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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 15 — Jul. 19, 2010
  • pp: 16279–16288
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Ultra-large multi-region photon sieves

Zhifeng Chen, Chinhua Wang, Donglin Pu, Jin Hu, and Linsen Chen  »View Author Affiliations


Optics Express, Vol. 18, Issue 15, pp. 16279-16288 (2010)
http://dx.doi.org/10.1364/OE.18.016279


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Abstract

A novel method of designing ultra-large photon sieves in visible regime with multi-region structure is proposed and experimentally demonstrated. Design principle that is based on both phase matching and total pinhole area matching among regions is introduced. The focusing properties of the multi-region structure and the conventional monolithic structure of the same numerical aperture in terms of energy efficiency and the sidelobe suppression are compared. Two photon sieves of focal length 500mm and diameters 50mm and 125mm with respectively 3 and 4 regions at working wavelength 632.8nm are fabricated using UV lithography to validate the proposed method. Good performance of the multi-region photon sieves are evaluated by imaging test. The extension of the proposed method suggests a new concept of ring-to-ring design in terms of pinhole size and density of each individual ring for photon sieves with superior suppressed sidelobes towards ultra-large dimension, high numerical aperture that can be implemented with UV lithography which is otherwise impossible with e-beam technique.

© 2010 OSA

1. Introduction

2. Design principle of a multi-region photon sieve

To introduce the design principle of a multi-region photon sieve, the conventional photon sieve theory is briefly reviewed. A photon sieve is essentially a FZP on which the rings have been broken up into isolated circular holes. The positions of the micro-holes are located at the center of the rings given by the radial distancernwhich is a distance from the center of the element to center of the nth ring of the zone plate, n is an integer representing the sequential of the rings:
rn2=2nfλ+n2λ2
(1)
The width of each zone iswn:
wn=λf2rn
(2)
The minimum pinhole size of the photon sievedmin:
dmin=λ2NA
(3)
where f is the focal length at wavelength λ, NA is the numerical aperture. In Eq. (3), it is clear that the minimum pinhole size is determined by the numerical aperture and the working wavelength. In its simplest version, the photon sieve consists of holes of diameterwnlocated at a corresponding radial distancern. Kipp et al [6

6. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001). [CrossRef] [PubMed]

] have shown that the size of the pinholes can be increased beyond the underlying zone width while still maintaining the constructive interference. The effective contribution of the light energy to the focal point from an enlarged pinhole diameter of d is given by an oscillating function:
FdwJ1(πd2w)
(4)
whereJ1is a first-order Bessel function. When F>0, light passing through the holes is making a positive contribution to the focused light, while F<0 the transmitted light is acting to reduce the focused intensity. From Eq. (4), it is seen that the focusing energy at the focal point can be maximized with appropriate ratios of d/w which appears at 1.53, 3.51, 5.51, 7.51, and so on in Fig. 1
Fig. 1 The effect of ratio factor d/w on the focused amplitude at focal point
. The flexibility of the enlargement of the pinhole size suggests that the minimum manufacturable pinhole at the outer rings can be controlled by the ratio factor (d/w), the problem, however, is that if the pinhole size of a photon sieve is enlarged by a fixed ratio over the entire photon sieve, for example, 9.51 at the 5th peak in Fig. 1, such that the minimum pinhole size at the outer rings is manufacturable with UV lithography, eg, >10μm for certain accuracy, the pinhole size at the inner part would be larger than several millimeters or centimeters especially in the case of photon sieves with large numerical aperture size, which would significantly reduce the diffraction efficiency and increase the directly transmitted background light intensity. Menon et al proposed an alternating-phase photon sieve or a phase photon sieve to suppress the zero order background diffraction, in which either the pinholes in alternate zones have a π phase shift or the pinholes have a phase shift of π with respect to the rest of the aperture [10

10. R. Menon, D. Gil, G. Barbastathis, and H. I. Smith, “Photon-sieve lithography,” J. Opt. Soc. Am. A 22(2), 342–345 (2005). [CrossRef]

], but no detailed analysis was given.

