIC microscopes.

#### 3.2. Image of a dielectric sphere

To demonstrate the power of the DIC imaging model, the DIC image of a dielectric sphere was calculated. The sphere refractive index was varied between 1 and 2 and was embedded in a medium with refractive index 1.5. The sphere had a radius of *λ*
_{0}/4 with the wavelength of the illumination, *λ*
_{0}, equal to 632.8 nm. A 1.3 NA (oil immersion) 100× objective was employed. The image was calculated for a range of values of *ϕ*_{b}
and the shear was kept constant at 0.5.

The animation in Fig. 8 demonstrates the salient features of the model which employs the Mie theory to calculate light scattered by the dielectric sphere. This Fig. shows the case of *ϕ*_{b}
= 0 and sphere refractive index 1. The left most set of axes in Fig. 8 show the sphere as it is scanned through the focused illumination. Note that this is not intended to depict how the sphere interacts with the incident field. The middle diagram reveals how the total field distribution (i.e. directly incident and scattered) looks like one wavelength beyond the center of the sphere in the axial direction. The field at this plane is used to construct the equivalent dipole plane which is in turn used to calculate the field at the detector shown in the right most set of axes. Results for other refractive indices are not shown as the general form of the image changes little with the refractive index of the sphere.

An image is built up by recording the detector signal at each scan position. Figure 9 is an animation of confocal images on the *xy* and *xz* planes, both through the origin, of a dielectric sphere for values of *ϕ*_{b}
ranging from 0 to *π*/2. The sphere had a refractive index 1. Note that two sets of axes have been used simply to enable both halves of both planes to be visualised. This animation illustrates the imaging properties of a confocal DIC microscope, in particular the three typical regimes of *ϕ*_{b}
=0, *π*/4 and *π*/2. Each frame has been normalised separately to enhance the image contrast.

The plot in Fig. 10 shows a line scan along the *x*-axis for the above case with *ϕ*_{b}
, taking the values 0, *π*/4 and *π*/2. Scans for confocal detection and conventional detection (100 *μm* radius detector) are shown. Note that each line scan has been normalised by the detector signal obtained in the absence of a scatterer in the case of *ϕ*_{b}
=*π*/2 for the detector type used. Consider first the case of *ϕ*_{b}
=0. The confocal and conventional scans differ substantially. This is to be expected from existing theory of confocal microscopy. Since the directly transmitted light is rejected, the detector signal is composed only of the field scattered by the dielectric sphere. This scattering results in two distinct field distributions, one due to the focused *x*-polarised beam and one due to the *y*. These distributions are permitted to interfere and the well known difference in resolution between conventional and confocal microscopes [9] leads to the substantial difference in the DIC image of a dielectric sphere for *ϕ*_{b}
= 0 as shown in Fig. 10. The asymmetry in this plot is due to the asymmetry in the components of the incident field as illustrated in Fig. 5.

The case of *ϕ*_{b}
=*π*/4 exhibits the differential signal typical of DIC images. There is very little difference between the confocal and conventional line scans. The case of *ϕ*_{b}
=*π*/2 does however reveal a significant difference between the confocal and conventional line scans. The distribution for the confocal case is broader than the conventional case suggesting that the resolution for the conventional case is better than for the confocal case. This is contrary to the case in bright field microscopy [9]. This observation may be explained by noting that the detector signal is composed of scattered and direct field components. Due to the polariser before the detector lens the *x* and *y*-components of the detector field are equal in magnitude but opposite in sign. Thus, since it was assumed that the *z*-component of the detector field is 0, we may describe the field at the detector as the sum of two scalar quantities *E* = *E*_{sc}
+ *E*_{d}
where *E*_{sc}
and *E*_{d}
represent the scattered and direct components of the field respectively. The detector signal, *I*_{det}
, may then be written from Eq. 11 as:

where ℜ denotes the real part, ^{*} denotes complex conjugation. The terms *I*_{sc}
, *I*_{d}
and *I*_{int}
represent the scattered, direct and interference components of the detector signal.

Figure 11 shows plots of *I*_{int}
+ *I*_{d}
and *I*_{sc}
as well as *I*_{tot}
for the conventional and confocal cases for *ϕ*_{b}
= *π*/2. The interference term (*I*_{int}
) appears to be independent of the detector size. The scattered component *I*_{sc}
does however vary with the detector size though this is well understood from the theory of bright field confocal microscopy [9]. The narrower scattered component of the confocal case simply leads to a broader total distribution when added to the interference term. This suggests that it may be better to employ a large detector when performing DIC imaging for large bias values. This is very similar to a phenomenon observed in phase contrast microscopy [18, 10].

#### 3.3. Step like surface profile features

As a final set of examples we show the image of step like features created in perfectly conducting material. In all of the following simulations, a wavelength of *λ*=632.8 nm and a 0.85NA (dry), 100× objective was employed. The objects under study have surface profile variation in only one direction thus only line scans are shown. This is not due to a restriction in our model that would permit the computation of an object of arbitrary complexity but rather our attempt to simplify this demonstration. The first example considers the confocal DIC line scan across a ridge of height Δ = *λ*/4 and width *λ*/2. The top of the ridge was positioned in the plane *z*=0 the plane normal to the optical axis containing the geometrical focus of the objective.

