## Abstract

Performance of two types of differential interference contrast objectives, i.e., the XOR pattern and the zone-plate doublet, is quantitatively characterized and compared using modulation transfer function. Effects of partial coherence, finite absorption and phase in a complex object, as well as bias retardation are also examined.

© 2011 OSA

## 1. Introduction

High resolution soft x-ray microscopy is a useful analytic tool in nanosciences [1–3]. Its nanoscale resolving power, combined with available x-ray phase imaging techniques, has been successfully used to image otherwise transparent samples with nanometer resolution [4–9]. Comparing with other phase contrast methods available at the x-ray wavelengths such as ptychography [10, 11], interferometry [12–14], and Zernike phase plates [7–9], x-ray differential interference contrast (DIC) microscopy is advantageous in that DIC objectives can be readily implemented in existing x-ray microscopes, thus forgoing dedicated experimental apparatus [4–6]. In addition, x-ray DIC microscopy has been demonstrated with both synchrotron and compact plasma x-ray sources, making it accessible to a wider range of users [15, 16]. Currently, two types of DIC objectives, i.e., the exclusive OR (XOR) patterns [4, 16, 17] and the zone-plate doublets (ZPD) [5, 18, 19], have been demonstrated. In both objective designs, arbitrary bias retardation and lateral shear can be incorporated to improve image contrast. The XOR pattern’s fabrication method is similar to that of a Fresnel zone-plate (FZP) and as a result the same finest outermost zone width, thus spatial resolution, can be achieved. ZPD on the other hand can be used for non-iterative quantitative x-ray phase imaging [19], although its nanofabrication process requires a more stringent placement accuracy. Figure 1 schematically illustrates the difference in imaging performance between conventional FZP, XOR, and ZPD objectives for a complex sinusoidal grating object. A conventional FZP reproduces the object’s intensity, defined as the absolute square of the object transmittance function; and the effect of object’s phase is not reflected in its image. In contrast, XOR and ZPD use the object’s phase, in addition to its absorption, for their image formation and therefore higher contrast can be obtained. Clearly, XOR and ZPD based imaging systems are more effective in generating image contrast for complex objects.

Despite the usefulness of x-ray DIC microscopy, little effort has been devoted to quantify its imaging performance. Modulation transfer function (MTF) is the standard evaluation criterion for imaging system responses, and it has been widely used to quantify absorption-based microscopes [20, 21]. An absorption grating is typically used as the object for MTF analyses in the case of absorption contrast microscopes. Evaluation of phase contrast microscopes requires either a pure phase object with no absorption [22, 23] or a complex object which introduces both phase shift and absorption to the transmitted wave-front. Since most biological and magnetic samples have complex refractive indices at the soft x-ray wavelengths, analyses using complex grating objects render more realistic performance characterization of x-ray phase contrast microscopes. Theoretical analyses using complex objects exist only for conventional absorption contrast and Zernike phase contrast microscopes [24, 25], but not for DIC microscopes. Following the classical approach [20, 26, 27], here we systematically quantify and compare the performance of XOR and ZPD based x-ray DIC microscopes using MTF. The DIC microscopes studied here are configured to be space invariant in order to allow for the exact transfer function analysis without approximation [28, 29]. Performance of the DIC microscopes are evaluated with complex grating objects using varying levels of partial coherence and bias retardation.

## 2. Modulation Transfer Function for X-ray Microscopy

#### 2.1. Mathematical Description of the Image Formation Process on an X-ray Microscope

A typical partially coherent cascade imaging system is illustrated in Fig. 2. Coordinate systems for the source, object, objective lens, and the image planes are denoted by (*α*, *β*), (*$\tilde{\xi}$*, *$\tilde{\eta}$*), (*x,y*), and (*u*, *v*) and the distances between the source, condenser, object, objective lens, and image are *z*
_{1}, *z*
_{2}, *z _{o}*, and

