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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 13 — Jun. 18, 2012
  • pp: 14284–14291
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Talbot effect beyond the paraxial limit at optical frequencies

Yi Hua, Jae Yong Suh, Wei Zhou, Mark D. Huntington, and Teri W. Odom  »View Author Affiliations


Optics Express, Vol. 20, Issue 13, pp. 14284-14291 (2012)
http://dx.doi.org/10.1364/OE.20.014284


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Abstract

This paper reports the experimental and theoretical investigation of the Talbot effect beyond the paraxial limit at optical frequencies. Au hole array films with periodicity a 0 comparable to the wavelength of coherent illumination λ were used to study the non-paraxial Talbot effect. Significant differences from the paraxial (classical) Talbot effect were observed. Depending on the ratio of a 0 / λ , the interference pattern in the direction perpendicular to the hole array was not necessarily periodic, and the self-image distances deviated from the paraxial Talbot distances. Defects within the hole array film or above the film were healed in the self-images as the light propagated from the surface.

© 2012 OSA

The Talbot effect is a ubiquitous, lensless self-imaging characteristic of periodic arrays [1

1. M. Masud, Classical Optics and its Applications (Cambridge University Press, 2002), Chap. 18.

]encountered in a range of research areas from atom optics [2

2. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995). [CrossRef] [PubMed]

] plasmonics [3

3. M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express 15(15), 9692–9700 (2007). [CrossRef] [PubMed]

].Under coherent illumination with wavelength λ at normal incidence, an array with periodicity a0will form self-images at integer multiples of the Talbot distance zT=2a02/λ [4

4. L. Rayleigh, “On copying diffraction gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).

]. In addition to self-images, patterns with periodicity of a0/qcan be observed at fractional Talbot distancespzT/2q, where p and q are relative prime numbers [5

5. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996). [CrossRef]

]. Self-images and fractional images have been used in various applications such as lithography [6

6. A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27(6), 2931–2937 (2009). [CrossRef]

], interferometry [7

7. Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24(19), 3162–3166 (1985). [CrossRef] [PubMed]

], optical trapping [8

8. Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007). [CrossRef]

], and array illumination [9

9. A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29(29), 4337–4340 (1990). [CrossRef] [PubMed]

]. At Talbot distances, self-images of the periodic array can be formed even in the presence of defects; this property has been used to generate defect-free self-images from a defective mask [10

10. H. Dammann, G. Groh, and M. Kock, “Restoration of faulty images of periodic objects by means of self-imaging,” Appl. Opt. 10(6), 1454–1455 (1971). [CrossRef] [PubMed]

].

Scalar wave theory is traditionally used for the Talbot effect in the paraxial limit, where the light pattern through a periodic hole array at z = 0 (the initial structure) can be represented by a scalar wave ψ and Fourier expanded as a sum of plane waves in the direction of light propagation (z>0) [17

17. T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, “Exact self-imaging of transversely periodic fields,” J. Opt. Soc. Am. A 21(8), 1424–1429 (2004). [CrossRef] [PubMed]

]
ψ(r,λeff)=m1=m2=Am1m2(ω)×exp[ikxm1x+ikym2y+ikzm1,m2z]
(1)
r = (x, y, z) is the spatial position, λeff = λ/ n is the effective wavelength in the medium with refractive index n,Am1m2(ω) is the complex amplitude of the plane wave, and m1and m2are integers. The wave vectork=(kxm1,kym2,kzm1,m2)=(2πm1/a0,2πm2/a0,kzm1,m2)satisfies |k|=2π/λeffandkzm1,m2>0.

In the paraxial limit (a0 >>λeff), Taylor expansion of kz results in
kzm1,m2=(|k|2kxm12kym22)1/22π[1(m12+m22)λeff2/2a02]/λeff
(2)
If this approximation is applied to Eq. (1), the following relation can be derived
ψ(x,y,z+szT,λeff)=exp[i2πszT/λeff]ψ(x,y,z,λeff)
(3)
where zT=2a02/λeff is the Talbot distance and s is an integer. Thus, exact self-images will be observed at integer multiples of zTbecause the intensities satisfyI(x,y,z+szT,λeff)=I(x,y,z,λeff)=|ψ(x,y,z,λeff)|2

