Research on partially coherent light propagation through zone plates

Junchao Ren; Yong Wang; Yong Wang; Yong Wang; Xiangyu Meng; Xiangyu Meng; Xiangyu Meng; Weihong Sun; Jiefeng Cao; Junqin Li; Renzhong Tai; Renzhong Tai; Renzhong Tai

doi:10.1364/OE.442230

1. Introduction

A Fresnel zone plate (FZP) consists of a series of transparent and opaque concentric rings. A wavefront generates a phase with an integral multiple of 2π between different transparent rings when coherent light propagates through the zone plate. The complex amplitude through the zone plate is coherently focused into one spot in the manner of a lens [1]. Zone plates are widely used in synchronous radiation microscopy and can characterize different elements and chemical states. Transmission X-ray microscopes (TXMs) use condenser and objective zone plates to achieve full-field imaging, which has the advantages of high photon flux and fast imaging [2–4]. Scanning transmission X-ray microscopes (STXMs) use Fresnel zone plates to obtain submicron-focused probes and scan samples to achieve imaging, which has the advantages of low radiation dosage and high spatial resolution [5,6]. Scanning fluorescence X-ray microprobes (SFXMs) are similar to STXMs except that fluorescent X-rays are collected by energy resolution detectors for trace element detection [7,8].

The focusing capability of the zone plate determines the spatial resolution of microscopy. By raising the fabrication accuracy of the zone plate to reduce the width of the outermost zone, the focusing capability of the zone plate can be improved [9–11]. Combining advanced precision in scanning control and careful optical design, a spatial resolution of 7 nm for STXM can be realized [12]. The zone plate needs coherent light to achieve nano focus. However, synchrotron radiation is partially coherent light. Full coherent light can be obtained by sacrificing the photon flux, which reduces the signal-to-noise ratio (SNR) and damages the microscopy quality. Fully coherent light is not necessary for zone plate focusing, but the focusing capability is influenced by the coherence property. Much work has been done to analyze the focus of the zone plate [10,13,14], but there is no quantitative analysis on the relationship between the focusing capability of the zone plate and the coherence property of the light.

In this work, the MOI model is extended to quantitatively simulate the mutual optical intensity propagation through the zone plate. The MOI model is based on the principle of statistical optics and uses mutual optical intensity to describe partially coherent light [15–17]. The MOI model is used to analyze the dependence of the focusing capability of the zone plate on the coherence property. The partially coherent light propagation through the STXM beamline at the Shanghai Synchrotron Radiation Facility (SSRF) is simulated by using the MOI model. By optimizing the exit slit size, we can obtain the balance between the focal spot size and photon flux.

2. Establishment of the MOI model for the zone plate

A detailed introduction to the MOI model has been given in our previous work [15–17]. Here, we briefly introduce the MOI model and give the mathematical model to describe the mutual optical intensity propagation through the zone plate in detail. The MOI model uses mutual optical intensity to describe partially coherent light. The propagation of mutual optical intensity in free space can be expressed as [1,18,19]

(1)$$J({{w_1},{w_2}} )= \smallint\!\!\!\smallint J({{x_1},\;{x_2}} )\exp \left[ { - i\frac{{2\pi }}{\lambda }({{r_2} - {r_1}} )} \right]\frac{{\chi ({{\gamma_1}} )}}{{\sqrt {\lambda {r_1}} }}\frac{{\chi ({{\gamma_2}} )}}{{\sqrt {\lambda {r_2}} }}d{x_1}d{x_2},$$

where $\lambda $ is the wavelength; ${x_1}$ and ${x_2}$ are any two points at the source plane; .. and ${w_2}$ are any two points at the image plane, respectively; ${r_1}$ and ${r_2}$ are the ${x_1}$-to-${w_1}$ and ${x_2}$-to-${w_2}$ distances, respectively; $\chi ({{\gamma_1}} )$ and $\chi ({{\gamma_2}} )$ are the inclination factors for the inclination angles ${\gamma _1}$ and ${\gamma _2}$, respectively, and $J({{x_1},\;{x_2}} )$ and $J({{w_1},{w_2}} )$ are mutual optical intensity at the object and subject planes, respectively.

