Abstract

It is shown that the propagation and transformation of a simply astigmatic Gaussian beam by an optical system with a characteristic ABCD matrix can be modeled by relatively simple equations whose terms consist solely of the heights and slopes of two paraxial rays. These equations are derived from the ABCD law of Gaussian beam transformation. They can be used in conjunction with a conventional automatic optical design program to design and optimize Gaussian beam optical systems. Several design examples are given using the CODE-V optical design package.

© 1983 Optical Society of America

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References

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  1. H. W. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  2. H. W. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  3. J. A. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
    [CrossRef] [PubMed]
  4. J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).
  5. W. H. Steier, Appl. Opt. 5, 1229 (1966).
    [CrossRef] [PubMed]
  6. J. A. Arnaud, Appl. Opt. 8, 1909 (1969).
    [CrossRef] [PubMed]
  7. J. A. Arnaud, “Hamiltonian Theory of Beam Mode Propagation,” in Progress in Optics, Vol. 11, E. Wolf, Ed. (North-Holland, Amsterdam, 1973).
    [CrossRef]
  8. CODE-V is a product of Optical Research Associates.
  9. H. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U.P.London, 1970).
  10. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), pp. 216–243.
  11. P. J. Sands, J. Opt. Soc. Am. 62, 369 (1972).
    [CrossRef]
  12. P. J. Sands, J. Opt. Soc. Am. 58, 1365 (1968).
    [CrossRef]
  13. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 43.
  14. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 435–438.
  15. R. P. Herloski, S. Marshall, R. L. Antos, J. Opt. Soc. Am. 72, 1106 (1982).

1982 (1)

R. P. Herloski, S. Marshall, R. L. Antos, J. Opt. Soc. Am. 72, 1106 (1982).

1972 (1)

1970 (1)

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

1969 (2)

1968 (1)

1966 (2)

1965 (1)

H. W. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Antos, R. L.

R. P. Herloski, S. Marshall, R. L. Antos, J. Opt. Soc. Am. 72, 1106 (1982).

Arnaud, J. A.

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

J. A. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
[CrossRef] [PubMed]

J. A. Arnaud, Appl. Opt. 8, 1909 (1969).
[CrossRef] [PubMed]

J. A. Arnaud, “Hamiltonian Theory of Beam Mode Propagation,” in Progress in Optics, Vol. 11, E. Wolf, Ed. (North-Holland, Amsterdam, 1973).
[CrossRef]

Buchdahl, H.

H. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U.P.London, 1970).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 435–438.

Herloski, R. P.

R. P. Herloski, S. Marshall, R. L. Antos, J. Opt. Soc. Am. 72, 1106 (1982).

Kogelnik, H.

Kogelnik, H. W.

H. W. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

H. W. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Li, T.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), pp. 216–243.

Marshall, S.

R. P. Herloski, S. Marshall, R. L. Antos, J. Opt. Soc. Am. 72, 1106 (1982).

Sands, P. J.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 43.

Steier, W. H.

Appl. Opt. (4)

Bell Syst. Tech. J. (2)

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

H. W. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

J. Opt. Soc. Am. (3)

P. J. Sands, J. Opt. Soc. Am. 62, 369 (1972).
[CrossRef]

P. J. Sands, J. Opt. Soc. Am. 58, 1365 (1968).
[CrossRef]

R. P. Herloski, S. Marshall, R. L. Antos, J. Opt. Soc. Am. 72, 1106 (1982).

Other (6)

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 43.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 435–438.

J. A. Arnaud, “Hamiltonian Theory of Beam Mode Propagation,” in Progress in Optics, Vol. 11, E. Wolf, Ed. (North-Holland, Amsterdam, 1973).
[CrossRef]

CODE-V is a product of Optical Research Associates.

H. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U.P.London, 1970).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), pp. 216–243.

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of the transformation of a Gaussian beam by a thin lens.

Fig. 2
Fig. 2

Diagram of one equivalent plane of an orthogonal optics system.

Fig. 3
Fig. 3

Diagram of the ray trace of two arbitrary paraxial rays.

Fig. 4
Fig. 4

Input beam representation technique for CODE-V.

Fig. 5
Fig. 5

CODE-V user-defined constraint code for Gaussian beam representation.

Fig. 6
Fig. 6

Sketch of the designed anamorphic beam converter giving the system parameters and error functions.

Fig. 7
Fig. 7

Diagram of the spot size minimization problem.