Beyond the paraxial limit, however, when a0/λeff approaches unity, the Taylor expansion in Eq. (2) is not valid. The light patterns can still be calculated based on the plane wave summation, but the approximation about kz can no longer be used. To calculate the accurate phase change of the propagating plane waves (the exp[ikzz] term in Eq. (1)), we consider kz independently for each (m1,m2)plane wave beyond the paraxial limit. Since kz is imaginary for evanescent waves with kx2+ky2>(2π/λeff)2, we therefore simplified the calculation of light patterns in the far field by neglecting the exponentially decaying evanescent waves. Thus, only the propagating plane waves, graphically depicted as points with coordinates (m1,m2)inside the circle with radiusa0/λeff in k space (Figs. 1(A)
Fig. 1 Self-image distances of the non-paraxial Talbot effect are different from the classical Talbot distances. (A-B)Fourier space spectrum of plane waves emitting from the hole array, where the circle with radius a0/λeff separates the propagating waves and evanescent waves. Light patterns from scalar wave calculations with (C) a0/λeff = 1.230 (a0 = 600 nm, λeff = 488 nm), (D)a0/λeff = 1.90 (a0 = 1.2 μm, λeff = 633 nm). Experimental measurement with (E) a0/λeff = 1.230, (F)a0/λeff = 1.90. Light patterns from FDTD simulations with (G) a0/λeff = 1.230, (H)a0/λeff = 1.90.
1(B)), are considered. In addition, the condition for achieving exact self-images requires all the plane waves to be in phase at the same distance. If the center frequency (0,0,kz0,0)is chosen as the reference, this condition can be written as:
exp[ikzm1,m2zR]=exp[ikz0,0zR]
(4)
for all propagating waves(m1,m2), where zR is the self-image distance. We divided the non-paraxial Talbot effect into three different regimes based on the number of propagating waves: (i) a0/λeff<1, (ii)1a0/λeff<2 and (iii)a0/λeff2.

In regime (i), only the center spatial frequency point in k space can propagate to the far field. Since there are no other plane waves inside the circlea0/λeffto interfere with this center wave, no structured light pattern is observed. In regime (ii), four nearest neighbors are considered when calculating the light pattern in addition to the center spatial frequency (Fig. 1(A)).Because the four nearest neighbors have the same kz component, the self-image distance can be easily derived from Eq. (4):

zR=λeff/(11(λeff/a0)2)
(5)

Equation (5) is similar to the self-image distances for 1D gratings with 1a0/λeff<2 [13

13. E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98(1-3), 132–140 (1993). [CrossRef]

]. In regime (iii), additional plane waves with higher spatial frequencies contribute to the far-field light pattern (Fig. 1(B)). The increased ratio of a0/λeff results in an increase in the number of plane waves that need to be considered in Eq. (4).

We simulated the light patterns at the different regimes above using the scalar wave theory (Figs. 1(C)1(D)). Since transmission through the hole array is not considered in this model, we assumed that light was uniformly transmitted through the holes and not through the film. In experiment, however, the transmission is enhanced by surface plasmons and depends strongly on the material property and geometry of the structures (Figs. 1(E)1(F)).For a more accurate calculation and to compare with experiments, we carried out 3D finite difference time domain (FDTD) simulations (Lumerical® software package) .In the FDTD simulations (Figs. 1(G)1(H)), we used plane wave illumination polarized along the x axis and a uniform mesh of 4 nm. Periodic boundaries were used in the x and y directions and perfectly matched layer (PML) was used in the z direction. The optical constants of Au were taken from Johnson and Christy [18

18. P. B. Johnson and R. W. Christy, “Optical-constants of Noble-metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

].Since we are only interested in the self-image distances, we only focused on the light patterns on a relative intensity scale; for all figures, yellow represents the strongest intensity.

To compare the light patterns in the three regimes, we used Au square hole arrays with a0 = 600 nm and 1200 nm, and continuous wave laser illumination λeff = 488 nm, 543 nm and 633 nm. Large-area (>1 cm2) Au hole array films with film thicknesses of 130 nm and hole diameters of 150 nm were fabricated using the PEEL technique [19

19. J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006). [CrossRef] [PubMed]

]. Homogenous illumination was obtained by passing the laser light, delivered by an optical fiber, through a collimator. The 3D light patterns generated from the hole arrays were then imaged using a confocal microscope (Nikon® D-Eclipse C1) with z steps of 0.1 μm. The light was collected through a pinhole in the confocal microscope, which removed out of focus light and improved the resolution of the images.