*z*, respectively. When the relative positions of the condenser, object, and objective lens are properly set, this partially coherent imaging system becomes exactly space invariant and the resultant image intensity distribution can be analytically expressed in terms of transfer functions without low coherence approximation and additional phase correcting lenses [28, 29]. This illumination apparatus is especially useful for x-ray DIC microscopy where a high degree of illumination coherence is generally required. Image intensity distribution in this case is given by [27, 28],

_{i}*, 𝕂, and 𝕋*

_{o}*are the Fourier spectra of the illumination mutual intensity*

_{o}**J**

*, the space invariant amplitude spread function*

_{o}**K**, and the object transmittance function

**t**

*, respectively. Note that following prior classical MTF analyses [20, 27], a circular finite extent incoherent source*

_{o}*I*(

_{s}*α*,

*β*) is used here to illuminate the microscope, i.e.,

*I*is a constant and

_{o}*R*is the source radius. Fourier spectrum 𝕁

_{s}*of this source is thus,*

_{o}*λ*is the wavelength,

*M*=

*z*/

_{i}*z*is the magnification,

_{o}*f*is the focal length of the condenser, and

_{c}*κ*≡

*λ*

^{2}/

*π*. Similarly, we have

**P**is the pupil function of the objective.

#### 2.2. Pupil Functions of the Objective Lenses

The objective lens plays a critical role in the image forming process of the microscope. Three types of objective lenses, i.e., Fresnel zone-plates (FZP) [21, 30], exclusive OR (XOR) patterns [4, 31], and zone-plate doublets (ZPD) [19], are examined here and their pupil functions **P**(*x,y*) are summarized in Table 1. Note that 2Δ*θ* is the bias retardation, Δ*s* is the lateral shear, and the parameter *γ* is defined as *π*/[Δ*r*(2*R _{o}* – Δ

*r*)] where Δ

*r*and

*R*are the outermost zone width and radius, respectively. These pupil functions are used in Eq. (4) for simulations performed in the following sections. Figure 3 shows examples of XOR and ZPD objectives with bias retardation values of 2Δ

_{o}*θ*= 0,

*π*/2, and

*π*, and Fig. 4 shows their corresponding pupil functions. For comparison, a conventional FZP objective and its pupil function are also shown in the figures. Note that the pupil function of a visible-light DIC microscope is identical to that of ZPD [19, 32], and the MTF analyses presented here for ZPD are also valid for visible-light DIC microscopy.

#### 2.3. Periodic Objects: Gratings

Gratings that introduce both absorption and phase variations are needed in order to evaluate the performance of DIC microscopes. Here three types of gratings, i.e., absorption sinusoidal, phase sinusoidal, and complex sinusoidal (Fig. 5), are used and their transmittance functions **t**
* _{o}* as well as Fourier coefficients

*a*are summarized in Table 2 (Detailed derivations in Appendix A). Note that the refractive index

_{n}*ñ*at the soft x-ray spectral region is in general complex and typically denoted by

*ñ*= 1 –

*δ*+

*jβ*where (1 –

*δ*) determines the phase shift introduced and

*β*the absorption. Denoting the maximum thickness variation of the complex sinusoidal grating as

*t*and the attenuation length as

_{o}*t*= 1/(2

_{a}*kβ*) where

*k*= 2

*π*/

*λ*is the vacuum wavenumber, the parameter

*η*is defined as the ratio of

*t*to

_{o}*t*. Characteristics of the complex object can then be modeled by varying the values of

_{a}*δ*/

*β*and

*η*.

#### 2.4. Implementing the Numerical Evaluation of the Image Integral

Given a grating with period *T*, its object transmittance function **t**
* _{o}*(

*ξ′*,

*η′*) can be expressed in a Fourier series as,

*ν*≜ 1/

_{o}*T*is the fundamental spatial frequency of the grating and

*a*is the

_{n}*nth*order Fourier coefficient (Table 2). Note that Eq. (5) is expressed in the magnified coordinate (−

*M$\tilde{\xi}$*, −

*M$\tilde{\eta}$*) = (

*ξ′*,

*η′*), representing the geometric image of the object. Fourier transform of the object transmittance function

**t**

*(*

_{o}*ξ′*,

*η′*) is given by,

*δ*is the Dirac delta function.

