Abstract

The propagation of stationary wave fields that exhibit simultaneously lateral and longitudinal periodicity is investigated. As a model, we use a Fabry–Perot resonator with periodically structured mirrors under monochromatic plane wave illumination. The resonator leads to a longitudinal periodicity, the grating mirrors to a lateral periodicity. The angular spectrum of the transmitted wave field is given as the product of two terms, one related to the lateral, the other to the longitudinal properties. Its modal structure can vary significantly depending on the ratio of the lateral and longitudinal periods and the reflectivity of the resonator's mirrors. For example, it is possible to generate bandgap behavior despite the fact that the periods may be significantly larger than the wavelength. The results of this investigation apply to the design of phase-coupled array resonators and multiplexers.

© 2009 Optical Society of America

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References

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  1. M. von Laue, “Die Freiheitsgrade von Strahlenbündeln,” Ann. Phys. (Leipzig) 44, 1197-1212 (1914).
  2. R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90, 222-244 (2002).
    [CrossRef]
  3. K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).
    [CrossRef]
  4. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125-127(1984).
    [CrossRef] [PubMed]
  5. J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771-1773 (1988).
    [CrossRef]
  6. F. X. D'Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816-818 (1989).
    [CrossRef]
  7. L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28, 4668-4678(1989).
    [CrossRef] [PubMed]
  8. M. Wrage, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, and A. P. Napartovich, “Phase locking in a multicore fiber laser by means of a Talbot resonator,” Opt. Lett. 25, 1436-1438 (2000).
    [CrossRef]
  9. Q. Li, P. Zhao, and W. Guo, “Amplitude compensation of a diode laser array phase locked with a Talbot cavity,” Appl. Phys. Lett. 89, 231120 (2006).
    [CrossRef]
  10. V. Eckhouse, A. A. Ishaaya, L. Shimshi, N. Davidson, and A. Friesem, “Intracavity coherent addition of lasers,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker. eds. (SPIE Press, 2008), Chap. 6.
    [CrossRef]
  11. A. W. Lohmann, D. Mendlovic, and G. Shabtay, “Talbot (1836), Montgomery (1967), Lau (1948) and Wolf (1955) on periodicity in optics,” Pure Appl. Opt. 7, 1121-1124 (1998).
    [CrossRef]
  12. H. F. Talbot, “Facts relating to optical science, No. IV,” Phil. Mag. 9, 401-407 (1836).
  13. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772-778 (1967).
    [CrossRef]
  14. A. W. Lohmann and D. A. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
    [CrossRef]
  15. K. Patorski and P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23-31 (1984).
    [CrossRef]
  16. K. Banaszek, K. Wodkiewicz, and W. P. Schleich, “Talbot effect in phase space: compact summation formula,” Opt. Express 2, 169-172 (1998).
    [CrossRef] [PubMed]
  17. J. Jahns, E. El Joudi, D. Hagedorn, and S. Kinne, “Talbot interferometer as a time filter,” Optik (Jena) 112, 295-298(2001).
    [CrossRef]
  18. J. Jahns and A. W. Lohmann, “Temporal filtering by double diffraction,” Appl. Opt. 43, 4339-4344 (2004).
    [CrossRef] [PubMed]
  19. H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67-73 (2007).
    [CrossRef]
  20. G. Indebetouw, “Self-imaging through a Fabry-Pérot interferometer,” Opt. Acta 30, 1463-1471 (1983).
    [CrossRef]
  21. S. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Commun. (to be published).

2007

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67-73 (2007).
[CrossRef]

2006

Q. Li, P. Zhao, and W. Guo, “Amplitude compensation of a diode laser array phase locked with a Talbot cavity,” Appl. Phys. Lett. 89, 231120 (2006).
[CrossRef]

2004

2002

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90, 222-244 (2002).
[CrossRef]

2001

J. Jahns, E. El Joudi, D. Hagedorn, and S. Kinne, “Talbot interferometer as a time filter,” Optik (Jena) 112, 295-298(2001).
[CrossRef]

2000

1998

K. Banaszek, K. Wodkiewicz, and W. P. Schleich, “Talbot effect in phase space: compact summation formula,” Opt. Express 2, 169-172 (1998).
[CrossRef] [PubMed]

A. W. Lohmann, D. Mendlovic, and G. Shabtay, “Talbot (1836), Montgomery (1967), Lau (1948) and Wolf (1955) on periodicity in optics,” Pure Appl. Opt. 7, 1121-1124 (1998).
[CrossRef]