The Fresnel zone plate consists of a series of transparent and opaque concentric rings, as shown in Fig. 1. The distribution of the radius r of the transparent and opaque ring of the zone plate satisfies the following relationship [1]

(2)$$\sqrt {r_n^2 + {f^2}} - f = n\lambda /2,$$

where f is the focal length, n is an odd number corresponding to an opaque ring, and n is an even number corresponding to a transparent ring. Expanding the square root and neglecting high terms, the width of the outermost ring of the zone plate can be obtained:

(3)$$\Delta r = \frac{\lambda f}{2{r_n}}.$$

Fig. 1. Schematic diagram of zone plate focusing setup

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According to the distribution of transparent and opaque rings, the zone plate is divided into a large number of elements, and the mutual optical intensity propagation is calculated from the element on the transparent rings to the element at the image plane. By summing all the mutual optical intensities from every element on the transparent rings, the mutual optical intensity distribution at the image plane can be obtained:

(4)$$J({{w_1},{w_2}} )= \mathop \sum \nolimits_{trans} \left[ A{({{x_2},{w_2}} )}^{\ast}\left(\mathop \sum \nolimits_{trans} J({{x_1},\;{x_2}} )A({{x_1},{w_1}}) \right) \right].$$

$A({{x_1},{w_1}} )$ is the transfer function from element ${x_1}$ on the transparent ring to element ${w_1}$ at the image plane.

(5)$$A({{x_1},{w_1}} )= \smallint \exp \left( {i\frac{{2\pi }}{\lambda }{r_1}} \right)\frac{{\chi ({{\theta_1}} )}}{{\sqrt {\lambda {r_1}} }}\textrm{d}x = \smallint \exp \left[ {i\frac{{2\pi }}{\lambda }\left( {{r_{01}} - \frac{{{w_1}x}}{{{r_{01}}}} + \frac{1}{2}\frac{{{x^2}}}{{{r_{01}}}}} \right)} \right]\frac{{\chi ({{\theta_1}} )}}{{\sqrt {\lambda {r_{01}}} }}\textrm{d}x,$$

where ${r_{01}}$ is the distance between the central points of the two elements ${x_1}$ and ${w_1}$. $\mathop \sum \nolimits_{trans} J({{x_1},\;{x_2}} ) A({{x_1},{w_1}} )$ means summing all of the mutual optical intensity propagation from every element on the transparent ring. Using formula (4), we can calculate the propagation of partially coherent light through the zone plate.

3. Propagation of partially coherent light through the zone plate

3.1 Effect of the coherence property on the zone plate focusing

The developed MOI model is used to analyze partially coherent light propagation through the zone plate. For simplicity, the source uses the Gaussian Schell Model (GSM) [18],

(6)$${J_{12}} = {I_0}\exp\left( { - \frac{{{{({x_1} - {x_2})}^2}}}{{2{\mathrm{\chi }^2}}}} \right)\exp\left( { - \frac{{x_1^2 + x_2^2}}{{4{\sigma^2}}}} \right),$$

where ${J_{12}}$ is the mutual optical intensity between any two points ${x_1}$ and ${x_2}$ at the source plane, $\mathrm{\chi }$ represents the coherence length, and $\sigma $ represents the intensity root mean square (rms) size, and ${I_0}$ is the intensity of the central point. The wavelength is $\lambda $ = 2.55 nm. The diameter of the zone plate is ${D_{ZP}}$ = 200 µm, and the width of the outermost ring is $\Delta r$ = 30 nm. Without the central stop and order selecting aperture (OSA), the intensity distribution at the focal plane focused by the zone plate (ZP) is calculated under different coherence lengths, as shown in Fig. 2(a). The focus spot size can be described by the Airy disk radius, which is defined as the distance between the central point of the main peak and the first dark ring. The Airy disk radius with various coherence lengths is shown in Fig. 2(b), where the focal length f = 2.35 mm. The Airy disk radius is 186.73 nm with a coherence length of 10 µm. The brightness of the intensity peak increases and the focal spot size decreases when the coherence length increases. The intensity distribution tends to stabilize at $\mathrm{\chi }$ = 200 µm. The corresponding Airy disk radius is 28.13 nm, which is close to the outermost ring width $\Delta r$ = 30 nm. In addition, as the coherence length increases, the diffraction effect caused by the limited size of the zone plate gradually increases. When $\mathrm{\chi }$ = 200 µm, apparent diffraction peaks are observed.

Fig. 2. Without a central stop and OSA, (a) intensity distribution at the focal plane with different coherence lengths and (b) the Airy disk radius as a function of coherence length. With both central stop and OSA sizes of 80 µm, (c) intensity distribution with different coherence lengths and (d) the Airy disk radius as a function of coherence length.