Image intensity distribution of this periodic grating formed by a microscope can therefore be obtained by substituting 𝕋* _{o}* into Eq. (1), i.e.,

The integral in Eq. (7) depends solely on the properties of the imaging system, i.e., the illumination 𝕁* _{o}* and the amplitude spread function 𝕂 of the microscope, while

*a*and

_{m}*a*are the Fourier coefficients of the object being imaged. Since 𝕁

_{n}*and 𝕂 are given by Eq. (3) and Eq. (4) as scaled versions of*

_{o}*I*and

_{s}**P**, respectively, the integral can be numerically evaluated by overlaying scaled

*I*,

_{s}**P**, and

**P***, i.e.,

To simplify the notations, we define

*ν*is the cutoff frequency and

_{c}*ν*is the extent of source angular distribution [33]. The effect of partial coherence can be characterized by the

_{s}*σ*value, defined as [27, 34], As

*σ*increases, the degree of illumination coherence on the object decreases. Coherent and incoherent imaging processes are therefore denoted by

*σ*= 0 and

*σ*= ∞, respectively. The normalized object spatial frequency

*α*is defined by normalizing the object’s fundamental spatial frequency

*ν*with respect to the cutoff frequency

_{o}*ν*, i.e., where

_{c}*M*is the magnification. Note that

*ν*is the fundamental spatial frequency of the grating as seen in the image plane while the actual grating spatial frequency is

_{o}*Mν*.

_{o}#### 2.5. Modulation Transfer Function and Visibility

Modulation transfer function (MTF) of an optical imaging system is defined by,

*V*(

_{image}*α*) and

*V*(

_{object}*α*) are the visibilities of the image and object, respectively [27]. Note that the object’s intensity is defined as the absolute square of the object transmittance function, i.e., |

**t**

*(*

_{o}*ξ′*,

*η′*)|

^{2}, and the visibility for a phase object is therefore zero by definition (see Appendix B). As a result, MTF in Eq. (13) cannot be used in this case, and instead image visibility

*V*is used directly to evaluate the performance of the imaging systems when imaging pure phase objects.

_{image}## 3. Quantitative Characterization using Modulation Transfer Function

Numerical evaluations of the performance of XOR and ZPD based x-ray DIC microscopes are presented in this section. Bias retardation values 2Δ*θ* = 0, *π*/6, *π*/2, 2*π*/3, and *π* are used for both DIC objectives. Lateral shear Δ*s* is fixed at Δ*s* = 2Δ*r*, where Δ*r* is the outermost zone width, in all cases. Effect of partial coherence is examined by varying the value of *σ*. Performance of the FZP objective is also shown for comparison.

Each grating is characterized by its periodicity, cross-sectional shape, and its material properties, i.e., the real and imaginary parts of the refractive index *ñ*. A sinusoidal absorption grating is imaged with degree of partial coherence varying from *σ*= 0 to 2. Phase sinusoidal gratings with phase variations from *φ* = 0 to *φ* = *π*/3 are examined under *σ* = 0, 0.1, 0.2, and ∞. Note again that for phase objects, image visibility *V _{image}* is used instead of MTF since the object’s visibility

*V*is zero by definition (see Appendix B). Complex objects with both phase and absorption effects are also examined, noting that such complex objects more realistically resemble samples at the x-ray wavelengths. Performance of the imaging system is examined by varying the magnitude of the object’s phase variation while maintaining a constant absorption, i.e., varying the value of

_{object}*δ/β*while keeping the value of

*η*constant. Specifically, the

*δ/β*value is increased from 0 to 10 while the value of

*η*is fixed at 0.1 in order to isolate the effect of phase on image contrast.