1989

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).
[CrossRef]

F. X. D'Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816-818 (1989).
[CrossRef]

L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28, 4668-4678(1989).
[CrossRef] [PubMed]

1988

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771-1773 (1988).
[CrossRef]

1984

K. Patorski and P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23-31 (1984).
[CrossRef]

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125-127(1984).
[CrossRef] [PubMed]

1983

G. Indebetouw, “Self-imaging through a Fabry-Pérot interferometer,” Opt. Acta 30, 1463-1471 (1983).
[CrossRef]

1971

A. W. Lohmann and D. A. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
[CrossRef]

1967

1914

M. von Laue, “Die Freiheitsgrade von Strahlenbündeln,” Ann. Phys. (Leipzig) 44, 1197-1212 (1914).

1836

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

Banaszek, K.

D'Amato, F. X.

F. X. D'Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816-818 (1989).
[CrossRef]

Davidson, N.

V. Eckhouse, A. A. Ishaaya, L. Shimshi, N. Davidson, and A. Friesem, “Intracavity coherent addition of lasers,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker. eds. (SPIE Press, 2008), Chap. 6.
[CrossRef]

Eckhouse, V.

V. Eckhouse, A. A. Ishaaya, L. Shimshi, N. Davidson, and A. Friesem, “Intracavity coherent addition of lasers,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker. eds. (SPIE Press, 2008), Chap. 6.
[CrossRef]

El Joudi, E.

J. Jahns, E. El Joudi, D. Hagedorn, and S. Kinne, “Talbot interferometer as a time filter,” Optik (Jena) 112, 295-298(2001).
[CrossRef]

Fischer, D.

Friesem, A.

V. Eckhouse, A. A. Ishaaya, L. Shimshi, N. Davidson, and A. Friesem, “Intracavity coherent addition of lasers,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker. eds. (SPIE Press, 2008), Chap. 6.
[CrossRef]

Glas, P.

Grunwald, R.

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67-73 (2007).
[CrossRef]

Guo, W.

Q. Li, P. Zhao, and W. Guo, “Amplitude compensation of a diode laser array phase locked with a Talbot cavity,” Appl. Phys. Lett. 89, 231120 (2006).
[CrossRef]

Hagedorn, D.

J. Jahns, E. El Joudi, D. Hagedorn, and S. Kinne, “Talbot interferometer as a time filter,” Optik (Jena) 112, 295-298(2001).
[CrossRef]

Helfert, S.

S. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Commun. (to be published).

Huneke, B.

S. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Commun. (to be published).

Indebetouw, G.

G. Indebetouw, “Self-imaging through a Fabry-Pérot interferometer,” Opt. Acta 30, 1463-1471 (1983).
[CrossRef]

Ishaaya, A. A.

V. Eckhouse, A. A. Ishaaya, L. Shimshi, N. Davidson, and A. Friesem, “Intracavity coherent addition of lasers,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker. eds. (SPIE Press, 2008), Chap. 6.
[CrossRef]

Jahns, J.

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67-73 (2007).
[CrossRef]

J. Jahns and A. W. Lohmann, “Temporal filtering by double diffraction,” Appl. Opt. 43, 4339-4344 (2004).
[CrossRef] [PubMed]

J. Jahns, E. El Joudi, D. Hagedorn, and S. Kinne, “Talbot interferometer as a time filter,” Optik (Jena) 112, 295-298(2001).
[CrossRef]

S. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Commun. (to be published).

Kapon, E.

Katz, J.

Kinne, S.

J. Jahns, E. El Joudi, D. Hagedorn, and S. Kinne, “Talbot interferometer as a time filter,” Optik (Jena) 112, 295-298(2001).
[CrossRef]

Knuppertz, H.

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67-73 (2007).
[CrossRef]

Leger, J. R.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771-1773 (1988).
[CrossRef]

Leitner, M.

Li, Q.

Q. Li, P. Zhao, and W. Guo, “Amplitude compensation of a diode laser array phase locked with a Talbot cavity,” Appl. Phys. Lett. 89, 231120 (2006).
[CrossRef]

Liu, L.

Lohmann, A. W.

J. Jahns and A. W. Lohmann, “Temporal filtering by double diffraction,” Appl. Opt. 43, 4339-4344 (2004).
[CrossRef] [PubMed]

A. W. Lohmann, D. Mendlovic, and G. Shabtay, “Talbot (1836), Montgomery (1967), Lau (1948) and Wolf (1955) on periodicity in optics,” Pure Appl. Opt. 7, 1121-1124 (1998).
[CrossRef]

A. W. Lohmann and D. A. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
[CrossRef]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, and G. Shabtay, “Talbot (1836), Montgomery (1967), Lau (1948) and Wolf (1955) on periodicity in optics,” Pure Appl. Opt. 7, 1121-1124 (1998).
[CrossRef]

Montgomery, W. D.

