Abstract

By means of experiment and simulation, we achieve unprecedented insights into the formation of Talbot images to be observed in transmission for light diffracted at wavelength-scale amplitude gratings. Emphasis is put on disclosing the impact and the interplay of various diffraction orders to the formation of Talbot images. They can be manipulated by selective filtering in the Fourier plane. Experiments are performed with a high-resolution interference microscope that measures the amplitude and phase of fields in real-space. Simulations have been performed using rigorous diffraction theory. Specific phase features, such as singularities found in the Talbot images, are discussed. This detailed analysis helps to understand the response of fine gratings. It provides moreover new insights into the fundamental properties of gratings that often find use in applications such as, e.g., lithography, sensing, and imaging.

© 2012 OSA

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2011

X.-B. Song, H.-B. Wang, J. Xiong, K. Wang, X. Zhang, K.-H. Luo, and L.-A. Wu, “Experimental observation of quantum Talbot effects,” Phys. Rev. Lett. 107(3), 033902 (2011).
[CrossRef] [PubMed]

T. Paul, C. Rockstuhl, and F. Lederer, “Integrating cold plasma equations into the Fourier modal method to analyze second harmonic generation at metallic nanostructures,” J. Mod. Opt. 58(5-6), 438–448 (2011).
[CrossRef]

F. J. Torcal-Milla, L. M. Sanchez-Brea, and J. Vargas, “Effect of aberrations on the self-imaging phenomenon,” J. Lightwave Technol. 29(7), 1051–1057 (2011).
[CrossRef]

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express 19(11), 10206–10220 (2011).
[CrossRef] [PubMed]

M.-S. Kim, T. Scharf, M. T. Haq, W. Nakagawa, and H. P. Herzig, “Subwavelength-size solid immersion lens,” Opt. Lett. 36(19), 3930–3932 (2011).
[CrossRef] [PubMed]

2010

2009

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11(3), 033021 (2009).
[CrossRef]

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]

S. Cherukulappurath, D. Heinis, J. Cesario, N. F. van Hulst, S. Enoch, and R. Quidant, “Local observation of plasmon focusingin Talbot carpets,” Opt. Express 17(26), 23772–23784 (2009).
[CrossRef] [PubMed]

2008

2007

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]

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90(9), 091119 (2007).
[CrossRef]

2006

C. Rockstuhl, I. Märki, T. Scharf, M. Salt, H. P. Herzig, and R. Dändliker, “High resolution interference microscopy: a tool for probing optical waves in the far-field on a nanometric length scale,” Curr. Nanosci. 2(4), 337–350 (2006).
[CrossRef]

2005

Y. Lu, C. Zhou, and H. Luo, “Talbot effect of a grating with different kinds of flaws,” J. Opt. Soc. Am. A 22(12), 2662–2667 (2005).
[CrossRef] [PubMed]

H. Gundlach, “From the history of microscopy: Abbe’s diffraction trials,” Innovation, The Magazine from Carl Zeiss 15, 18–23 (2005).

2002

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).
[CrossRef]

G. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A: Pure Appl. Opt 4(6), S376–S380 (2002).
[CrossRef]

1997

1996

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

1995

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]

1994

1993

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

1990

1989

1987

1985

A. Ko?odziejczyk, “Realization of Fourier images without using a lens by sampling the optical object,” J. Mod. Opt. 32, 74–746 (1985).

1983

1982

S. Yokozeki, “Moiré fringes,” Opt. Lasers Eng. 3(1), 15–27 (1982).
[CrossRef]

1981

H. Köhler, “On Abbe’s theory of image formation in the microscope,” Opt. Acta (Lond.) 28(12), 1691–1701 (1981).
[CrossRef]

1973

1969

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” J. Mod. Opt. 16, 281–287 (1969).

1965

1881

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

1873

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech 9(1), 413–418 (1873).
[CrossRef]

1836

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Abbe, E.

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech 9(1), 413–418 (1873).
[CrossRef]

Ambrosini, D.

G. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A: Pure Appl. Opt 4(6), S376–S380 (2002).
[CrossRef]

Baruchel, J.

Berry, M.

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

Bhattacharya, J. C.

Bryngdahl, O.

Burow, R.

Cerrina, F.

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]

Cesario, J.

Chang, R.-C.

Chapman, M. S.

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]

Chen, Y.

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90(9), 091119 (2007).
[CrossRef]

Cheng, C.

Cheng, Y. C.

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]

Cheng, Y.-S.

Cherukulappurath, S.

Cloetens, P.

Cronin, A. D.

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11(3), 033021 (2009).
[CrossRef]

Dändliker, R.

C. Rockstuhl, I. Märki, T. Scharf, M. Salt, H. P. Herzig, and R. Dändliker, “High resolution interference microscopy: a tool for probing optical waves in the far-field on a nanometric length scale,” Curr. Nanosci. 2(4), 337–350 (2006).
[CrossRef]

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).
[CrossRef]

David, C.

de Abajo, F.

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90(9), 091119 (2007).
[CrossRef]

De Martino, C.

Dennis, M. R.

Edgar, R. F.

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” J. Mod. Opt. 16, 281–287 (1969).

Eiju, T.

Ekstrom, C. R.

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]

Elssner, K.-E.

Enoch, S.

García de Abajo, F. J.

Grzanna, J.

Guigay, J. P.

Gundlach, H.

H. Gundlach, “From the history of microscopy: Abbe’s diffraction trials,” Innovation, The Magazine from Carl Zeiss 15, 18–23 (2005).

Hammond, T. D.

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]

Haq, M. T.

Hariharan, P.

Harzendorf, T.

Heinis, D.

Herzig, H. P.

M.-S. Kim, T. Scharf, M. T. Haq, W. Nakagawa, and H. P. Herzig, “Subwavelength-size solid immersion lens,” Opt. Lett. 36(19), 3930–3932 (2011).
[CrossRef] [PubMed]

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express 19(11), 10206–10220 (2011).
[CrossRef] [PubMed]

M.-S. Kim, T. Scharf, and H. P. Herzig, “Small-size microlens characterization by multiwavelength high-resolution interference microscopy,” Opt. Express 18(14), 14319–14329 (2010).
[CrossRef] [PubMed]

M.-S. Kim, T. Scharf, and H. P. Herzig, “Amplitude and phase measurements of highly focused light in optical data storage systems,” Jpn. J. Appl. Phys. 49(8), 08KA03 (2010).
[CrossRef]

C. Rockstuhl, I. Märki, T. Scharf, M. Salt, H. P. Herzig, and R. Dändliker, “High resolution interference microscopy: a tool for probing optical waves in the far-field on a nanometric length scale,” Curr. Nanosci. 2(4), 337–350 (2006).
[CrossRef]

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).
[CrossRef]

Huang, F.

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90(9), 091119 (2007).
[CrossRef]

Isoyan, A.

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]

Jiang, F.

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]

Kim, M.-S.

Klein, S.

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

Köhler, H.

H. Köhler, “On Abbe’s theory of image formation in the microscope,” Opt. Acta (Lond.) 28(12), 1691–1701 (1981).
[CrossRef]

Kolodziejczyk, A.

A. Ko?odziejczyk, “Realization of Fourier images without using a lens by sampling the optical object,” J. Mod. Opt. 32, 74–746 (1985).

Kurtsiefer, Ch.

Lederer, F.

T. Paul, C. Rockstuhl, and F. Lederer, “Integrating cold plasma equations into the Fourier modal method to analyze second harmonic generation at metallic nanostructures,” J. Mod. Opt. 58(5-6), 438–448 (2011).
[CrossRef]

Li, L.

Lohmann, A. W.

Lu, Y.

Luo, H.

Luo, K.-H.

X.-B. Song, H.-B. Wang, J. Xiong, K. Wang, X. Zhang, K.-H. Luo, and L.-A. Wu, “Experimental observation of quantum Talbot effects,” Phys. Rev. Lett. 107(3), 033902 (2011).
[CrossRef] [PubMed]

Marconi, M.