Figure 12 shows an animation which shows how the field scattered by the ridge changes as the ridge is scanned beneath the beam and also the field at the detector. As has been explained previously, the scattered field shown in Fig. 12 is imaged onto the detector by the application of two further steps. Figure 13 shows DIC microscope line scans for the ridge for bias values of *ϕ*_{b}
=0, *π*/4 and *π*/2 for both a point detector (confocal) and a 100*μ*m radius detector (conventional). Consider first the case of *ϕ*_{b}
=0. The detector signals for the respective detectors have been normalised by the detector signal which would be measured if the ridge was removed with *ϕ*_{b}
=*π*/2, leaving only a mirror defocused by Δ. It is evident that the confocal case achieves better extinction, i.e., the detector signal is very small in the absence of an edge. The other two cases demonstrate that confocal detection is more sensitive than conventional detection. One must however remember that the reduced signal strength, and therefore degraded signal to noise ratio, obtained when using confocal detection may offset the observed benefits of using confocal detection.

We have also modeled a single step object. The step height, *x*_{s}
and *ϕ*_{b}
were all varied. In each case the step was aligned with the *y*-axis of the object space and scanned along the *x*-direction The step was constructed such that when it is located at *x* = 0, the surface of the step is at *z* = 0 for *x* < 0 and *z* = Δ for *x* > 0.

Figure 14 shows the confocal detector signal for three different values of *ϕ*_{b}
and five different values of Δ. Each signal is normalised by the detector signal obtained for a mirror placed at *z* = Δ for *ϕ*_{b}
=*π*/2. Each successive step height varies by only *λ*/20 yet the detector signals show significant contrast between the various step heights. Note that in plots (a) and (b), the signal for *x* > 0 is in general lower than for *x* < 0. This is because for *x* > 0 the incident beam is essentially reflecting off a defocused mirror and so a lower signal is measured by the detector.

Next consider the plots in Fig. 15 which show the confocal detector signal as the shear is altered between 0.1 and 0.5. The step height used was Δ = *λ*/4. Again, each detector signal is normalised by the signal obtained with a mirror placed at *z* = Δ using *ϕ*_{b}
= *π*/2. The plots indicate that for a small value of *ϕ*_{b}
the smaller *x*_{s}
reduces the signal strength for a given step height. For larger values of *ϕ*^{b}
, approaching *π*/2, the shear doesn’t seem to affect the signal strength significantly. It is of course expected that the resolution of the system reduces for larger values of shear.

## 4. Conclusions

We have shown that vectorial analysis is vital to the study of high numerical aperture DIC microscopes because it reveals significant deviations from the scalar analysis. This is due to the complex nature of polarisation in the vicinity of the focus of a beam focused by a high numerical aperture lens and also due to scattering by arbitrary objects being vectorial in nature. We have also shown that in transmission light DIC microscopy a detector pinhole improves resolution for low bias values but actually degrades resolution for bias values approaching *π*/2. This is a particularly important observation given the difficulty of pinhole alignment in confocal microscopy. The amazing sensitivity to surface profile variations has been demonstrated thus providing the basis for a method of performing a rigorous examination of surface profile resolution in scanned DIC microscopy. Finally, it was shown that confocal scanned DIC microscopy is more sensitive to surface profile variations than conventional DIC scanning microscopes.

## Acknowledgments

The discussions on the FDTD program with Dr Emmanouil Kriezis are gratefully acknowledged. Dr Kriezis was also kind enough to provide us with a working version of his FDTD code that helped up to develop our version.

## References and links

**1. **D. Lessor, J. Hartman, and R. Gordon, “Quantitative surface topography determination by Nomarski reflection microscopy. I. Theory,” J. Opt. Soc. Am. **69**, 357–366 (1979). [CrossRef]

**2. **W. Galbraith, “The image of a point of light in differential interference contrast microscopy: Computer simulation,” Microsc. Acta **85**, 233–254 (1982).

**3. **T. Holmes and W. Levy, “Signal-processing characteristics of differential interference-contrast microscopy.” Appl. Opt. **26**, 3929–3939 (1987). [CrossRef] [PubMed]

**4. **C. J. 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]

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

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

**7. **B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. (London) A **253**, 358–379 (1959). [CrossRef]

**8. **M. Born and E. Wolf, *Principles of Optics*, seventh ed. (Cambridge University Press, Cambridge, 1999).

**9. **P. Török, P. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. **45**, 1681–1698 (1998). [CrossRef]

**10. **P. Munro and P. Török, “Vectorial, high-numerical-aperture study of phase-contrast microscopes,” J. Opt. Soc. Am. A **21**, 1714–1723 (2004). [CrossRef]

**11. **P. Török, P. Higdon, and T. Wilson, “On the general properties of polarising conventional and confocal microscopes,” Opt. Commun. **148**(4–6), 300–315 (1998). [CrossRef]

**12. **A. Taflove and S. Hagness, *Computational electrodynamics*, second edition (Artech House, 2000).

**13. **K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**(3), 302–307 (1966). [CrossRef]

**14. **P. Török, P. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers,” Opt. Commun. **155**(4–6), 335–341 (1998). [CrossRef]

**15. **C. Bohren and D. Huffman, *Absorption and scattering of light by small particles* (Wiley Interscience, 1983).

**16. **G. Arfken, *Mathematical Methods for Physicists*, 3rd ed. (Academic Press, Boston, 1985).

**17. **P. Török and C. Sheppard, *High numerical aperture focusing and imaging* (Adam Hilger, (to pe published)).

**18. **P. Munro and P. Török, “Effect of detector size on optical resolution in phase contrast microscopes,” Opt. Lett. **29**, 623–625 (2004). [CrossRef] [PubMed]