#### 3.1. Absorption Sinusoidal Grating/Absorption Objects

We first examine the sinusoidal absorption grating which has only three diffraction orders, i.e., 0 and ±1. Under coherent illumination, the ±1 diffraction orders are allowed to pass the pupil if *α* < 1. With FZP, the pupil acts as a low-pass filter allowing all harmonics below the cut-off frequency *ν _{c}* to pass without modification. Therefore, within

*α*< 1, the object’s waveform is reconstructed without loss of information on the image plane (Fig. 6(a)) [20]. On the other hand, performance of XOR and ZPD depends on their bias retardation since the ±1 diffraction orders are modified by the patterns within their pupils. Figures 6(b)–6(f) and Figs. 7(b)–7(f) respectively show MTFs obtained using XOR and ZPD under coherent illumination with various bias retardation values. These results show that XOR and ZPD are not optimized for absorption objects since their MTF is in general lower than that of FZP. For XOR, the low MTF region gradually widens as Δ

*θ*increases from 0 to

*π*/2, while the low MTF region for ZPD remains fixed for 0 < Δ

*θ*<

*π*/2. Note that XOR and ZPD with Δ

*θ*=

*π*/2 always result in

*MTF*= 1 for

*α*< 1 because in these cases value of the pupil function at the origin is zero, and the zeroth order is consequently eliminated. As a result, minimum intensity in the image is suppressed to zero and MTF becomes one for

*α*< 1. Since the pupil function of XOR consists of a binary grating while that of ZPD has a cosine structure (Fig. 4), the transitions between low and high MTF regions are steep for XOR and smooth for ZPD.

For partially coherent illumination, i.e., *σ* > 0, non-zero MTF values are observed beyond *α* = 1 as expected. Again, for absorptive objects performance of the XOR and ZPD are inferior to FZP especially for XOR with large Δ*θ* values (Figs. 6 and 7). Effect of bias retardation on MTF is relatively small when Δ*θ* is smaller than *π*/12 for ZPD, as can be seen by comparing Figs. 7(b) and 7(c). Overall, DIC microscopy is not suited for imaging absorption objects and the use of a conventional x-ray microscope is recommended for such objects.

#### 3.2. Phase Sinusoidal Grating/Phase Objects

In this section, we use image visibility *V _{image}* to evaluate the performance of DIC microscopes for pure phase objects. The first columns of Figs. 8 and 9 show the image visibilities of a phase sinusoidal grating under coherent illumination. In general, a phase object is nearly invisible with FZP when its phase variation magnitude

*φ*is small. However, such phase grating objects can become visible even with FZP when

*φ*is large, as shown in Fig. 8(a). This is due to the fact that, unlike the absorption sinusoidal grating which has only three diffraction orders, the phase sinusoidal grating has an infinite number of diffraction orders (Table 2). When

*α*is small, i.e.,

*α*< 0.5, a large number of diffraction orders pass though the pupil in the image plane. This allows the pure phase object to be reconstructed without loss of information. Consequently, the image, as a higher fidelity representation of the phase object, lacks contrast in intensity. As

*α*increases beyond 0.5, the finite extent of the pupil prevents the passing of higher order harmonics. As a result, only lower order harmonics contribute to the reconstructed image and the image becomes visible at the cost of missing higher order harmonics. However, image visibility in this case remains relatively low (

*V*= 0.43) even for

_{image}*φ*=

*π*/3.

As seen in Figs. 8(b)–8(f) and Figs. 9(b)–9(f), XOR and ZPD under coherent illumination are able to produce large image contrast for phase objects. Bias retardation is essential for the performance of XOR and ZPD. Certainly, given the same phase object, the resultant image visibility varies significantly with different Δ*θ* values and larger Δ*θ* generally results in higher image visibility over a wider range of *α*. As in the case of absorption objects, the transitions between low and high image visibility regions are steep for XOR and smooth for ZPD. Also note that XOR and ZPD with Δ*θ* = *π*/2 always yield *V _{image}* = 1 below the cutoff frequency due to the elimination of the zeroth order component.

