Abstract

Talbot imaging is a well-known effect that causes sinusoidal patterns to be reimaged by diffraction with characteristic period that varies inversely with both wavelength and the square of the spatial frequency. This effect is treated using the Fresnel diffraction integral for fields with sinusoidal ripples in amplitude or phase. The periodic nature is demonstrated and explained, and a sinusoidal approximation is made for the case where the phase or amplitude ripples are small, which allows direct determination of the field for arbitrary propagation distance. Coupled with a straightforward method for calculating the effect in a diverging or converging beam, the Talbot method provides a useful approximation for a class of diffraction problems.

© 2010 Optical Society of America

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References

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  1. W. H. F. Talbot, “Facts relating to optical sciences. No. IV,” Philos. Mag. 9, 401–407 (1836).
  2. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  3. J. Ojeda-Castaneda and E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
    [CrossRef]
  4. M. Testorf and J. Ojeda-Castaneda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. A 13, 119–125(1996).
    [CrossRef]
  5. Q. Lu and C. Zhou, “Rigorous electromagnetic analysis of Talbot effect with the finite-difference time-domain method,” Proc. SPIE 5638, 108–116 (2005).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).
  7. C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L (2007).
    [CrossRef]
  8. P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790 (2010).
  9. J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
    [CrossRef]
  10. P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
    [CrossRef]
  11. T. Milster, “505 Classnotes,” University of Arizona, http://www.optics.arizona.edu/Milster/.
  12. H. O. Carmesin and D. Goldbeck, “Depth map by convergent 3D-Talbot-interferometry,” Optik (Jena) 108, 3 (1998).
  13. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  14. R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
    [CrossRef]
  15. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  16. http://www.breault.com/.

2010

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790 (2010).

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

2009

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

2007

C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L (2007).
[CrossRef]

2005

Q. Lu and C. Zhou, “Rigorous electromagnetic analysis of Talbot effect with the finite-difference time-domain method,” Proc. SPIE 5638, 108–116 (2005).
[CrossRef]

1998

H. O. Carmesin and D. Goldbeck, “Depth map by convergent 3D-Talbot-interferometry,” Optik (Jena) 108, 3 (1998).

1996

1985

J. Ojeda-Castaneda and E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

1983

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

1965

1962

1836

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Burge, J. H.

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790 (2010).

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L (2007).
[CrossRef]

Carmesin, H. O.

H. O. Carmesin and D. Goldbeck, “Depth map by convergent 3D-Talbot-interferometry,” Optik (Jena) 108, 3 (1998).

Goldbeck, D.

H. O. Carmesin and D. Goldbeck, “Depth map by convergent 3D-Talbot-interferometry,” Optik (Jena) 108, 3 (1998).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Jozwicki, R.

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

Keller, J. B.

Lu, Q.

Q. Lu and C. Zhou, “Rigorous electromagnetic analysis of Talbot effect with the finite-difference time-domain method,” Proc. SPIE 5638, 108–116 (2005).
[CrossRef]

Milster, T.

T. Milster, “505 Classnotes,” University of Arizona, http://www.optics.arizona.edu/Milster/.

Ojeda-Castaneda, J.

M. Testorf and J. Ojeda-Castaneda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. A 13, 119–125(1996).
[CrossRef]

J. Ojeda-Castaneda and E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

Sicre, E. E.

J. Ojeda-Castaneda and E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

Talbot, W. H. F.

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

Testorf, M.

Winthrop, J. T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Worthington, C. R.

Zhao, C.

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790 (2010).

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L (2007).
[CrossRef]

Zhou, C.

Q. Lu and C. Zhou, “Rigorous electromagnetic analysis of Talbot effect with the finite-difference time-domain method,” Proc. SPIE 5638, 108–116 (2005).
[CrossRef]

Zhou, P.

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790 (2010).

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

J. Ojeda-Castaneda and E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

Optik (Jena)

H. O. Carmesin and D. Goldbeck, “Depth map by convergent 3D-Talbot-interferometry,” Optik (Jena) 108, 3 (1998).

Philos. Mag.

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

Proc. SPIE

Q. Lu and C. Zhou, “Rigorous electromagnetic analysis of Talbot effect with the finite-difference time-domain method,” Proc. SPIE 5638, 108–116 (2005).
[CrossRef]

C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L (2007).
[CrossRef]

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790 (2010).

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Other

T. Milster, “505 Classnotes,” University of Arizona, http://www.optics.arizona.edu/Milster/.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

http://www.breault.com/.

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Figures (8)

Fig. 1
Fig. 1

Talbot effect illustrated as three-beam interference. The angular spectrum of a laser beam transmitted through a periodic object consists of three plane waves, which combine together to form the periodic replication of the original periodic structure along the z axis.

Fig. 2
Fig. 2

Simulation of the phase and amplitude distribution using the method of three-beam interference. The original phase ripple has an amplitude of 0.05 λ .

Fig. 3
Fig. 3

Propagation in a converging space is converted to equivalent propagation in a collimated space.

Fig. 4
Fig. 4

Transfer function due to a defocused image plane for interferometric measurement of a 50 mm optic with 500 mm defocus error. The wavelength is 632.8 nm . The defocus causes loss of information (phase smoothing) for a frequency above 40 cycles / diameter and causes severe problems for higher frequencies, which show phase reversal.

Fig. 5
Fig. 5

(a) Amplitude and (b) phase variation from a knife edge.

Fig. 6
Fig. 6

(a) Period of the edge diffraction ripples, (b) the amplitude envelop, and (c) the phase envelop due to edge diffraction.

