Abstract

This paper shows that a fundamental Gaussian beam propagating in a lenslike medium with cylindrical symmetry can be generated by the rotation about its axis of a skew ray which obeys the laws of geometrical optics. A complex representation: X(z) = ξ(z) + (z), where ξ(z) and η(z) are the projections of the skew ray on two perpendicular meridional planes, is discussed. It is found that the beam radius is equal to the modulus of X(z) and the on-axis phase to the phase of X(z). Using this representation, we derive a general expression for the on-axis phase shift ΔΦ experienced by a beam with an input complex beam parameter q through an optical system whose ray matrix is [ABCD]:ΔΦ=phase of(A+B/q). When the beam is matched to the optical system (output q = q), ΔΦ can be written cos–1(A + D)/2. This representation also provides a useful beam tracing method which is demonstrated and a simple interpretation for the known representation of Gaussian modes by ray packets.

© 1985 Optical Society of America

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References

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  1. H. W. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  2. H. W. Kognelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [CrossRef]
  3. H. W. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).
  4. V. P. Bykov, L. A. Vainshtein, “Geometric Optics of Open Resonators,” Sov. Phys. JETP 20, 338 (1965).
  5. W. K. Kahn, “Geometric Optical Derivation of Formula for the Variation of the Spot Size in a Spherical Mirror Resonator,” Appl. Opt. 4, 758 (1965).
    [CrossRef]
  6. W. H. Steier, “The Ray Packet Equivalent of a Gaussian Light Beam,” Appl. Opt. 5, 1229 (1966).
    [CrossRef] [PubMed]
  7. S. A. Collins, “Analysis of Optical Resonators Involving Focusing Elements,” Appl. Opt. 3, 1263 (1964);T. Li, “Dual Forms of the Gaussian Beam Chart,” Appl. Opt. 3, 1315 (1964);J. P. Gordon, “A Circle Diagram for Optical Resonator,” Bell Syst. Tech. J. 43, 1826 (1964);T. S. Chu, “Geometrical Representation of Gaussian Beam Propagation,” Bell Syst. Tech. J. 45, 287 (1966).
    [CrossRef]
  8. G. A. Deschamps, P. E. Mast, in Proceedings, Symposium on Quantum Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).
  9. P. Laures, “Geometrical Approach to Gaussian Beam Propagation,” Appl. Opt. 6, 747 (1967).
    [CrossRef] [PubMed]
  10. J. A. Arnaud, H. Kogelnik, “Gaussian Beams with General Astigmatism,” unpublished work. See, however, a revised version in Ref. 15.
  11. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), Chap. 4.
  12. J. A. Arnaud, “Degenerate Optical Cavities,” Appl. Opt. 8, 189 (1969).The relevant footnote is on p. 190; Note that this paper was submitted in May 1968.
    [CrossRef] [PubMed]
  13. J. A. Arnaud, “Degenerate Optical Cavities. 2: Effect of Misalignments,” Appl. Opt. 8, 1909 (1969). The principle of complex coordinate shifts is stated in the footnote on p. 1910.
    [CrossRef] [PubMed]
  14. G. A. Deschamps, “The Gaussian Beam as a Bundle of Complex Rays,” Electron. Lett. 7, 684 (1971).
    [CrossRef]
  15. J. A. Arnaud, H. Kogelnik, “Gaussian Light Beams with General Astigmatism,” Appl. Opt., 8, 1687 (1969).
    [CrossRef] [PubMed]
  16. I. Montrosset, R. Orta, “La méthode des rayons complexes,” Ann. Télécommun. 38, 1 (1983).
  17. D. Weingarten, “Complex Symmetries of Electrodynamics,” Ann. Phys. 76, 510 (1973).
    [CrossRef]
  18. J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.
  19. N. Aoyagi, S. Yamaguchi, “Generalized Fresnel Transformations and Their Properties,” Jpn. J. Appl. Phys. 12, 1343 (1973).
    [CrossRef]
  20. J. A. Arnaud, “A Theory of Gaussian Pulse Propagation,” Opt. Quantum Electron. 16, 125 (1984).
    [CrossRef]
  21. R. Herloski, S. Marshall, R. Antos, “Gaussian Beam Ray-Equivalent Modeling and Optical Design,” Appl. Opt. 22, 1168 (1983).
    [CrossRef] [PubMed]
  22. Y. Kravtsov, “Complex Rays and Complex Caustics,” paper presented at the Fourth All-Union Symposium on Diffraction of Waves, Moscow, U.S.S.R. (Nauka, Moscow, 1967).