To solve the problem, a multi-region photon sieve design principle is proposed and experimentally demonstrated. The essentials of the design principle is that a photon sieve is divided into several annular regions, the pinhole size in each region can be enlarged with different ratio factors considering the criteria of manufacturable but yet well within the diffraction range. Different regions must meet the phase matching and the total pinhole area matching conditions for the optimized intensity distribution at the focal point. The detailed design theory and principle is as follows:

  • 1. A photon sieve is divided into several regions depending on the numerical aperture and the minimum manufacturable pinhole size. Different regions take different ratio factors (d/w) with values at maxima of amplitude in Fig. 1, located at 1.53, 3.51, 5.51, 9.51 and so on for the maximum diffraction contribution to the focal point. The enlarged pinhole size in each region should be limited within the best diffraction range of ~10-1000λ.
  • 2. As shown in Fig. 1, if the ratio of d/w is taken at positive amplitude, the radial position of the nth ring of micro-holes is rn2=2nfλ+n2λ2. Otherwise if the negative amplitude is taken, the position of the nth ring micro-hole is rn2=2(n+0.5)fλ+(n+0.5)2λ2, which is shifted half ring position from the positive one.
  • 3. The amplitude distribution of the focal spot with a central peak and various orders of sidelobes is determined by coherent contributions from each divided annular sub-regions which is defined by the locations of borders of the sub-regions. The contributions from different sub-regions show different behaviors. The outer region or a region of more rings will generate a narrower central amplitude distribution and narrower sidelobes in space than the inner region or the region of fewer rings. These characteristics form the phase matching condition among different regions, in which the sequential number of the start and the finish ring of each region will be optimized such that the sidelobes from each region will be out of phase (i.e., destructive interference is fulfilled) to minimize the overall interferometric sidelobes.
  • 4. Although the change of ratio d/w and the area of pinholes in each region will not affect their own normalized amplitude distribution on the focal plane, the total amplitude distribution on focal plane is a coherent interferometric combination from all sub-regions. The absolute amplitude distributions of the sub-regions do affect the total interferometric amplitude distribution on the focal plane. Therefore, the condition of total pinhole area matching among regions must be met in order to obtain the optimized and suppressed sidelobes on the focal spot. In conventional FZP, the area of each open ring isπλf [6

    6. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001). [CrossRef] [PubMed]

    ], which is a constant over the entire Fresnel zone plate. In a multi-region photon sieve, the total area of the pinholes on each ring in each region can be controlled and optimized to suppress the sidelobes, in which a weighing factor (WF) can be defined and be applied to the areaπλfto optimize the total area which essentially determines the absolute amplitude of the transmission from each region, the number of pinhole on each ring can thus be determined according to the ratio of total area to single pinhole area of πd 2/4. Physically, the weighing factor acts as an apodizing function upon which the sidelobes are dependent.

To give a more detailed illustration of the theory, a concrete design example with an aperture of 125mm, focal length of 500mm and working wavelength of 632.8nm is given. The total ring number of the photon sieve in conventional monolithic type is 6149 and the minimum pinhole size is 2.53μm according to Eq. (1) and Eq. (3). The photon sieve is divided into three annular regions along the radial direction. In region 1, the ratio factor d/w is taken as 1.53, and two adjacent rings are occupied for the enlarged pinholes such that the enlarged pinholes do not overlap with those on other rings. The centre of enlarged pinholes is at the center of corresponding open ring. In region 2, the ratio factor d/w is taken as 3.51 and three adjacent rings are occupied. The centre of enlarged pinholes is set at the center of corresponding opaque ring due to the negative amplitude in Fig. 1 and half a ring shift is applied. In region 3, the ratio factor d/w is taken as 5.51 and four adjacent rings are occupied. With the applied ratio factors in each region, the minimum pinhole size in region 1, 2, and 3 are enlarged from 6.13 to 9.44μm, 3.78 to 13.32μm and 2.53 to 13.95μm, respectively, while the maximum pinhole sizes in each region are still maintained at 304.27μm, 21.63μm, 20.77μm, which is well within the diffraction regime.