Fig. 7
Fig. 7

(a) Phase and (b) amplitude distribution at x = 0 as a weak phase ripple propagates one Talbot distance. The results from the Fourier analysis were validated with numerical simulation using Asap.

Fig. 8
Fig. 8

(a) Amplitude and (b) phase distributions at some fractional Talbot distance in a converging spherical beam. The calculated equivalent propagation distance for converging the wavefront with a phase grating matches the simulation using Asap. The data from the Asap simulation are shown at appropriate propagation distances. It is clear that the results have a characteristic period defined using the equivalent propagation distance. Also, the presence of higher harmonics is seen in the nonsinusoidal shape at z T / 8 . The amplitude variation is scaled by the inverse area in this figure, and the phase variation is also scaled to show its variation. Only ten cycles at the middle of the aperture are shown in both pictures. The numbers in the figures have units of millimeters.

Equations (30)

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u ( x ) = 1 + α sin ( 2 π x p ) .
U z = 0 ( ξ ) = δ ( ξ ) + 1 2 j α [ δ ( ξ + ξ 0 ) δ ( ξ ξ 0 ) ] ,
H z ( ξ ) = e j 2 π z λ 1 ( λ ξ ) 2 e j 2 π z λ e j π z λ ξ 2 .
U z ( ξ ) = e j 2 π z λ { δ ( ξ ) + 1 2 j α e j π z λ ξ 0 2 [ δ ( ξ + ξ 0 ) δ ( ξ ξ 0 ) ] } = e j 2 π z λ { δ ( ξ ) + 1 2 j α [ cos ( π z λ ξ 0 2 ) j sin ( π z λ ξ 0 2 ) ] [ δ ( ξ + ξ 0 ) δ ( ξ ξ 0 ) ] } .
u z ( x ) = 1 + α [ cos ( π z λ ξ 0 2 ) j sin ( π z λ ξ 0 2 ) ] sin ( 2 π x p ) = 1 + α cos ( π z λ ξ 0 2 ) sin ( 2 π x p ) j α sin ( π z λ ξ 0 2 ) sin ( 2 π x p ) .
A z ( x ) = 1 + 2 α cos ( π z λ ξ 0 2 ) sin ( 2 π x p ) + α 2 sin 2 ( 2 π x p ) 1 + α cos ( 2 π z z T ) sin ( 2 π x p ) + 1 2 α 2 sin 2 ( 2 π z z T ) sin 2 ( 2 π x p ) + ,
ψ z ( x ) = arctan [ α sin ( π z λ ξ 0 2 ) sin ( 2 π x p ) 1 + α cos ( π z λ ξ 0 2 ) sin ( 2 π x p ) ] α sin ( 2 π z z T ) sin ( 2 π x p ) + 1 2 α 2 sin ( 4 π z z T ) sin 2 ( 2 π x p ) + ,
A α cos ( 2 π z z T ) .
W = W sin ( 2 π z z T ) = α 2 π sin ( π z λ p 2 ) ,
u z = 0 ( x ) = e j α sin ( 2 π x p ) = q = + J q ( α ) · e j 2 π q x p ,
u z ( x ) = q = + J q ( α ) · e j π z λ ( q p ) 2 · e j 2 π x q p .
u z = 0 ( x ) = e j α sin ( 2 π x p ) 1 + j α sin ( 2 π x p ) α 2 2 sin 2 ( 2 π x p ) + ,
u z ( x ) = 1 α 2 4 + j α · e j π λ z p 2 sin ( 2 π x p ) + α 2 4 · e j 4 π λ z p 2 cos ( 4 π x p ) .
ψ z ( x ) = α cos ( 2 π z z T ) sin ( 2 π x p ) α 2 4 [ sin ( 8 π z z T ) cos ( 4 π x p ) + 2 sin ( 4 π z z T ) sin 2 ( 2 π x p ) ] + .
A z ( x ) = 1 α 2 4 + α sin ( 2 π z z T ) sin ( 2 π x p ) + α 2 4 [ cos ( 8 π z z T ) cos ( 4 π x p ) + 2 cos 2 ( 2 π z z T ) sin 2 ( 2 π x p ) ] + .
ψ z ( x ) = α cos ( 2 π z z T ) sin ( 2 π x p ) ,
A z ( x ) = 1 + α sin ( 2 π z z T ) sin ( 2 π x p ) .
W = W cos ( 2 π z z T ) = W cos ( π z λ p 2 ) .
W W ( 1 π 2 λ 2 z 2 2 p 2 ) .
Δ Z = Z 2 Z 1 = f 2 ( 1 R 2 1 R 1 ) ,
p = f R 1 p 1 .
W = W · cos ( π λ · Δ Z p 2 ) = W · cos ( π λ R 1 · ( R 1 R 2 ) R 2 · p 1 2 ) .
L e = R 1 ( R 1 R 2 ) R 2 ,
W = W · cos ( π λ · L e p 1 2 ) .
W = W cos ( π λ L e f normalized 2 4 a 1 2 ) .
N f = a 1 2 λ L e ,
W = W cos ( π f normalized 2 4 N f ) .
T F = W W = cos ( π f normalized 2 4 N f ) .
T F = cos ( π λ · Δ Z p 2 ) = cos ( π λ R 1 2 p 1 2 ( 1 R 2 1 R 1 ) ) = cos ( π λ F n 2 f normalized 2 ( 1 R 2 1 R 1 ) ) .
5 λ L e 2 < 0.1 a , N f = a 2 λ L e > 1250.

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