Abstract

A method is presented that expands the scheme of physical optics propagation beyond the Fresnel approximation to include beams that are nonparaxial. The formalism retains most of the calculation advantages of the Fresnel approach; i.e., it is based on a single Fourier transform step. The kernel of the new transformation is no longer separable in Cartesian coordinates; thus the formalism can account for astigmatic coupling effects originating purely from diffraction. The validity limits of the proposed algorithm are explored. Analytical expressions, numerical simulation results, and experimental data are compared.

© 2004 Optical Society of America

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References

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  1. A. Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, I. Oishi, “Efficient coupling between laser diodes with a highly elliptic field and single-mode fibers by means of GIO fibers,” IEEE Photonics Technol. Lett. 13, 1191–1193 (2001).
    [CrossRef]
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  4. Y. M. Engelberg, S. Ruschin, “Coma aberration in diffraction from a narrow slit,” in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE5178, 112–123 (2003).
    [CrossRef]
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  7. Zemax User’s Manual, http://www.zemax.com/updates/index.html ; download file: ZEMAX̲Manual̲2003-02-04.exe.
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    [CrossRef]
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  14. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. L. Frank, “The properties of the Sommerfeld diffraction integral for a large aperture converging beam,” Optik (Stuttgart) 43, 149–157 (1975).
  19. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  20. P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
    [CrossRef]
  21. A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).

2001 (1)

A. Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, I. Oishi, “Efficient coupling between laser diodes with a highly elliptic field and single-mode fibers by means of GIO fibers,” IEEE Photonics Technol. Lett. 13, 1191–1193 (2001).
[CrossRef]

1999 (1)

1998 (1)

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

1997 (2)

1996 (1)

1992 (1)

1991 (1)

1989 (1)

1983 (1)

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981 (1)

1975 (1)

L. Frank, “The properties of the Sommerfeld diffraction integral for a large aperture converging beam,” Optik (Stuttgart) 43, 149–157 (1975).

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Alonso, M. A.

An, Y.

Asatryan, A. A.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Butler, D. J.

Engelberg, Y. M.

Y. M. Engelberg, S. Ruschin, “Coma aberration in diffraction from a narrow slit,” in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE5178, 112–123 (2003).
[CrossRef]

Fiengold, A.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Forbes, G. W.

Fradkin, Z.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Frank, L.

L. Frank, “The properties of the Sommerfeld diffraction integral for a large aperture converging beam,” Optik (Stuttgart) 43, 149–157 (1975).

Geron, A.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gordon, R. L.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1974).

Hrynevych, M.

Kuchiki, S.

A. Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, I. Oishi, “Efficient coupling between laser diodes with a highly elliptic field and single-mode fibers by means of GIO fibers,” IEEE Photonics Technol. Lett. 13, 1191–1193 (2001).
[CrossRef]

Landry, M. J.

Lawrence, G. N.

G. N. Lawrence, “Optical modeling,” in Applied Optics and Optical Engineering Series, R. R. Shanon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1992), Vol. XI, pp. 125–200.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Levy, J.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Liang, C.

Majer, D.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Matmon, G.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Mittas, A.

Ogura, A.

A. Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, I. Oishi, “Efficient coupling between laser diodes with a highly elliptic field and single-mode fibers by means of GIO fibers,” IEEE Photonics Technol. Lett. 13, 1191–1193 (2001).
[CrossRef]

Ohta, K.

A. Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, I. Oishi, “Efficient coupling between laser diodes with a highly elliptic field and single-mode fibers by means of GIO fibers,” IEEE Photonics Technol. Lett. 13, 1191–1193 (2001).
[CrossRef]

Oishi, I.

A. Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, I. Oishi, “Efficient coupling between laser diodes with a highly elliptic field and single-mode fibers by means of GIO fibers,” IEEE Photonics Technol. Lett. 13, 1191–1193 (2001).
[CrossRef]

Rafaeli, E.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Rudman, M.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Rupert, J. W.

Ruschin, S.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Y. M. Engelberg, S. Ruschin, “Coma aberration in diffraction from a narrow slit,” in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE5178, 112–123 (2003).
[CrossRef]

Rutt, H. N.

Shekel, E.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Sheppard, C. J. R.

Shiraishi, K.

A. Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, I. Oishi, “Efficient coupling between laser diodes with a highly elliptic field and single-mode fibers by means of GIO fibers,” IEEE Photonics Technol. Lett. 13, 1191–1193 (2001).
[CrossRef]

Southwell, W. H.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Steane, A. M.

Tidhar, G.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Török, P.

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Varga, P.

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Vecht, J.

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).

Zeng, X.

Appl. Opt. (2)

IEEE Photonics Technol. Lett. (1)

A. Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, I. Oishi, “Efficient coupling between laser diodes with a highly elliptic field and single-mode fibers by means of GIO fibers,” IEEE Photonics Technol. Lett. 13, 1191–1193 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Optik (Stuttgart) (1)

L. Frank, “The properties of the Sommerfeld diffraction integral for a large aperture converging beam,” Optik (Stuttgart) 43, 149–157 (1975).

Phys. Rev. A (1)

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Other (9)

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1974).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

E. Shekel, A. Fiengold, Z. Fradkin, A. Geron, J. Levy, G. Matmon, D. Majer, E. Rafaeli, M. Rudman, G. Tidhar, J. Vecht, S. Ruschin, “64×64fast optical switching module,” in Optical Fiber Communication Conference, Vol. 1 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 27–29.