The division of the sub-regions must follow the phase matching condition illustrated in step 3. To be of generality, the division can be implemented in terms of numerical aperture(NA) and the number of sub-regions, so that the design can be independent of the individual absolute diameter of the photon sieves. For a 3-region photon sieve, for example, two locations of the border to define the 3 sub-regions are to be determined once the aperture is given. A trial location of the first border (i.e., to define the inner most region, region 1) can be set first, the amplitude contribution to the focal spot from the inner-most region (region 1) can thus be calculated, as shown by black line in Fig. 2(a)
Fig. 2 (a) Amplitude distribution on focal plane from region 1, 2, and 3, as well as the summation of amplitude of region 1 and region 2, respectively; (b) Total intensity on focal plane of the optimized 3-region PS (black line), radius being equally divided PS (red line), and ring number being equally divided PS (blue line). The diameter of the photon sieve is 125mm, focal length is 500mm, and the working wavelength is 632.8nm.
. The second location of the border (to define the middle region, region 2 and the out-most region, region 3) can then be optimized such that the contribution from region 2 (red line in Fig. 2a) is out of phase to the contribution of region 1(i.e., the first valley of black line is coincident with the first peak of the red line), and also the interferometric contribution from region 1 and 2 (green line in Fig. 2(a)) is out of phase to that from region 3(blue line in Fig. 2(a)). During the iteration process of optimizing locations of the two borders, the area matching conditions among the regions have also to be met such that the amplitude summation at the first valley of black line(from region 1), the first peak of red line (from region 2) and the second valley of blue line (region 3) has to be minimized (to minimize the sidelobes) by adjusting the weighing factors(WF) in each region. After several iterations, the optimized border locations and the weighing factors in each region can be obtained.

The optimized locations of two borders of the 3-region photon sieve is located at 0.411 and 0.671 of the numerical aperture(NA), which can be further transformed to sequential ring number as seen in Table 1

Table 1. The detailed design results of a 50mm- and a 125mm-aperture size with 3 sub-regions

table-icon
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. It can also be shown that WF in each region is fixed as long as the number of region is fixed due to the fact that the WF is actually a relative apodization window function among each region. The detailed design results are shown in Table 1 together with the results of a 3-region 50mm-diameter photon sieve.

Figure 2(a) shows the optimized amplitude distribution on the focal plane from region 1, region 2, region 3, and also the summation of region 1 and region 2, respectively. Figure 2(b) shows the optimized overall intensity distribution from all the 3-region interference (black line). As a comparison, the intensity distributions on the focal plane from 3-region photon sieves of the same diameter which are segmented by either equally divided radius (red line) or the equally divided total ring number(blue line) are also given. It is seen that the optimized intensity distribution in Fig. 2(b) (black line) is much better than those with arbitrarily divided segments in terms of the well suppressed sidelobes(< 35dB), and the reduced focal spot size.

Figure 3
Fig. 3 (a) Comparison of the normalized intensity distribution on focal plane between a conventional monolithic photon sieve and a 3-region photon sieve; (b) comparison of pinhole size between a 3-region and a conventional monolithic photon sieve. Parameters are: diameter 50mm, focal length 500mm, wavelength 632.8 nm. Conventional photon sieve: black line; 3-region photon sieve: red line. (c) Comparison of the peak intensity on focal plane between a conventional monolithic photon sieve and a 3-region photon sieve with the same minimum pinhole size.
shows the detailed comparison of the designed 3-region photon sieve (design parameters are shown in Table 1) with a conventional monolithic region photon sieve of the same aperture size 50mm and focal length 500mm. In the conventional monolithic sieve, a Gaussian apodization window for the pinhole density modulation on each ring is applied to suppress the sidelobes [6

6. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001). [CrossRef] [PubMed]

]. The minimum pinhole size is enlarged from 6.33μm in monolithic photon sieve to 23.69μm in the 3-region photon sieve (Fig. 3(b)). It is seen that performance of the multi-region photon sieve reaches or is even better than that of the conventional photon sieves in terms of full width at half magnitude (FWHM) and the suppressed sidelobes with the much enlarged pinhole sizes (Fig. 3(a)). When the diffraction efficiency between the conventional monolithic and a 3-region photon sieve are compared, it is seen, from Fig. 3(c), that the peak intensity of the 3-region photon sieve is improved by 26% from conventional 66020(arbitrary unit, a.u.) to the 3-region 83190(a.u.) if the pinholes in the conventional monolithic photo sieve are enlarged 3.51x on the whole basis so that the minimum pinholes in the monolithic one are the same as those in the 3-region one (i.e., 23.69μm). Accordingly, the diffraction efficiency of the 3-region photon sieve is increased to 2.6% from 2.1% of the conventional monolithic photon sieve, in which the diffraction efficiency is defined by the ratio of the total energy on the focal plane to the total energy incident onto the photon sieve.