In regime (i) (a0/λeff<1), we tested the casea0/λeff=0.948 (a0 = 600 nm, λeff = 633 nm).As expected, no periodic structure was observed because there was no interference of plane waves (data not shown). In regime (ii) (1a0/λeff<2), we tested a0/λeff=1.230 (Fig. 1(A)). The corresponding scalar wave simulation showed periodic self-images in the z direction (Fig. 1(C)) that agreed with experiment, where we measured the self-image distances to be 1.2 μm, about 20% less than the Talbot distance zT=2a02/λeff=1.48μm (Fig. 1(E)). Based on the analytical expression given by Eq. (5), the self-image distance was calculated to be zR=1.2μm, which agreed with the measurement. The light pattern from the FDTD simulation also matched with our measurements (Fig. 1(G)).

In regime (iii) (a0/λeff2), self-images satisfying I(x, y, z + zR) = I(x, y, z) can only be observed when the initial structure satisfies certain specific conditions [17

17. T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, “Exact self-imaging of transversely periodic fields,” J. Opt. Soc. Am. A 21(8), 1424–1429 (2004). [CrossRef] [PubMed]

]. Although exact self-images do not exist for the periodic hole arrays we used, we found that light patterns at certainzRdo not deviate significantly from the initial structure. Since these differences are barely distinguishable from noise from experimental misalignment of the light source, we still designate these planes “self-image planes”. For periodic hole arrays with a0/λeff=1.9(Fig. 1(B)), we observed three self-images within 10 μm from the film surface in the measurement (Fig. 1(F)): the first self-image appeared at 4 μm, the second at 7.1 μm, and the third at 9.6 μm. Compared to the first and the second self-images, the third self-image was of weaker intensity, and deviations from the initial structure were also more pronounced. The measurement was very different from the paraxial Talbot effect, where the Talbot distance zT=2a02/λeff=4.55μm indicates two repeats within 10 μm from the surface of the film with the second repeat at z=2zT=9.1μm. We note that the measured self-images were not periodic in the longitudinal direction, and the fractional Talbot planes commonly observed in the paraxial Talbot effect were also missing. These differences can be explained by the breakdown of the paraxial approximation, which affects the interference patterns at both the Talbot planes and the fractional Talbot planes. Both scalar wave (Fig. 1(D)) and FDTD simulations (Fig. 1(H)) showed similar self-imaging behavior.

Besides resolving the 3D light pattern from a perfect initial structure, we investigated how the presence of defects would affect the transmitted light pattern. Understanding the impact of defects on the 3D light pattern can potentially benefit applications that rely on the Talbot effect, including lithography and imaging. First, we studied the influence of a 2D defect, a 1.5-µm diameter hole, within the a0 = 1.2µm periodic Au hole array film in regime (iii) witha0/λeff = 2.20. We oversaturated the intensity at the position of the defect so that the light pattern from the hole arrays and the defect could be measured at the same time. When focused to the hole array film (z = 0), the defect appeared as a bright spot approximately 1.5 µm in diameter (Fig. 3(A)
Fig. 3 2D defects on the film were healed gradually in the self-image planes. Light patterns at self-image distances (A)z = 0 μm, (B)z = 4.4 μm, (C)z = 8.4 μm and (D)z = 13.1 μm. (E) the yz cross-section. As the light propagated in the z direction, the intensity of the defect decreased while the size of the defect increased. (a0 = 1.2 μm, λeff = 543 nm).
). The presence of the defect resulted in the disappearance of severalsmall bright spots corresponding to the holes in the hole array film. At the first self-image distance, some of the missing holes began to be restored (Fig. 3(B)). At the second (Fig. 3(C)) and third self-image planes (Fig. 3(D)), more missing holes were reconstructed, and the intensity of the defect continuously diminished. This process where defects in the initial structure are healed in the self-images can be seen clearly in the y-z cross-section (Fig. 3(E)).In the presence of the 2D defect, the hole array still generated self-images, and the self-image distances were not affected. Effectively, the defect showed behavior similar to a point source emitting spherical waves, whereas the size of the defect became larger, the intensity decreased in the propagation direction. Light through the defect and the periodic hole array appeared to evolve independently from each other. At self-image planes farther than 10 µm, the influence of the defect became very weak, and defect-free self-images were again observed.

