Abstract

This paper considers the propagation and diffraction of coherent light beams through nonorthogonal optical systems such as sequences of astigmatic lenses oriented at oblique angles to each other. The fundamental (gaussian) mode has elliptical light spots in each beam cross section and ellipsoidal (or hyperboloidal) wavefronts near the axis. It is found that the orientation of the light spot differs from that of the wavefront, and changes continuously by as much as π radians as the beam propagates through free space. A theory of these general astigmatic beams is given and simple experimental observations are described. The coupling factor between two such beams is also given.

© 1969 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Los Angeles, 1964), Chap. 4.
  2. M. M. Popov, Opt. Spectrosc. 25, 213 (1968);J. Arnaud, Appl. Opt. 8, 189 (1969).
    [Crossref] [PubMed]
  3. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, No. 2, 129 (1965).
    [Crossref]
  4. Y. Suematsu, H. Fukinuki, Bull. Tokyo Inst. Technol., 88, 33 (1968).
  5. See, for example, H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [Crossref] [PubMed]
  6. W. K. Kahn, in Proceedings of the Symposium on Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967).
  7. H. Kogelnik, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).

1968 (2)

M. M. Popov, Opt. Spectrosc. 25, 213 (1968);J. Arnaud, Appl. Opt. 8, 189 (1969).
[Crossref] [PubMed]

Y. Suematsu, H. Fukinuki, Bull. Tokyo Inst. Technol., 88, 33 (1968).

1966 (1)

1965 (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, No. 2, 129 (1965).
[Crossref]

Fukinuki, H.

Y. Suematsu, H. Fukinuki, Bull. Tokyo Inst. Technol., 88, 33 (1968).

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, No. 2, 129 (1965).
[Crossref]

Kahn, W. K.

W. K. Kahn, in Proceedings of the Symposium on Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967).

Kogelnik, H.

See, for example, H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[Crossref] [PubMed]

H. Kogelnik, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).

Li, T.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Los Angeles, 1964), Chap. 4.

Popov, M. M.

M. M. Popov, Opt. Spectrosc. 25, 213 (1968);J. Arnaud, Appl. Opt. 8, 189 (1969).
[Crossref] [PubMed]

Suematsu, Y.

Y. Suematsu, H. Fukinuki, Bull. Tokyo Inst. Technol., 88, 33 (1968).

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, No. 2, 129 (1965).
[Crossref]

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, No. 2, 129 (1965).
[Crossref]

Appl. Opt. (1)

Bull. Tokyo Inst. Technol. (1)

Y. Suematsu, H. Fukinuki, Bull. Tokyo Inst. Technol., 88, 33 (1968).

Opt. Spectrosc. (1)

M. M. Popov, Opt. Spectrosc. 25, 213 (1968);J. Arnaud, Appl. Opt. 8, 189 (1969).
[Crossref] [PubMed]

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, No. 2, 129 (1965).
[Crossref]

Other (3)

W. K. Kahn, in Proceedings of the Symposium on Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967).

H. Kogelnik, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Los Angeles, 1964), Chap. 4.

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Figures (7)

Fig. 1
Fig. 1

Spot ellipse and phase ellipse.

Fig. 2
Fig. 2

Circle diagram for the parameters q1 and q2.

Fig. 3
Fig. 3

Plot of φw and φR for λ = 1μm, p1 = −0.5 m; p2 = 1.5 m, z01 = 0.5 m, and z02 = −0.5 m; sh2α = 0.2 and 1.0 (i.e., α =0.10 and 0.44), respectively.

Fig. 4
Fig. 4

Plot of the principal axes of the spot ellipse, wξ and wη, for λ = 1μm, p1 = 0.5 m, p2 = 1.5 m, z01 = 0.5 m, and z02 = −0.5 m, sh2α = 0.4, 0.8 and 1.0 (i.e., α = 0.20, 0.37 and 0.44), respectively.

Fig. 5
Fig. 5

Coupling factor between a beam with simple astigmatism (α = 0) and a beam with general astigmatism (a). For both beams p1 = 0.5 m, p2 = 1.5 m, z01 = 0.5 m, z02 = −0.5 m, β = 0.

Fig. 6
Fig. 6

Photograph of the spot ellipse at various distances z from the second cylindrical lens (z = 0). The beam parameters at z = 0 (calculated from the measured beam waist before the first cylindrical lens) are: q1 = −665 + j 0.61 mm, q2 = 0.00005 + j 0.19 mm, φ = 0.27 × 10−7 + j 0.47 × 10−3.

Fig. 7
Fig. 7

Theoretical and experimental (circles) values for the orientation of the intensity ellipse (φw) as a function of the axial distance z. The beam parameters at z = 0 are q1 = −660 + j 52.1 mm, and q2 = 0.406 + j 16.4 mm, φ = 0.00202 + j 0.041.