In order to show the capability of the proposed design principle of multi-region photon sieves, Fig. 4
Fig. 4 (a) Comparison of the normalized intensity distribution on focal plane between a 3-region and a 4-region photon sieve with the same diameter; and (b) Comparison of pinhole size between a 3-region and a 4-region photon sieve as well as a conventional monolithic photon sieve. The parameters are: diameter 125mm, focal length 500mm and wavelength 632.8 nm. Conventional photon sieve: black line; 3-region photon sieve: red line; and 4-region photon sieve: blue line.
shows the normalized intensity on the focal plane of a photon sieve of diameter 125mm with 4 regions using the phase matching and total pinhole area matching principle, and the result is compared with that of the counterpart of the same diameter with 3 regions. The optimized locations of three borders of the 4-region photon sieve is located at 0.252, 0.407 and 0.655 of the numerical aperture, which can be further transformed to sequential ring number as seen in Table 2

Table 2. The detailed design results of 125mm and 420mm-aperture size with 4 regions.

table-icon
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. The detailed design parameters are shown in Table 2.

In Fig. 4, it is seen that the performance of the 3-region and 4-region photon sieves are comparable in terms of sidelobe suppression and FWHM(Fig. 4(a)), the minimum pinhole diameter of the 4-region sieve is enlarged from 2.53μm to 15.16μm in contrast to 9.44μm from 2.53μm in the 3-region photon sieve (Fig. 4(b)).

Figure 5
Fig. 5 (a) Normalized intensity distribution on focal plane; and (b) Pinhole size of a 4-region photon sieve and a conventional monolithic photon sieve. The parameters are: diameter 420mm, focal length 500mm, and wavelength 632.8 nm. Conventional: black line; 4-region: red line.
shows a 4-region design result of a photon sieve with an ultra-large aperture size of 420mm, the largest aperture size reported so far. The focal length is 500mm and working wavelength is 632.8nm for both photon sieves. In Fig. 5(a), it is seen that the radius of the focal point is reduced to 1.224μm which is close to radius of Airy disk 0.919μm due to the increased aperture size and the sidelobes of intensity distribution is well suppressed below 30dB. The minimum pinhole size is enlarged from 0.75μm in conventional monolithic photon sieve to 4.58μm in the 4-region photon sieve (Fig. 5(b)). The weighing factor WF in each region is the same as that used in the photon sieve with 125mm diameter. The other parameters are summarized in Table 2.

3. Experimental characterization and demonstration

To demonstrate the proposed methodology for the ultra-large photon sieve design, we experimentally fabricated four photon sieves, i.e. a monolithic and a 3-region photon sieve with an aperture size of 50mm, a 3-region and a 4-region photon sieve with an aperture size of 125mm, respectively, to validate the theory using the UV lithography. The parameters used in the design and fabrication for the photon sieves are listed in Table 1 and Table 2.

The photon sieves were fabricated on a silica substrate with an opaque chromium film(300nm thick) and a photo-resist layer spin-coated on top of the chromium film. The designed pinholes of the photon sieves were saved as *.BMP format and were input to a digital mirror device (DMD) system for UV lithography. A 405nm laser was used to UV exposure of the *.BMP picture on the photo resist, and after the exposition of the UV laser, the resist is developed and the chromium is etched. Figure 6
Fig. 6 (a)Typical photo of a 3-region photon sieve with an aperture of 125mm; and (b)the microscopic imaging of pinholes showing the transition from region 2 to region 3.
shows a typical photo of the photon sieve with an aperture of 125mm (Fig. 6(a)), which was taken with a lamp back illumination. 3 regions with different d/w ratios are clearly shown in the photo. Figure 6(b) shows the detailed microscopic imaging of pinholes at the transition boundary from region 2 to region 3. The pinhole diameter of region 2 and region 3 at the boundary is 21.38μm and 33.48μm, respectively.

To evaluate the performance of the photon sieves, an experimental setup was built as shown in Fig. 7
Fig. 7 Experimental setup for photon sieve characterization
for the purpose of focal spot characterization and target imaging test. The incident laser beam (632.8nm) is focused by a microscopic objective and passes through a pinhole filter. For focal spot characterization, a lens is used for the beam collimation, i.e., the dashed block in Fig. 7 is replaced by a collimation lens, the collimated beam is then focused by the photon sieve under test. The focal spot is magnified with a microscope objective (20x) in order to acquire sufficient spatial resolution and is subsequently received by a CCD (pixel size 5.2μm). For imaging test, the collimation lens was replaced by a rotating diffuser (to remove speckles) and a test target (USAF 1951), as shown in Fig. 7. The image of the target is directly received by the CCD.