1984 (1)

J. A. Arnaud, “A Theory of Gaussian Pulse Propagation,” Opt. Quantum Electron. 16, 125 (1984).
[CrossRef]

1983 (2)

R. Herloski, S. Marshall, R. Antos, “Gaussian Beam Ray-Equivalent Modeling and Optical Design,” Appl. Opt. 22, 1168 (1983).
[CrossRef] [PubMed]

I. Montrosset, R. Orta, “La méthode des rayons complexes,” Ann. Télécommun. 38, 1 (1983).

1973 (2)

D. Weingarten, “Complex Symmetries of Electrodynamics,” Ann. Phys. 76, 510 (1973).
[CrossRef]

N. Aoyagi, S. Yamaguchi, “Generalized Fresnel Transformations and Their Properties,” Jpn. J. Appl. Phys. 12, 1343 (1973).
[CrossRef]

1971 (1)

G. A. Deschamps, “The Gaussian Beam as a Bundle of Complex Rays,” Electron. Lett. 7, 684 (1971).
[CrossRef]

1969 (3)

1967 (1)

1966 (2)

1965 (4)

H. W. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

V. P. Bykov, L. A. Vainshtein, “Geometric Optics of Open Resonators,” Sov. Phys. JETP 20, 338 (1965).

W. K. Kahn, “Geometric Optical Derivation of Formula for the Variation of the Spot Size in a Spherical Mirror Resonator,” Appl. Opt. 4, 758 (1965).
[CrossRef]

H. W. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
[CrossRef]

1964 (1)

Antos, R.

Aoyagi, N.

N. Aoyagi, S. Yamaguchi, “Generalized Fresnel Transformations and Their Properties,” Jpn. J. Appl. Phys. 12, 1343 (1973).
[CrossRef]

Arnaud, J.

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.

Arnaud, J. A.

Bykov, V. P.

V. P. Bykov, L. A. Vainshtein, “Geometric Optics of Open Resonators,” Sov. Phys. JETP 20, 338 (1965).

Collins, S. A.

Deschamps, G. A.

G. A. Deschamps, “The Gaussian Beam as a Bundle of Complex Rays,” Electron. Lett. 7, 684 (1971).
[CrossRef]

G. A. Deschamps, P. E. Mast, in Proceedings, Symposium on Quantum Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).

Herloski, R.

Kahn, W. K.

Kogelnik, H.

J. A. Arnaud, H. Kogelnik, “Gaussian Light Beams with General Astigmatism,” Appl. Opt., 8, 1687 (1969).
[CrossRef] [PubMed]

J. A. Arnaud, H. Kogelnik, “Gaussian Beams with General Astigmatism,” unpublished work. See, however, a revised version in Ref. 15.

Kogelnik, H. W.

H. W. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

H. W. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
[CrossRef]

Kognelnik, H. W.

Kravtsov, Y.

Y. Kravtsov, “Complex Rays and Complex Caustics,” paper presented at the Fourth All-Union Symposium on Diffraction of Waves, Moscow, U.S.S.R. (Nauka, Moscow, 1967).

Laures, P.

Li, T.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), Chap. 4.

Marshall, S.

Mast, P. E.

G. A. Deschamps, P. E. Mast, in Proceedings, Symposium on Quantum Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).

Montrosset, I.