In addition to defects within the hole array film, we studied the influence of 3D defects away from the surface of the film. We used anisotropic Au pyramid particles [19

19. J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006). [CrossRef] [PubMed]

] approximately 3 μm in diameter and embedded them in a uniform poly(dimethylsiloxane) (PDMS) matrix as scattering defects. The pyramids were distributed randomly 11 μm away from the surface of the film. At the film surface (z = 0 µm), we only observed the periodic hole array; the particle was not visible (Fig. 4(A)
Fig. 4 Defect away from the surface of the film was healed in the self-images of the hole array pattern. The light patterns at (A) the film surface z = 0 μm, (B) the position of the particles z = 11.2 μm(C) the self-image planes at z = 13.5 μm and (D) the self-image planes at z = 29 μm. (E) yz cross-section of the 3D light pattern showed the particle had a very local influence on the light pattern. (a0 = 1.2 μm, λ = 543 nm, n = 1.4)
). When we focused to z = 11.2 µm (Fig. 4(B)), two bright spots in the light pattern were missing because of scattering by the pyramidal particle. The missing bright spots were restored in the self-images as the light propagated in the z direction, and only one bright spot was missing at the self-image plane 2.3 μm from the particle (Fig. 4(C)). As the propagation distance increased, the periodic pattern was completely restored as if there were no defect in the light path (Fig. 4(D)). In the y-z cross-section, the self-imaging property of the periodic hole array was unchanged, and the influence of the particle was limited to a very local region in the longitudinal direction (Fig. 4(E)).

The healing effect observed for the two types of defects can be explained based on the lack of constructive interference of plane waves. Defects and the periodic hole array are composed of plane waves with different spatial frequencies, where the periodic pattern only has plane waves with spatial frequencies k = (2πm1/a0,2πm2/a0,kzm1,m2).In contrast, finite-size defects can be decomposed into plane waves with many other different spatial frequencies. At the self-image distances, the hole array pattern reappears because plane waves with k=(2πm1/a0,2πm2/a0,kzm1,m2)are in phase and interference constructively. Most of the plane waves composing the defects, however, will have large deviations from the in-phase condition and destructively interfere. With increased propagation distance, deviations from the in-phase conditions become larger, which lead to the gradual disappearance of the defects.

In conclusion, we demonstrated the non-paraxial Talbot effect at optical frequencies. We found that self-images of the initial 2D periodic structure were not necessarily periodic in the longitudinal direction and that phase matching curves could be used to understand and predict changes in the non-paraxial self-image distances compared to the classical Talbot effect. Interestingly, the non-paraxial Talbot effect shared the same healing property as the paraxial Talbot effect. Defects with in the initial hole array structure as well as defects above the film surface were healed in the self-images of the hole array. Although this work was performed at optical frequencies, we anticipate our findings can be applied to other frequency ranges as well as other fields of physics.

Acknowledgments

This work was supported by the NIH-Director’s Pioneer Award (DP1OD003899). This work made use of the NUANCE Center facilities, supported by NSF-MRSEC, NSF-NSEC and the Keck Foundation and the Center of Nanoscale Materials supported by the US Department of Energy, Office of Basic Energy Sciences (contract no. DE-AC02-06CH11357).

References and links

1.

M. Masud, Classical Optics and its Applications (Cambridge University Press, 2002), Chap. 18.

2.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995). [CrossRef] [PubMed]

3.

M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express 15(15), 9692–9700 (2007). [CrossRef] [PubMed]

4.

L. Rayleigh, “On copying diffraction gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).

5.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996). [CrossRef]

6.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27(6), 2931–2937 (2009). [CrossRef]

7.

Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24(19), 3162–3166 (1985). [CrossRef] [PubMed]

8.

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007). [CrossRef]

9.

A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29(29), 4337–4340 (1990). [CrossRef] [PubMed]

10.