Figure 8
Fig. 8 (a) Intensity distributions of focal spots of a monolithic and a 3-region photon sieve with the same diameter of 50mm; (b) Target image produced by the conventional monolithic PS; and (c) Target image produced by 3-region PS of the same diameter as in (b). The other parameters are focal length 500 mm and working wavelength 632.8nm.
shows the intensity distribution of the focal spot and the imaging test result of the 3-region photon sieve with a diameter of 50mm and the comparison with the monolithic counterpart of the same diameter. The measured FWHMs are 7.91μm and 8.23μm respectively for the monolithic and the 3-region sieves, which are very close to the theoretical value of 7.49μm measured from Fig. 8(a). The target images produced by the monolithic and the 3-region 50mm-diameter photon sieve at 632.8 nm are shown in Fig. 8(b) and Fig. 8(c), respectively. The resolutions of the two images are comparable which implies the 3-region photon sieve works very well when compared with the conventional monolithic one. Detailed examinations show the resolution limit for this photon sieve is about 71.8 lp/mm, which is approximately consistent with the measured focal spot size on the order of 16μm.

The similar results can be seen in Fig. 9
Fig. 9 (a) Intensity distributions of focal spots of a 3-region and 4-region photon sieve with the same diameter of 125mm; (b) Target image produced by the 3-region PS; and (c) Target image produced by the 4-region PS of the same diameter as in (b). The other parameters are focal length 500 mm and working wavelength 632.8nm.
which shows the intensity distributions of the focal spot and the imaging test results of a 3-region and 4-region photon sieves with the same diameter of 125mm, respectively. The measured FWHMs are 3.75μm and 4.08μm respectively for the 3-region and 4-region sieves, which are both close to the theoretical value of 3.12μm, Fig. 9(a). When the image quality is evaluated, it is seen that both the 3-region and 4-region photon sieves produce the similar target images, which is consistent with the design results shown in Fig. 4(a). From Fig. 9(b) and 9(c), it is estimated that the resolution limit for the sieves is about 128 lp/mm, which is fairly consistent with the designed value(corresponding to 8um focal spot). The discrepancy between the designed and the measured is probably due to the imperfect fabrication processes such as the mechanical positioning error especially in the case of large aperture size.

4. Conclusion

We have proposed a novel method of designing ultra-large photon sieves with pinhole sizes both manufacturable and well within the diffraction limit in visible regime using multi-region structures and experimentally demonstrated the feasibility using UV lithography. It is shown that the performance of the multi-region photon sieve can reach or even be better than that of the conventional monolithic photon sieves of the same diameter. A 4-region photon sieve with an ultra-large aperture size of 420mm is designed for the first time in which the pinhole size is well limited in the range of 4-300μm. The extension of this method suggests a new concept of ring-to-ring design in terms of pinhole size and density of each individual ring for photon sieves towards ultra-large dimension and high numerical aperture that can be implemented with UV lithography which is otherwise impossible with e-beam technique.

Acknowledgments

The work is supported in part by China 863 project (2006AAJ304), University Scientific Research Foundation of Jiangsu Province (09KJA140004), Suzhou High-Tech Research Program (ZXG0712) and Pre-research Project of Soochow University.

References and links

1.

R. A. Hyde, “Eyeglass. 1. Very large aperture diffractive telescopes,” Appl. Opt. 38(19), 4198–4212 (1999). [CrossRef]

2.

I. M. Barton, J. A. Britten, S. N. Dixit, L. J. Summers, I. M. Thomas, M. C. Rushford, K. Lu, R. A. Hyde, and M. D. Perry, “Fabrication of large-aperture lightweight diffractive lenses for use in space,” Appl. Opt. 40(4), 447–451 (2001). [CrossRef]

3.

G. Andersen and D. Tullson, “Broadband antihole photon sieve telescope,” Appl. Opt. 46(18), 3706–3708 (2007). [CrossRef] [PubMed]

4.

A. B. Meinel and M. P. Meinel, “Parametric dependencies of high-diffraction-order achromatized aplanatic configurations that employ circular or crossed-linear diffractive optical elements,” Appl. Opt. 41(34), 7155–7166 (2002). [CrossRef] [PubMed]

5.

A. B. Meinel and M. P. Meinel, “Large membrane space optics: imagery and aberrations of diffractive and holographic achromatized optical elements of high diffraction order,” Opt. Eng. 41(8), 1995 (2002). [CrossRef]

6.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001). [CrossRef] [PubMed]

7.