Napartovich, A. P.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).
[CrossRef]

K. Patorski and P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23-31 (1984).
[CrossRef]

Piestun, R.

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90, 222-244 (2002).
[CrossRef]

Roychoudhuri, C.

F. X. D'Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816-818 (1989).
[CrossRef]

Schleich, W. P.

Scott, M. L.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771-1773 (1988).
[CrossRef]

Shabtay, G.

A. W. Lohmann, D. Mendlovic, and G. Shabtay, “Talbot (1836), Montgomery (1967), Lau (1948) and Wolf (1955) on periodicity in optics,” Pure Appl. Opt. 7, 1121-1124 (1998).
[CrossRef]

Shamir, J.

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90, 222-244 (2002).
[CrossRef]

Shimshi, L.

V. Eckhouse, A. A. Ishaaya, L. Shimshi, N. Davidson, and A. Friesem, “Intracavity coherent addition of lasers,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker. eds. (SPIE Press, 2008), Chap. 6.
[CrossRef]

Siebert, E. T.

F. X. D'Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816-818 (1989).
[CrossRef]

Silva, D. A.

A. W. Lohmann and D. A. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
[CrossRef]

Szwaykowski, P.

K. Patorski and P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23-31 (1984).
[CrossRef]

Talbot, H. F.

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

Veldkamp, W. B.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771-1773 (1988).
[CrossRef]

von Laue, M.

M. von Laue, “Die Freiheitsgrade von Strahlenbündeln,” Ann. Phys. (Leipzig) 44, 1197-1212 (1914).

Vysotsky, D. V.

Wodkiewicz, K.

Wrage, M.

Yariv, A.

Zhao, P.

Q. Li, P. Zhao, and W. Guo, “Amplitude compensation of a diode laser array phase locked with a Talbot cavity,” Appl. Phys. Lett. 89, 231120 (2006).
[CrossRef]

Ann. Phys. (Leipzig)

M. von Laue, “Die Freiheitsgrade von Strahlenbündeln,” Ann. Phys. (Leipzig) 44, 1197-1212 (1914).

Appl. Opt.

Appl. Phys. Lett.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771-1773 (1988).
[CrossRef]

F. X. D'Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816-818 (1989).
[CrossRef]

Q. Li, P. Zhao, and W. Guo, “Amplitude compensation of a diode laser array phase locked with a Talbot cavity,” Appl. Phys. Lett. 89, 231120 (2006).
[CrossRef]

J. Eur. Opt. Soc. Rapid Commun.

S. Helfert, B. Huneke, and J. Jahns, “Self-imaging effect in multimode waveguides with longitudinal periodicity,” J. Eur. Opt. Soc. Rapid Commun. (to be published).

J. Opt. Soc. Am.

Opt. Acta

G. Indebetouw, “Self-imaging through a Fabry-Pérot interferometer,” Opt. Acta 30, 1463-1471 (1983).
[CrossRef]

K. Patorski and P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23-31 (1984).
[CrossRef]

Opt. Commun.

A. W. Lohmann and D. A. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
[CrossRef]

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67-73 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Jena)

J. Jahns, E. El Joudi, D. Hagedorn, and S. Kinne, “Talbot interferometer as a time filter,” Optik (Jena) 112, 295-298(2001).
[CrossRef]

Phil. Mag.

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

Proc. IEEE

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90, 222-244 (2002).
[CrossRef]

Prog. Opt.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).
[CrossRef]

Pure Appl. Opt.

A. W. Lohmann, D. Mendlovic, and G. Shabtay, “Talbot (1836), Montgomery (1967), Lau (1948) and Wolf (1955) on periodicity in optics,” Pure Appl. Opt. 7, 1121-1124 (1998).
[CrossRef]

Other

V. Eckhouse, A. A. Ishaaya, L. Shimshi, N. Davidson, and A. Friesem, “Intracavity coherent addition of lasers,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker. eds. (SPIE Press, 2008), Chap. 6.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Graphical description of the addressed problem: (a) The shown optical device with reflective gratings, M 1 and M 2 , in a resonator configuration leads to a wave field u ( x , y , z ) that is periodic in both the lateral and the longitudinal direction. (b) Equivalent situation in a planar multimode waveguide with width W p x / 2 (different shadings indicate different refractive indices). The width is not constant with z, but it gets modulated in the z direction with a longitudinal period p z , as indicated here by the notches.