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In the operational zone plate focusing experiment, the central stop and OSA are required to block zero-order and higher-order diffraction light. This is necessary to analyze the light propagation through zone plates with various central stops and OSA sizes. The distance between the OSA and the zone plate is $l = f \times \frac{{{D_{stop}}}}{{{D_{ZP}}}}$. The length l is 1.41 mm when the size of the central stop ${D_{stop}}$ and the OSA size are both 80 µm. The intensity distributions at the focal plane with different coherent lengths are shown in Fig. 2(c). Due to the diffraction effect of the central stop and limited size of the zone plate, the diffraction peaks gradually increase as the coherence length increases. The rms of the intensity distribution within 0.5 µm is a statistical result which considers the multiple peaks contribution. The Airy disk radius is approximately proportional to the intensity rms. Therefore, the Airy disk radius instead of the intensity size rms is used to define the spot size. The Airy disk radius with various coherence lengths is shown in Fig. 2(d). When the coherence length is greater than 200 µm, the Airy disk radius tends to be 21.56 nm.

3.2 Effect of the central stop on the zone plate

In the STXM experiment, the central stop of the zone plate and OSA are combined to block the zero- and higher-order diffraction light. The background noise at the whole focal plane is depressed, and thus the STXM technology quality can be improved. When the coherence length is 200 µm, the intensity distribution at the focal plane within 300 µm is shown in Fig. 3. Without a central stop and OSA (black line), the Airy disk radius is 31.60 nm. The effective photon flux (within 2 Airy disk radius) and (within 4 Airy disk radius) at the focal plane are 171.57 and 183.98, which are only 19.83% and 21.27% of the total flux at the focal plane, respectively. This indicates that the zero- and higher-order diffraction light has a greater effect on the flux. With central stop and OSA sizes of both 50 µm (red line), the Airy disk radius is 25.20 nm. The effective flux (within 2 Airy disk radius) and (within 4 Airy disk radius) at the focal plane are 85.20 and 121.36, which are 57.41% and 81.78% of the total flux, respectively. Most flux covers the focal plane with a 4 times Airy disk radius. The Airy disk radius with a central stop and OSA is smaller than that without a central stop and OSA. The zero and higher-order diffraction light can be depressed effectively by the central stop and OSA. Finally, the SNR and spatial resolution for STXM are improved.

Fig. 3. The intensity distribution at the ZP focal plane of 300 µm.

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With a coherence length of 200 µm, the intensity distributions with different central stop sizes are calculated, as shown in Fig. 4(a)–4(d), where the size and position of the OSA change with the size of the central stop. As the central stop size increases, the effective size of the zone plate decreases, and the diffraction effect gradually increases, resulting in a more apparent secondary peak. In addition, the flux through the zone plate decreases with increasing central stop size. The two aspects lead to low brightness for the focused spot. Figure 4(e) shows that the Airy disk radius changes with various sizes of the central stop (black line), and the SNR (the photon flux in the Airy disk divided by the photon flux outside the Airy disk) changes with various central stop sizes (red line). As the size of the central stop increases, the Airy disk radius becomes small, and the SNR becomes low. The reason is that the double-slit diffraction effect from the finite size of the zone plate blocked by the central stop is improved with increasing central stop size. As a result, the central spot size becomes small, and the proportion of secondary peak flux increases, resulting in a low SNR. The diffraction efficiency of the zone plate with various central stop sizes, as shown in Fig. 4(f). Diffraction efficiency is defined as the photon flux at the focal plane divided by the photon flux of zone plate incident plane. Reducing the sizes of the central stop and OSA slightly increases the spot size, but the diffraction efficiency and SNR are significantly improved, which is beneficial for improving the imaging quality of the zone plate.

Fig. 4. The ZP diameter is 200 µm. (a-d) Intensity distribution at the focal plane when the size of the central stop is 20 µm, 50 µm, 80 µm, and 120 µm; (e) Airy disk radius (black line) and SNR (red line) at the focal plane and (f) diffraction efficiency of the zone plate.

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4. Zone plate focus in the STXM endstation

Taking the SSRF BL08U1A beamline as an example, the MOI model is used to calculate the partially coherent light propagation from the undulator source to the plane focused by the zone plate. A schematic diagram of the beamline layout is shown in Fig. 5. Soft X-rays are generated by an elliptically polarized undulator (EPU), propagate through the four-blade aperture, ellipsoidal cylindrical mirror, SX700-type monochromator and toroidal mirror, and horizontally and vertically focus at the exit slit plane. At an energy of 486 eV, the intensity rms size is 35.8 µm × 11.1 µm (H×V) at the exit slit plane, and the corresponding coherence length is 4.2 µm × 21.6 µm (H×V) [15]. The zone plate is located 2 m downstream of the exit slit. The material of the zone plate ring is 180 nm thick Au with a diameter of 200 µm and an outermost ring width of Δr = 30 nm. The central stop material is Au with a thickness of 10 µm and diameter of 80 µm. The substrate is 100 nm thick Si3N4. At an energy of 486 eV, the focal length of the zone plate $f$ = 2.353 mm.