I. Montrosset, R. Orta, “La méthode des rayons complexes,” Ann. Télécommun. 38, 1 (1983).

Orta, R.

I. Montrosset, R. Orta, “La méthode des rayons complexes,” Ann. Télécommun. 38, 1 (1983).

Steier, W. H.

Vainshtein, L. A.

V. P. Bykov, L. A. Vainshtein, “Geometric Optics of Open Resonators,” Sov. Phys. JETP 20, 338 (1965).

Weingarten, D.

D. Weingarten, “Complex Symmetries of Electrodynamics,” Ann. Phys. 76, 510 (1973).
[CrossRef]

Yamaguchi, S.

N. Aoyagi, S. Yamaguchi, “Generalized Fresnel Transformations and Their Properties,” Jpn. J. Appl. Phys. 12, 1343 (1973).
[CrossRef]

Ann. Phys. (1)

D. Weingarten, “Complex Symmetries of Electrodynamics,” Ann. Phys. 76, 510 (1973).
[CrossRef]

Ann. Télécommun. (1)

I. Montrosset, R. Orta, “La méthode des rayons complexes,” Ann. Télécommun. 38, 1 (1983).

Appl. Opt. (10)

J. A. Arnaud, H. Kogelnik, “Gaussian Light Beams with General Astigmatism,” Appl. Opt., 8, 1687 (1969).
[CrossRef] [PubMed]

P. Laures, “Geometrical Approach to Gaussian Beam Propagation,” Appl. Opt. 6, 747 (1967).
[CrossRef] [PubMed]

J. A. Arnaud, “Degenerate Optical Cavities,” Appl. Opt. 8, 189 (1969).The relevant footnote is on p. 190; Note that this paper was submitted in May 1968.
[CrossRef] [PubMed]

J. A. Arnaud, “Degenerate Optical Cavities. 2: Effect of Misalignments,” Appl. Opt. 8, 1909 (1969). The principle of complex coordinate shifts is stated in the footnote on p. 1910.
[CrossRef] [PubMed]

H. W. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
[CrossRef]

H. W. Kognelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
[CrossRef]

W. K. Kahn, “Geometric Optical Derivation of Formula for the Variation of the Spot Size in a Spherical Mirror Resonator,” Appl. Opt. 4, 758 (1965).
[CrossRef]

W. H. Steier, “The Ray Packet Equivalent of a Gaussian Light Beam,” Appl. Opt. 5, 1229 (1966).
[CrossRef] [PubMed]

S. A. Collins, “Analysis of Optical Resonators Involving Focusing Elements,” Appl. Opt. 3, 1263 (1964);T. Li, “Dual Forms of the Gaussian Beam Chart,” Appl. Opt. 3, 1315 (1964);J. P. Gordon, “A Circle Diagram for Optical Resonator,” Bell Syst. Tech. J. 43, 1826 (1964);T. S. Chu, “Geometrical Representation of Gaussian Beam Propagation,” Bell Syst. Tech. J. 45, 287 (1966).
[CrossRef]

R. Herloski, S. Marshall, R. Antos, “Gaussian Beam Ray-Equivalent Modeling and Optical Design,” Appl. Opt. 22, 1168 (1983).
[CrossRef] [PubMed]

Bell Syst. Tech. J. (1)

H. W. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

Electron. Lett. (1)

G. A. Deschamps, “The Gaussian Beam as a Bundle of Complex Rays,” Electron. Lett. 7, 684 (1971).
[CrossRef]

Jpn. J. Appl. Phys. (1)

N. Aoyagi, S. Yamaguchi, “Generalized Fresnel Transformations and Their Properties,” Jpn. J. Appl. Phys. 12, 1343 (1973).
[CrossRef]

Opt. Quantum Electron. (1)

J. A. Arnaud, “A Theory of Gaussian Pulse Propagation,” Opt. Quantum Electron. 16, 125 (1984).
[CrossRef]

Sov. Phys. JETP (1)

V. P. Bykov, L. A. Vainshtein, “Geometric Optics of Open Resonators,” Sov. Phys. JETP 20, 338 (1965).

Other (5)

G. A. Deschamps, P. E. Mast, in Proceedings, Symposium on Quantum Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).

J. A. Arnaud, H. Kogelnik, “Gaussian Beams with General Astigmatism,” unpublished work. See, however, a revised version in Ref. 15.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), Chap. 4.