Equations (31)

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2 ψ x 2 + 2 ψ y 2 2 jk ψ z = 0 ,
ψ ( x , y , z ) = ( q 1 q 2 ) 1 2 { exp [ j k 2 ( x 2 q 1 + y 2 q 2 ) ] } ,
q 1 ( z ) = q 01 + z , q 2 ( z ) = q 02 + z ,
ψ ( x , y , z ) = ( q 1 q 2 ) 1 2 ( exp { j k 2 [ ( cos 2 φ q 1 + sin 2 φ q 2 ) x 2 + ( sin 2 φ q 1 + cos 2 φ q 2 ) y 2 + sin 2 φ ( 1 q 2 1 q 1 ) xy ] } ) .
j k 2 [ Q 1 x 2 + Q 2 y 2 + tan 2 φ ( Q 2 Q 1 ) xy ] ,
Q 1 = ( cos 2 φ / q 1 ) + ( sin 2 φ / q 2 ) Q 2 = ( sin 2 φ / q 1 ) + ( cos 2 φ / q 2 ) ,
Φ ( z ) = 1 2 [ Phase of ( q 1 ) + Phase of ( q 2 ) ] .
x = ξ w cos φ w η w sin φ w = ξ R cos φ R η R sin φ R , y = ξ w sin φ w + η w cos φ w = ξ R sin φ R + η R cos φ R .
tan 2 φ w = [ ( ρ 1 ρ 2 ) / ( ω 1 ω 2 ) ] th 2 α
tan 2 φ R = [ ( ω 1 ω 2 ) / ( ρ 1 ρ 2 ) ] th 2 α .
φ = j α , 1 / q i = ρ i j ω i = 1 / ( j p i + z i ) , i = 1 , 2 , ρ i ( z ) = z i / ( z i 2 + p i 2 ) ω i ( z ) = p i / ( z i 2 + p i 2 )
ψ = λ ( q 1 q 2 ) 1 2 { exp [ ( ξ w 2 w ξ 2 + η w 2 w η 2 ) j k 2 ( ξ R 2 R ξ + η R 2 R η ) ] } .
λ / π w ξ , η 2 = 1 2 { ω 1 + ω 2 ± [ ( ω 1 ω 2 ) 2 c h 2 2 α + ( ρ 1 ρ 2 ) 2 s h 2 2 α ] 1 2 }
( R ξ , η ) 1 = 1 2 { ρ 1 + ρ 2 ± [ ( ρ 1 ρ 2 ) 2 c h 2 2 α + ( ω 1 ω 2 ) 2 s h 2 2 α ] 1 2 }
tan 2 ( φ w φ R ) = sh 2 α ch 2 α { ρ 1 ρ 2 ω 1 ω 2 + ω 1 ω 2 ρ 1 ρ 2 } .
tan 2 φ w tan 2 φ R = t h 2 2 α
c h 2 2 α ( z 1 z 2 ) 2 + ( p 1 + p 2 ) 2 ( z 1 z 2 ) 2 + ( p 1 p 2 ) 2
ch 2 α | q 1 q 2 * q 1 q 2 | .
ρ i 1 / z i
ω i p i / z i 2
tan 2 φ w z 1 z 2 p 1 p 2 th 2 α ,
tan 2 φ R p 1 p 2 z 1 z 2 th 2 α .
tan 2 ( φ w φ R ) = sh 2 α ch 2 α ( z 1 z 2 p 1 p 2 + p 1 p 2 z 1 z 2 ) .
1 / θ ξ , η 2 = ( π / 2 λ ) { p 1 + p 2 ± [ ( p 1 p 2 ) 2 c h 2 2 α + ( z 1 z 2 ) 2 s h 2 2 α ] 1 2 } .
k 2 [ ( cos 2 v f 1 + sin 2 v f 2 ) x 2 + ( sin 2 v f 1 + cos 2 v f 2 ) y 2 + sin 2 v ( 1 f 2 1 f 1 ) xy ] k 2 [ F 1 x 2 + F 2 y 2 + tan 2 v ( F 2 F 1 ) xy ] ,
Q 1 = Q 1 F 1 , Q 2 = Q 2 F 2 , tan 2 φ = tan 2 φ ( Q 2 Q 1 ) tan 2 v ( F 2 F 1 ) ( Q 2 Q 1 ) ( F 2 F 1 ) .
2 / q 1 , 2 = Q 1 + Q 2 ± [ ( Q 1 Q 2 ) / cos 2 φ ] .
( q 1 1 + q 2 1 ) ( q 1 1 + q 2 1 ) * .
c ab = xy plane E a E b * ds ,
C ab = c ab / c aa 1 2 c bb 1 2 ,
c ab 2 = ( q 1 a q 2 b * ) ( q 1 b * q 2 a ) ( q 1 a q 2 a ) × ( q 1 b q 2 b ) * cos 2 ( φ a φ b * ) ,

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