Fig. 5. Schematic diagram of BL08U1A beamline at the SSRF.

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When the exit slit = 20 µm × 20 µm, the horizontal and vertical intensity distributions along the focal depth are shown in Fig. 6(a) and 6(b), where the x-axis is the horizontal direction, the y-axis is the vertical direction, and the ZP focal depth is along the optical axis (z-axis). The ideal focal plane of zone plate is at z = 0. The light at the exit slit plane can be seen as a secondary source, and the incident wavefront at the zone plate plane is a spherical wave with a curvature radius of 2 m. Therefore, the focus plane position moves backward to z = 2.353 µm. The horizontal and vertical focal depths are both 2.82 µm.

Fig. 6. The intensity distribution along the focal depth in the horizontal x axis (a) and vertical y axis (b) directions. The ideal focal plane position is at z = 0.

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By changing the size of the horizontal opening of the exit slit, the X-ray coherence property at the zone plate plane is adjusted to analyze the influence of the coherence property on the focusing capability of the zone plate. The global degree of coherence [20] at the zone plate plane with various exit slit opening sizes is shown in Fig. 7(a). When the exit slit opening is less than 10 µm, the global degree of coherence is larger than 0.95, and the incident light can be regarded as fully coherent light. The global degree of coherence decreases as the exit slit opening size increases. When the exit slit opening is greater than 200 µm, the global degree of coherence tends to 0.27, which is determined by the coherence property of the secondary source at the exit slit plane. The intensity distribution at the focal plane is shown in Fig. 7(b). The intensity rms size and photon flux at the focal plane change with various exit slit opening sizes, as shown in Fig. 7(c). When the exit slit is less than 10 µm, the secondary source is approximately a point source. The X-ray incident on the zone plate is almost fully coherent. The spot intensity rms size is constant at 46.95 nm. The corresponding Airy disk radius is 21.58 nm, which is similar to the theoretical radius according to the Airy disk radius formula 0.61×λ/NA = 37 nm for fully coherent light [1]. When the exit slit opening increases, the secondary source size increases and the global degree of coherence decreases. The corresponding focus spot intensity rms size increases. When the exit slit is greater than 230 µm, it is larger than the lateral spot size 6σ of 215 µm at the exit slit plane. The global degree of coherence at the zone plate plane is 0.27, and the intensity rms size at the focal plane converges to 57.05 nm. The X-ray with different energies have no coherence. The beam with limited energy resolution in the propagation through zone plate can be realized by summing intensity distribution from X-ray with different energies. For the BL08U1A beamline, the energy resolution at 486 eV is greater than 2000 when the exit slit size is 100 µm. The corresponding spot size increases by 10%. Reducing the exit slit opening size can improve the coherence property and decrease the spot size at the focal plane at the cost of photon flux. However, a higher photon flux is beneficial for improving the SNR of the detector. Therefore, it is necessary to balance the spatial resolution and the photon flux to achieve optimized zone plate focusing.

Fig. 7. The global degree of coherence at the zone plate plane changes with various exit slit sizes; (b) the intensity distribution at the focal plane with different exit slit sizes; (c) the spot rms size (black line) and flux (red line) at the focal plane changes with various exit slit sizes.

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5. Conclusion

The quantitative simulation on the propagation of the partially coherent light through the zone plate is performed for the first time by extending the MOI model. The model is used to calculate the mutual optical intensity propagation and analyze the influence of the X-ray coherence property on the focusing capability of the zone plate. As the coherence length increases, the focus spot size becomes small. However, clear diffraction peaks are observed in the intensity profile. By reducing the sizes of the central stop and OSA, the Airy disk radius can be reduced, and the focus spot SNR can be improved. Furthermore, we studied the influence of the exit slit opening on the spatial resolution of the STXM beamline at the SSRF. As the exit slit opening size decreases, the coherence property increases and the focus spot size decreases at the cost of photon flux. It is necessary to balance the spatial resolution and the photon flux to achieve optimized zone plate focusing.

Funding

National Key Research and Development Program of China (2016YFB0700402); National Natural Science Foundation of China (11775291, 11805260, 11875314); Ministry of Science and Technology of the People's Republic of China (2017YFA0403400).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Research on partially coherent light propagation through zone plates

Abstract

1. Introduction

2. Establishment of the MOI model for the zone plate

3. Propagation of partially coherent light through the zone plate

3.1 Effect of the coherence property on the zone plate focusing

3.2 Effect of the central stop on the zone plate

4. Zone plate focus in the STXM endstation

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (6)

Optics Express