H. Dammann, G. Groh, and M. Kock, “Restoration of faulty images of periodic objects by means of self-imaging,” Appl. Opt. 10(6), 1454–1455 (1971). [CrossRef] [PubMed]

11.

W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17(23), 20966–20974 (2009). [CrossRef] [PubMed]

12.

W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57(6), 772–775 (1967). [CrossRef]

13.

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98(1-3), 132–140 (1993). [CrossRef]

14.

J. D. Ring, J. Lindberg, C. J. Howls, and M. R. Dennis, “Aberration-like cusped focusing in the post-paraxial Talbot effect,” Opt. Lett. submitted.

15.

M. H. Chowdhury, J. M. Catchmark, and J. R. Lakowicz, “Imaging three-dimensional light propagation through periodic nanohole arrays using scanning aperture microscopy,” Appl. Phys. Lett. 91(10), 103118 (2007). [CrossRef] [PubMed]

16.

W. W. Zhang, C. L. Zhao, J. Y. Wang, and J. S. Zhang, “An experimental study of the plasmonic Talbot effect,” Opt. Express 17(22), 19757–19762 (2009). [CrossRef] [PubMed]

17.

T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, “Exact self-imaging of transversely periodic fields,” J. Opt. Soc. Am. A 21(8), 1424–1429 (2004). [CrossRef] [PubMed]

18.

P. B. Johnson and R. W. Christy, “Optical-constants of Noble-metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

19.

J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006). [CrossRef] [PubMed]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(110.6760) Imaging systems : Talbot and self-imaging effects
(240.6680) Optics at surfaces : Surface plasmons
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: April 18, 2012
Revised Manuscript: June 2, 2012
Manuscript Accepted: June 2, 2012
Published: June 12, 2012

Citation
Yi Hua, Jae Yong Suh, Wei Zhou, Mark D. Huntington, and Teri W. Odom, "Talbot effect beyond the paraxial limit at optical frequencies," Opt. Express 20, 14284-14291 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14284


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References

  1. M. Masud, Classical Optics and its Applications (Cambridge University Press, 2002), Chap. 18.
  2. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A51(1), R14–R17 (1995). [CrossRef] [PubMed]
  3. M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express15(15), 9692–9700 (2007). [CrossRef] [PubMed]
  4. L. Rayleigh, “On copying diffraction gratings, and on some phenomena connected therewith,” Philos. Mag.11, 196–205 (1881).
  5. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt.43(10), 2139–2164 (1996). [CrossRef]
  6. A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B27(6), 2931–2937 (2009). [CrossRef]
  7. Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt.24(19), 3162–3166 (1985). [CrossRef] [PubMed]
  8. Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett.90(3), 031107 (2007). [CrossRef]
  9. A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt.29(29), 4337–4340 (1990). [CrossRef] [PubMed]
  10. H. Dammann, G. Groh, and M. Kock, “Restoration of faulty images of periodic objects by means of self-imaging,” Appl. Opt.10(6), 1454–1455 (1971). [CrossRef] [PubMed]
  11. W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express17(23), 20966–20974 (2009). [CrossRef] [PubMed]
  12. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am.57(6), 772–775 (1967). [CrossRef]
  13. E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun.98(1-3), 132–140 (1993). [CrossRef]
  14. J. D. Ring, J. Lindberg, C. J. Howls, and M. R. Dennis, “Aberration-like cusped focusing in the post-paraxial Talbot effect,” Opt. Lett.submitted.
  15. M. H. Chowdhury, J. M. Catchmark, and J. R. Lakowicz, “Imaging three-dimensional light propagation through periodic nanohole arrays using scanning aperture microscopy,” Appl. Phys. Lett.91(10), 103118 (2007). [CrossRef] [PubMed]
  16. W. W. Zhang, C. L. Zhao, J. Y. Wang, and J. S. Zhang, “An experimental study of the plasmonic Talbot effect,” Opt. Express17(22), 19757–19762 (2009). [CrossRef] [PubMed]
  17. T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, “Exact self-imaging of transversely periodic fields,” J. Opt. Soc. Am. A21(8), 1424–1429 (2004). [CrossRef] [PubMed]
  18. P. B. Johnson and R. W. Christy, “Optical-constants of Noble-metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
  19. J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res.39(4), 249–257 (2006). [CrossRef] [PubMed]

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