Q. Cao and J. Jahns, “Focusing analysis of the pinhole photon sieve: individual far-field model,” J. Opt. Soc. Am. A 19(12), 2387–2393 (2002). [CrossRef]

8.

Q. Cao and J. Jahns, “Nonparaxial model for the focusing of high-numerical-aperture photon sieves,” J. Opt. Soc. Am. A 20(6), 1005–1012 (2003). [CrossRef]

9.

G. Andersen, “Large optical photon sieve,” Opt. Lett. 30(22), 2976–2978 (2005). [CrossRef] [PubMed]

10.

R. Menon, D. Gil, G. Barbastathis, and H. I. Smith, “Photon-sieve lithography,” J. Opt. Soc. Am. A 22(2), 342–345 (2005). [CrossRef]

11.

F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express 14(25), 11958–11963 (2006). [CrossRef] [PubMed]

12.

Z. Gao, X. Luo, J. Ma, Y. Fu, and C. Du, “Imaging properties of photon sieve with a large aperture,” Opt. Laser Technol. 40(4), 614–618 (2008). [CrossRef]

13.

C. Zhou, X. Dong, L. Shi, C. Wang, and C. Du, “Experimental study of a multiwavelength photon sieve designed by random-area-divided approach,” Appl. Opt. 48(8), 1619–1623 (2009). [CrossRef] [PubMed]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(100.2960) Image processing : Image analysis
(230.4000) Optical devices : Microstructure fabrication

ToC Category:
Diffraction and Gratings

History
Original Manuscript: May 7, 2010
Revised Manuscript: June 4, 2010
Manuscript Accepted: July 6, 2010
Published: July 16, 2010

Citation
Zhifeng Chen, Chinhua Wang, Donglin Pu, Jin Hu, and Linsen Chen, "Ultra-large multi-region photon sieves," Opt. Express 18, 16279-16288 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-16279


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References

  1. R. A. Hyde, “Eyeglass. 1. Very large aperture diffractive telescopes,” Appl. Opt. 38(19), 4198–4212 (1999). [CrossRef]
  2. I. M. Barton, J. A. Britten, S. N. Dixit, L. J. Summers, I. M. Thomas, M. C. Rushford, K. Lu, R. A. Hyde, and M. D. Perry, “Fabrication of large-aperture lightweight diffractive lenses for use in space,” Appl. Opt. 40(4), 447–451 (2001). [CrossRef]
  3. G. Andersen and D. Tullson, “Broadband antihole photon sieve telescope,” Appl. Opt. 46(18), 3706–3708 (2007). [CrossRef] [PubMed]
  4. A. B. Meinel and M. P. Meinel, “Parametric dependencies of high-diffraction-order achromatized aplanatic configurations that employ circular or crossed-linear diffractive optical elements,” Appl. Opt. 41(34), 7155–7166 (2002). [CrossRef] [PubMed]
  5. A. B. Meinel and M. P. Meinel, “Large membrane space optics: imagery and aberrations of diffractive and holographic achromatized optical elements of high diffraction order,” Opt. Eng. 41(8), 1995 (2002). [CrossRef]
  6. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature 414(6860), 184–188 (2001). [CrossRef] [PubMed]
  7. Q. Cao and J. Jahns, “Focusing analysis of the pinhole photon sieve: individual far-field model,” J. Opt. Soc. Am. A 19(12), 2387–2393 (2002). [CrossRef]
  8. Q. Cao and J. Jahns, “Nonparaxial model for the focusing of high-numerical-aperture photon sieves,” J. Opt. Soc. Am. A 20(6), 1005–1012 (2003). [CrossRef]
  9. G. Andersen, “Large optical photon sieve,” Opt. Lett. 30(22), 2976–2978 (2005). [CrossRef] [PubMed]
  10. R. Menon, D. Gil, G. Barbastathis, and H. I. Smith, “Photon-sieve lithography,” J. Opt. Soc. Am. A 22(2), 342–345 (2005). [CrossRef]
  11. F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express 14(25), 11958–11963 (2006). [CrossRef] [PubMed]
  12. Z. Gao, X. Luo, J. Ma, Y. Fu, and C. Du, “Imaging properties of photon sieve with a large aperture,” Opt. Laser Technol. 40(4), 614–618 (2008). [CrossRef]
  13. C. Zhou, X. Dong, L. Shi, C. Wang, and C. Du, “Experimental study of a multiwavelength photon sieve designed by random-area-divided approach,” Appl. Opt. 48(8), 1619–1623 (2009). [CrossRef] [PubMed]

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