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]

Märki, I.

C. Rockstuhl, I. Märki, T. Scharf, M. Salt, H. P. Herzig, and R. Dändliker, “High resolution interference microscopy: a tool for probing optical waves in the far-field on a nanometric length scale,” Curr. Nanosci. 2(4), 337–350 (2006).
[CrossRef]

McMorran, B. J.

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11(3), 033021 (2009).
[CrossRef]

Menoni, C.

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]

Merkel, K.

Mühlig, S.

Nakagawa, W.

Nesci, A.

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).
[CrossRef]

Noponen, E.

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

Nowak, S.

Oreb, B. F.

Paoletti, D.

G. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A: Pure Appl. Opt 4(6), S376–S380 (2002).
[CrossRef]

Paul, T.

T. Paul, C. Rockstuhl, and F. Lederer, “Integrating cold plasma equations into the Fourier modal method to analyze second harmonic generation at metallic nanostructures,” J. Mod. Opt. 58(5-6), 438–448 (2011).
[CrossRef]

Pfau, T.

Pritchard, D. E.

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]

Quidant, R.

Rayleigh, L.

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

Rocca, J.

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]

Rockstuhl, C.

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express 19(11), 10206–10220 (2011).
[CrossRef] [PubMed]

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[CrossRef]

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X.-B. Song, H.-B. Wang, J. Xiong, K. Wang, X. Zhang, K.-H. Luo, and L.-A. Wu, “Experimental observation of quantum Talbot effects,” Phys. Rev. Lett. 107(3), 033902 (2011).
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Figures (16)

Fig. 1
Fig. 1

Schematic of the HRIM system. A Bertrand lens is inserted only to image the back focal plane of the objective (Fourier plane). The Talbot images will be recorded without the Bertrand lens. Images of the back focal plane are presented in the insets in order to visualize diffraction orders at different wavelengths: diffraction patterns of a 2-µm-period 1D amplitude grating for three different wavelengths (642 nm, 532 nm, and 405 nm).

Fig. 2
Fig. 2

The x-z slices of measured 3D intensity distributions, which exhibits Talbot images emerging from a 2-µm period grating for different wavelengths: (a) 642 nm, (b) 532 nm, and (c) 405 nm. The grating surface is set to z = 0 µm. The intensities are normalized.

Fig. 3
Fig. 3

The x-z slices of simulated 3D intensity distributions for a 2-µm period grating at different wavelengths (corresponding to Fig. 2): (a) 642 nm, (b) 532 nm, and (c) 405 nm. The intensities are normalized and the grating surface is at z = 0 µm.

Fig. 4
Fig. 4

Experimental arrangements for a 2-µm period grating: (a) in air and (b) in immersion oil. A schematic, the back focal plane of the objective, and the measured longitudinal intensity distributions are shown for each case. We can see the different diffraction angles, and therefore the number of collected diffraction orders differs due to the immersion effect with the fixed acceptance cone angle. The intensities are normalized and the grating surface is at z = 0 µm.

Fig. 5
Fig. 5

The variation of the NA and measured Talbot images of both intensity and phase distributions for the 2-µm grating in oil immersion: (a) NA = 0.7 and (b) NA = 1.4. The upper row shows the schematic and the back focal plane of each NA case. The lower row shows the corresponding longitudinal intensity and phase distributions of each case. The intensities are normalized and the grating surface is at z = 0 µm.

Fig. 6
Fig. 6

(a) A single grating-like pattern represents a snapshot of a plane wave propagating in the positive z direction. (b) Moiré patterns are formed as tilt fringes of the two plane-wave interference. Please note that it does not support the sinusoidal fringe patterns. For three plane waves, as in (c), repetitive modulations in the z direction appear: a representation of the conventional Talbot image. Two self-image planes are indicated and the distance between them leads to the half Talbot length (ZT/2) since the half of the lateral period is shifted.