Y. M. Engelberg, S. Ruschin, “Coma aberration in diffraction from a narrow slit,” in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE5178, 112–123 (2003).
[CrossRef]

G. N. Lawrence, “Optical modeling,” in Applied Optics and Optical Engineering Series, R. R. Shanon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1992), Vol. XI, pp. 125–200.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Zemax User’s Manual, http://www.zemax.com/updates/index.html ; download file: ZEMAX̲Manual̲2003-02-04.exe.

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Figures (15)

Fig. 1
Fig. 1

Coordinate description for POP of a diverging beam.

Fig. 2
Fig. 2

Coordinate description for POP of a converging beam.

Fig. 3
Fig. 3

Validity limits for far-field propagators (λ=1.31 μm, b=1). Maximum–minimum input aperture versus propagation distance.

Fig. 4
Fig. 4

Zoom-in on Fig. 3, for small propagation distances.

Fig. 5
Fig. 5

Sample optical system for evaluation of different propagation methods.

Fig. 6
Fig. 6

Far-field relative intensity profile at the first lens surface of the sample optical system: Gaussian input.

Fig. 7
Fig. 7

Far-field phase profile at the first lens surface of the sample optical system: Gaussian input.

Fig. 8
Fig. 8

Focal-plane spot size obtained for the sample optical system by use of different propagation methods.

Fig. 9
Fig. 9

Far-field (distance of 5000 waves) phase distribution for a 5-wave half-width aperture.

Fig. 10
Fig. 10

Far-field (distance of 5000 waves) phase distribution for a 20-wave half-width aperture.

Fig. 11
Fig. 11

Setup for measurement of slit diffraction far-field intensity pattern; dia, diameter.

Fig. 12
Fig. 12

Measured versus calculated far-field intensity distributions for a 3.5-μm (full-width) slit: TE.

Fig. 13
Fig. 13

Measured versus calculated far-field intensity distributions for a 3.5-μm (full-width) slit: TM.

Fig. 14
Fig. 14

Measured versus calculated far-field intensity distributions for a 1.75-μm (full-width) slit: TE.

Fig. 15
Fig. 15

Measured versus calculated far-field intensity distributions for a 1.75-μm (full-width) slit: TM.

Equations (41)

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

a(x, y, z)=FF-1{T(z)FF[a(x, y, 0)]}.
T(z)=exp{jkz[1-(λξ)2-(λη)2]1/2},
Tp(z)=exp(jkz)exp(-jπλzρ2),ρ2=η2+ξ2.
a(x2, y2, z)=exp(jkz)jλz q(r2, z)×FF[a(x1, y1, 0)q(r1, z)](z, ξ(x2), η(y2)),
x2=λξz,y2=ληz,
q(r, z)=exp(jkr2/2z),
a(x2, y2, z)=1jλa(x1, y1, 0) ×exp(jkR)Rcos θdx1dy1,
R=[z2+(x2-x1)2+(y2-y1)2]1/2.
cos θ=z/R.
R=z1+x2-x1z2+y2-y1z21/2z1+12x2-x1z2+12y2-y1z2.
R=R21+x12-2x1x2R22+y12-2y1y2R221/2R21-x1x2R22-y1y2R22,
R2=(z2+x22+y22)1/2.
a(x2, y2, z)=zjλR22exp(jkR2)×FF[a(x1, y1, 0)](z, ξ(x2, y2), η(x2, y2)),
x2=λR2ξ,y2=λR2η.
x2=λξz[1-λ2(ξ2+η2)]1/2,
y2=ληz[1-λ2(ξ2+η2)]1/2.
a(x2, y2, z)
=zjλR22exp(jkR2)×FF[a(x1, y1, 0)q(r1, z)](z, ξ(x2, y2), η(x2, y2)),
R=z1+x22-2x1x2z2+y22-2y1y2z21/2z1+x22+y222z2-x1x2z2-y1y2z2,
z=(z12+x12+y12)1/2=const.
a(x2, y2, 0)=exp(jkz)jλz q(r2, z)×FF[a(x1, y1)](ξ(x2), η(y2)).
π4λz3 [(x2-x1)2+(y2-y1)2]21.
x2(firstintensityminimum)=λz2x1max.
x2 max=b λzx1max,
x1max>120.4λz3π1/4-0.4λz3π1/2-4bλz1/2,
πλR2 (x12+y12)-π4λR23 [x1(x1-2x2)
  +y1(y1-2y2)]21.
x1max<(λz/10π)1/2.
πλR2 (x12+y12)-πλz (x12+y12)-π4λR23
  ×[x1(x1-2x2)+y1(y1-2y2)]21.
z>10πx1max2λ1+bλx1max2-3/2-1.
z>15πb2λ,
π4λz3 [x2(x2-2x1)+y2(y2-2y1)]21.
x2max<0.6(λz3)1/4.
NA>1.67b(λ/z)3/4.
2E-n2c22Et2=0.
a(x2, z)=zλexp-j π4a(x1, 0) exp(jkR)R3/2dx1,
R=[z2+(x2-x1)2]1/2.
a(x2, z)=exp(jkz)λzexp-j π4q(x2, z)×FF[a(x1, 0)q(x1, z)](z, ξ(x2)).
a(x2, z)=zλR23exp(-jπ/4)exp(jkR2)×FF[a(x1, 0)](z, ξ(x2)).
Ioutputplane|a(x2, y2, z)|2cos θ=|a(x2, y2, z)|2zR2.

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