Image visibility values *V _{image}* of a phase sinusoidal grating under partially coherent illuminations for

*σ*= 0.1,

*σ*= 0.2, and

*σ*= ∞ are also shown in Figs. 8 and 9. The image visibility gradually decreases with increasing

*σ*and completely fades at

*σ*= ∞. The range of high

*V*region depends on bias retardation and in general larger Δ

_{image}*θ*values produce higher

*V*for a wider range of

_{image}*α*. Note that the XOR with Δ

*θ*=

*π*/2 has lower visibility overall as shown in Fig. 8(f).

#### 3.3. Complex Sinusoidal Grating

A complex sinusoidal grating is used to examine the effect of absorption and phase concurrently. Results presented in the previous two sections show that XOR and ZPD with proper bias retardation are able to produce relatively large image contrast for sinusoidal phase gratings, whereas FZP is suited for absorption objects. By varying the value of *δ/β* from 0 to 10 while fixing *η* at 0.1, MTFs of the XOR and ZPD are examined and compared. Note that for *δ/β* = 0, the object introduces no phase effect and as a result characteristics of an absorption sinusoidal grating is observed as expected. As the *δ/β* value increases, the resultant images reflect features of both absorption and phase gratings.

MTFs under coherent illumination for complex sinusoidal objects are shown in the first columns of Figs. 10 and 11. Observed image intensity modulation in the FZP imaging system is mostly due to the absorption of the complex sinusoidal grating and the effect of the phase variation is negligible. As a result, the MTF value of FZP remains around unity even for *δ/β* = 10 (Fig. 10(a)). Similarly, image intensities obtained using XOR and ZPD with Δ*θ* = 0 barely reflect the presence of phase variations in the complex object, as shown in Figs. 10(b) and 11(b). These observations agree with the results presented in the previous section, i.e., phase effect on the image intensity is generally negligible for XOR and ZPD with Δ*θ* = 0. For 0 < Δ*θ* < *π*/2, image intensity of the complex object with large *δ/β* is visibly enhanced by XOR and ZPD. MTF values as high as 8.45 for XOR and 8.15 for ZPD can be obtained for the complex object with *δ/β* = 10 (Figs. 10(d)–10(f) and Figs. 11(c)–11(f)). Note that the values of MTF can be greater than 1 for complex objects imaged by DIC objectives because DIC image contrast in this case is enhanced by the object’s phase variations. As Δ*θ* increases, high MTF region widens and eventually extends up to the theoretical resolution limit, i.e., *α* = 1, for both XOR and ZPD. Note that both XOR and ZPD with Δ*θ* = *π*/2 produce *MTF* = 10 for all *α* < 1 (Figs. 10(f) and 11(f)). This is because in these cases the zeroth order Fourier coefficients are eliminated by the pupil function, and the resultant visibility in image intensity becomes 1 for all *δ/β* values. Since the object’s intensity visibility is *V _{object}* = 0.1 for the complex sinusoidal object with

*η*= 0.1 (see Appendix B),

*MTF*= 1/0.1 = 10 in this case.

Modulation transfer functions of complex objects under partially coherent illuminations are shown in Figs. 10 and 11 with *σ* = 0.1, 0.2, and ∞. As expected, the effect of phase in a complex object is minor for FZP but significant for XOR and ZPD. ZPD and XOR with Δ*θ* = 0 again show similar characteristics as FZP, as seen in Figs. 10(b) and 11(b). Increasing the value of Δ*θ* widens the high MTF region for both DIC objectives.