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.

Y. Kravtsov, “Complex Rays and Complex Caustics,” paper presented at the Fourth All-Union Symposium on Diffraction of Waves, Moscow, U.S.S.R. (Nauka, Moscow, 1967).

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Figures (5)

Fig. 1
Fig. 1

Gaussian beam represented by a skew paraxial ray rotating about the z axis. At any z value, the beam radius is the distance between the ray and the z axis. Furthermore the angular position of the ray gives the difference between the phase of the optical field and the geometrical phase shift.

Fig. 2
Fig. 2

This figure shows how one can trace the refraction of a Gaussian beam by a lens by tracing two paraxial rays using conventional procedures.

Fig. 3
Fig. 3

Procedure to determine the waist of the beam using an auxiliary projection on the xy plane.

Fig. 4
Fig. 4

Determination of the wave front curvature center C. Here the projections of two skew rays need to be traced.

Fig. 5
Fig. 5

Application of the method to a simple problem: The refraction by a lens of a beam whose waist coincides with the focal plane of the lens. The tracing readily shows that the beam waist after traversing the lens is located at the image focal plane.

Equations (20)

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Ψ ( x , y , z ) = 1 q x ( z ) q y ( z ) exp { j k 2 [ x 2 q x ( z ) + y 2 q y ( z ) ] } ,
q x ( z ) = z z x 0 + j k w x 0 2 2 ,
Ψ ( x , y , z ) = 1 n ( z ) X ( z ) Y ( z ) exp { j k 2 n ( z ) [ x 2 q x ( z ) + y 2 q y ( z ) ] } ,
q x ( z ) = X ( z ) dX / dz
Φ ( z ) = ½ [ phase of X ( z ) + phase of Y ( z ) ] .
M x = [ A x B x C x D y ] and M y = [ A y B y C y D y ]
Δ Φ Φ Φ = ½ [ phase of X X + phase of Y Y ] = ½ [ phase of A x X + B x X ˙ X + phase of A y Y + B y Y ˙ Y ] = ½ [ phase of ( A x + B x q x ) + phase of ( A y + B y q y ) ] ,
Δ Φ = ½ [ phase of q x q x + phase of q y q y ] ,
z = 0 ( q x = q y = j π w 0 2 λ ) ,
Δ Φ = tan 1 ( λ z π w 0 2 ) .
q x = A x q x + B x C x q x + D x
Δ Φ = ½ [ cos 1 A x + D x 2 + cos 1 A y + D y 2 ]
Δ Φ 2 = cos 1 ad ,
[ ξ ( z ) η ( z ) ] = [ cos θ sin θ sin θ cos θ ] [ 2 z kn w 0 w 0 ] ,
ξ ( z ) = w 0 sin θ + 2 z kn w 0 cos θ , η ( z ) = w 0 cos θ + 2 z kn w 0 sin θ .
n [ ξ ˙ ( z ) η ( z ) ξ ( z ) η ˙ ( z ) ] = 2 k
n ξ ( l ) η ˙ ( l ) = n ξ ( l ) η ˙ ( l ) ,
X ( z ) = ξ ( z ) + j η ( ξ ) .
X ( z ) = 2 exp ( j θ ) k w 0 n ( z + jkn w 0 2 2 ) = 2 exp ( j θ ) k w 0 n q .
Φ = phase X ( z ) = θ + phase of q ( z ) .

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