Fig. 2
Fig. 2

(a) Grating interferometer and (b) Fabry–Perot interferometer. The interferometric setup splits up the light from the real light source S into several virtual light sources in (a) the lateral and (b) the longitudinal direction.

Fig. 3
Fig. 3

Fabry–Perot resonator with structured mirrors M 1 and M 2 . The interferometric setup leads to a virtual array of light sources that is (quasi) periodic in lateral and longitudinal directions. The separation of these light sources depends on the geometry of the interferometer.

Fig. 4
Fig. 4

Grating diffraction and Ewald sphere. (a) Grating diffraction. (b) Construction of the angular spectrum. For simplicity, a 2D presentation is shown.

Fig. 5
Fig. 5

Fabry–Perot resonator: (a) Generation of longitudinal modes. For simplicity, only two reflections are shown. (b) Modal spectrum in k space. Notice that it is not generally true that a line with k z = n 2 π / p z crosses the Ewald sphere exactly at k x = 0 .

Fig. 6
Fig. 6

Fabry–Perot resonator with periodically structured mirrors. (a) Setup with structured mirrors. (b) Construction of the angular spectrum. The dots indicate positions on the Ewald sphere, where Talbot and Montgomery conditions are both (approximately) satisfied. Note that since in general, p x p z , it is usually Δ k x Δ k z .

Fig. 7
Fig. 7

3D representation of the Ewald sphere and the equations k x T = m 2 π / p x (green/horizontal lines) and k z M = n 2 π / p z (red/vertical lines). For simplicity, Δ k x = Δ k z in this figure.

Fig. 8
Fig. 8

Ewald sphere representation of the propagating ( k z > 0 ) and reflected ( k z < 0 ) modes in the paraxial domain of the FPI. Note: First, the angular range for the paraxial regime is exaggerated here. Second, the largest value of k z does not necessarily occur exactly at k x = 0 . However, for sufficiently large resonator length L, the modal separation Δ k z will be very small compared to the radius of the Ewald sphere.

Fig. 9
Fig. 9

Exemplary transfer functions for (a) grating and (b) Fabry–Perot interferometer. For simplicity, the transfer function of the GI is shown without a variation in the height of the maxima, since the main purpose is to consider the k x positions.

Fig. 10
Fig. 10

Simulation results for different values of the reflectivity R: (a) 0.01, (b) 0.1, (c) 0.5, and (d) 0.9. In each case, the total angular spectrum is shown in green (tall curves). The red curves show the filter function for the FPI in each case which modulate the transmitted angular spectrum of the GI. For low values of R, the transfer function of the GI is dominant. For large R, the transfer function of FPI becomes significant. In this simulation, p x = 100 μm , p z = 6700 μm , and λ = 1.0 μm .

Tables (1)

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Table 1 Classification of Optical Interferometers a

Equations (14)

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u 0 ( x ) = u 0 ( x + p x ) u z ( x ) = u z + z T ( x ) ,
z T = 2 p x 2 λ .
u z ( x ) = u z + p z ( x ) u 0 ( x ) = m A m e i k x , m x ,
k x , m 2 = ( 2 π ) 2 [ ( 1 λ ) 2 ( m p z ) 2 ] .
k x 2 + k y 2 + k z 2 = ( 2 π / λ ) 2 ,
k T , z m = ± 2 π ( 1 λ ) 2 ( m p x ) 2 , m = 0 , ± 1 , ± 2 , .
k M , x n = ± 2 π ( 1 λ ) 2 ( n p z ) 2 , n = ± 1 , ± 2 ,
( k x M ) 2 + ( k z T ) 2 = ( 2 π λ ) 2 ,
m 2 ( λ 2 L x ) 2 + n 2 ( λ 2 L z ) 2 = 1.
( k x M ) 2 + k y 2 + ( k z T ) 2 = ( 2 π λ ) 2 .
T GI ( k x ) = 1 2 π g ( x ) exp ( i k x x ) d x .
u FPI n ( x , z ) = ( 1 R ) · exp [ i ( k x n x + k z n z ) ] · j = 0 R j exp ( i k z j p z ) .
k x n ¯ ± 2 π p z ( 2 n ¯ p z λ n ¯ 2 ) 1 / 2 ,
T ( k x ) = T GI ( k x ) T FPI ( k x ) .

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