#### 3.4. Effect of Rotation on MTF

In the previous sections, we have examined the performance of DIC objectives whose orientation along the shear direction is perpendicular to the grating structure, i.e., the relative rotational angle between the DIC objective and the grating object *θ _{r}* = 0° (Fig. 12). Here we examine the effect of rotation on MTF by orienting each DIC objective with respect to the grating objects. Figures 13 and 14 show the results with rotation angles

*θ*= 0° ∼ 90° for various

_{r}*σ*values. Note that a complex sinusoidal grating with

*η*= 0.1 and

*δ/β*= 0 ∼ 10 is used as the object and the object spatial frequency

*α*is fixed at 0.6 for all cases. In general, MTF is largest at

*θ*= 0° and minimizes at

_{r}*θ*= 90°. When

_{r}*σ*= 0, the transitions between low and high MTF regions are again steep for XOR and smooth for ZPD.

## 4. Conclusion

Performance of x-ray DIC microscopes employing XOR and ZPD objectives was quantitatively examined using MTF. It has been demonstrated that x-ray DIC microscopy can utilize the phase information in the sample to produce large image contrast for both phase and complex objects. Effect of partial coherence was also quantitatively examined and shown that illumination coherence plays an important role in DIC image contrast.

## Appendix A Fourier Coefficients of Grating Objects

## A.1 Absorption Sinusoidal Grating

Object transmittance function **t**
* _{o}*(

*ξ′*,

*η′*) of the absorption sinusoidal grating, illustrated in Fig. 5(a), is defined as,

*n*order Fourier coefficients,

^{th}## A.2 Phase Sinusoidal Grating

Object transmittance function **t**
* _{o}*(

*ξ′*,

*η′*) of a phase sinusoidal grating shown in Fig. 5(d) is given by [29],

*φ*is the magnitude of the phase variation. Using the identity [29]

*J*is the

_{n}*n*order Bessel function of the first kind and denoting

^{th}*θ*= 2

*πν*, Eq. (A.3) can be rewritten as,

_{o}ξ′*δ*is the Dirac delta function and the

*n*order Fourier coefficient for a phase sinusoidal grating is given by,

^{th}## A.3 Complex Sinusoidal Grating

Refractive index *ñ* at x-ray wavelengths is in general complex and has the form *ñ* = 1 – *δ* + *j*
*β* where (1 – *δ*) determines the phase shift introduced by the material and *β* is the absorption. Here we assumed that the grating is sinusoidally shaped and its material has a refractive index *ñ* = 1 – *δ* + *j*
*β*. Assuming that the object is placed in vacuum (refractive index *n _{o}* = 1), the object transmittance function

**t**

*(*

_{o}*ξ*′,

*η*′) is given by,

*k*is the vacuum wavenumber, and

*t*(

*ξ*′,

*η*′) describes the grating thickness variation across the object plane (

*ξ*′,

*η*′), i.e.,

*t*is the maximum thickness variation in the grating. Denoting the attenuation length as

_{o}*t*= 1/(2

_{a}*kβ*),

*η*as the ratio of

*t*to

_{o}*t*, and considering only the spatially varying part of the grating, Eq. (A.8) becomes,

_{a}Using the identity in Eq. (A.4) and

*η*and the ratio of

*δ*to

*β*. Figure 5(c) shows the complex sinusoidal grating with

*η*= 1 and

*δ/β*= 1.

Fourier spectrum of the complex sinusoidal transmittance function is therefore,

*δ*is the Dirac delta function. Fourier coefficients of the complex sinusoidal grating are where

*z*is given by Eq. (A.11).

## Appendix B Visibility of Grating Objects

Visibility *V* of an intensity pattern is defined as [27],

*I*and

_{max}*I*are the maximum and the minimum of the intensity distribution, respectively. Here we define the object’s intensity distribution as the absolute square of its transmittance function, i.e., |

_{min}**t**

*(*

_{o}*ξ′*,

*η′*)|

^{2}. This definition follows the classical paper by Becherer and Parrent [20] and is widely used in MTF definition [29].

## B.1 Absorption Sinusoidal Grating

Intensity distribution of the absorption sinusoidal object is given by,

## B.2 Phase Sinusoidal Grating

Intensity distribution of the phase sinusoidal object is given by,

## B.3 Complex Sinusoidal Grating

Intensity distribution of the complex sinusoidal object is given by,

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