Fig. 7
Fig. 7

Measurements and simulations for a 1-µm-period grating in oil immersion at 642-nm wavelength when all propagating diffraction orders are contributing to the Talbot image. (a) The CCD image of the back focal plane of the objective with five propagating diffraction orders. The measured longitudinal (b) intensity and (c) phase distributions. The corresponding simulations for (d) intensity and (e) phase. The intensities are normalized and the grating surface is at z = 0 µm.

Fig. 8
Fig. 8

The measured intensity distribution emerging form a 1-µm-period grating in oil immersion at 642 nm wavelength when only the 0th order passes through the HRIM. (a) The CCD image of the back focal plane of the objective and (b) the longitudinal intensity distribution. The intensity is normalized and the grating surface is at z = 0 µm. The diaphragm cuts off the all higher diffraction orders except the 0th order.

Fig. 9
Fig. 9

The scenario for the 0th and +1st orders passing through the HRIM: (a) the illustration of the two-plane-wave interference, (b) the CCD image of the back focal plane, (c) the measured and (d) simulated longitudinal intensity distributions. The intensities are normalized and the grating surface is at z = 0 µm. The dark obstruction blocks the −1st order and the diaphragm cuts off the ±2nd orders.

Fig. 10
Fig. 10

The scenario for the +1st and +2nd orders passing through the HRIM: (a) the illustration of the two-plane-wave interference, (b) the CCD image of the back focal plane, (c) the measured and (d) simulated longitudinal intensity distributions. The intensities are normalized and the grating surface is at z = 0 µm. The dark obstruction blocks the 0th, −1st and −2nd orders.

Fig. 11
Fig. 11

The scenario for the ±1st orders passing through the HRIM. (a) The illustration of the two-plane-wave interference and (b) the CCD image of the back focal plane. The measured (c) intensity and (d) phase distributions along the longitudinal plane (x-z plane). The corresponding simulation results for the (d) intensity and (f) phase. The intensities are normalized and the grating surface is at z = 0 µm. The dark obstruction blocks the 0th order and the diaphragm cuts off the ±2nd orders.

Fig. 12
Fig. 12

The scenario for the ±2nd orders passing through the HRIM: (a) the CCD image of the back focal plane, (b) the measured intensity distributions along the longitudinal plane (x-z plane). The simulation results for the (c) intensity and (d) phase. The intensities are normalized and the grating surface is at z = 0 µm. The dark obstruction blocks the 0th and ±1st orders. Due to low contrast of interference fringes, the phase distribution was not properly measured.

Fig. 13
Fig. 13

The scenario for the −1st and +2nd orders passing through the HRIM: simulated (a) intensity and (b) phase distributions along the longitudinal plane (x-z plane). The intensity is normalized and the grating surface is at z = 0 µm. Due to low intensity, the fields data were not properly measured.

Fig. 14
Fig. 14

The scenario for the ±1st and ±2nd orders passing through the HRIM. (a) The CCD image of the back focal plane and the measured (b) intensity and (c) phase distributions along the longitudinal plane (x-z plane). The corresponding simulation results for the (d) intensity and (e) phase. The intensities are normalized and the grating surface is at z = 0 µm. The dark obstruction blocks the 0th order.

Fig. 15
Fig. 15

The scenario for the 0th, +1st and +2nd orders passing through the HRIM. (a) The CCD image of the back focal plane, (b) the measured and (c) the simulated longitudinal intensity distributions. The intensities are normalized and the grating surface is at z = 0 µm. The dark obstruction blocks the −1st and −2nd orders.

Fig. 16
Fig. 16

The scenario for the 0th and ±1st orders passing through the HRIM. (a) The CCD image of the back focal plane and the measured (b) intensity and (c) phase distributions along the longitudinal plane (x-z plane). The corresponding simulation results for the (d) intensity and (e) phase. The intensities are normalized and the grating surface is at z = 0 µm. The diaphragm cuts off the ±2nd orders.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

sin θ m = mλ Λ ,
Z T = λ 1 1 ( λ Λ ) 2 .
Z T = 2 Λ 2 λ .
λ medium = λ n .
ΔOPL=( n oil n air )l,

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