Tables (2)

Tables Icon

Table I Beam Converter Constraints and Variables

Tables Icon

Table II CODE-V Optimization: Selected Intermediate Cycle Results

Equations (50)

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ω ( z ) = ω 0 [ 1 + ( λ z / π ω 0 2 ) 2 ] 1 / 2 ,
ω 2 ( z ) = ω 0 2 + ( λ z / π ω 0 ) 2 ,
R ( z ) = z [ 1 + ( π ω 0 2 / λ z ) 2 ] 1 / 2 ,
1 / q ( z ) = 1 / R ( z ) j λ / π ω 2 ( z )
q ( z ) = z + j π ω 0 2 / λ .
q = ( A q + B ) / ( C q + D )
1 / q = ( C + D / q ) / ( A + B / q )
= ( C q + D ) / ( A q + B ) ,
1 / q = C / m + n / n m 2 R j λ / π ( m ω ) 2 .
x = F 1 ( x , y , u , υ ) ,
y = F 2 ( x , y , u , υ ) ,
u = F 3 ( x , y , u , υ ) ,
υ = F 4 ( x , y , u , υ ) .
x = M 11 x + M 12 y + M 13 u + M 14 υ + O ( 2 ) ,
y = M 21 x + M 22 y + M 23 u + M 24 υ + O ( 2 ) ,
u = M 31 x + M 32 y + M 33 u + M 34 υ + O ( 2 ) ,
υ = M 41 x + M 42 y + M 43 u + M 44 υ + O ( 2 ) ,
x = A 11 x + B 11 u + A 12 y + B 12 υ ,
u = C 11 x + D 11 u + C 12 y + D 12 υ ,
y = A 21 x + B 21 u + A 22 y + B 22 υ ,
υ = C 21 x + D 21 u + C 22 y + D 22 υ .
A 11 = x / x , B 21 = y / u , D 22 = υ / υ , etc . ,
x = A x x + B x u ,
u = C x x + D x u ,
y = A y y + B y υ ,
υ = C y y + D y υ ,
1 / q = [ B D + A C ( π ω 0 2 / λ ) 2 ] / [ B 2 + ( A π ω 0 2 / λ ) 2 ] j ( A D B C ) ( π ω 0 2 / λ ) / [ B 2 + ( A π ω 0 2 / λ ) 2 ] .
q = [ B D + A C ( π ω 0 2 / λ ) 2 ] / [ D 2 + ( C π ω 0 2 / λ ) 2 ] + j ( A D B C ) ( π ω 0 2 / λ ) / [ D 2 + ( C π ω 0 2 / λ ) 2 ] .
ω 2 = [ B 2 + ( A π ω 0 2 / λ ) 2 [ λ λ / ( π ω 0 ) 2 ] / ( A D B C ) ,
z = [ B D + A C ( π ω 0 2 / λ ) 2 ] / [ D 2 + ( C π ω 0 2 / λ ) 2 ] ,
ω 0 2 = ( A D B C ) 2 [ ω 0 2 ] / [ D 2 + ( C π ω 0 2 / λ ) 2 ] .
y 1 = A y 1 + B υ 1 ,
υ 1 = C y 1 + D υ 1 ,
y 2 = A y 2 + B υ 2 ,
υ 2 = C y 2 + D υ 2 .
A = ( y 1 υ 2 y 2 υ 1 ) / ( y 1 υ 2 y 2 υ 1 ) ,
B = ( y 1 y 2 y 2 y 1 ) / ( y 1 υ 2 y 2 υ 1 ) ,
C = ( υ 1 υ 2 υ 2 υ 1 ) / ( y 1 υ 2 y 2 υ 1 ) ,
D = ( y 1 υ 2 y 2 υ 1 ) / ( y 1 y 2 y 2 υ 1 ) ,
ω 2 = [ ( y 1 y 2 y 2 y 1 ) 2 + ( y 1 υ 2 y 2 υ 1 ) 2 ( π ω 0 2 / λ ) 2 ] [ λ λ / ( π ω 0 ) 2 ] / [ ( y 1 υ 2 y 2 υ 1 ) ( y 1 υ 2 y 2 υ 1 ) ( y 1 y 2 y 2 y 1 ) ( υ 1 υ 2 υ 2 υ 1 ) ] ,
z = [ ( y 1 y 2 y 2 y 1 ) ( y 1 υ 2 y 2 υ 1 ) + ( y 1 υ 2 y 2 υ 1 ) × ( υ 1 υ 2 υ 2 υ 1 ) ( π ω 0 2 / λ ) 2 ] / [ ( y 1 υ 2 y 2 υ 1 ) 2 + ( υ 1 υ 2 υ 2 υ 1 ) 2 ( π ω 0 2 / λ ) 2 ] ,
ω 0 2 = [ ( y 1 υ 2 y 2 υ 1 ) ( y 1 υ 2 y 2 υ 1 ) + ( y 1 y 2 y 2 y 1 ) × ( υ 1 υ 2 υ 2 υ 1 ) ] 2 ( ω 0 2 ) / { [ ( y 1 υ 2 y 2 υ 1 ) 2 + ( υ 1 υ 2 υ 2 υ 1 ) 2 ( π ω 0 2 / λ ) 2 ] ( y 1 υ 2 y 2 υ 1 ) 2 } ,
y w = y 1 = ω 0 , y d = y 2 = 0 , υ w = υ 1 = 0 , υ d = υ 2 = λ / π ω 0 .
ω = ( y d 2 + y ω 2 ) 1 / 2 ,
z 0 = ( y d υ d + y ω υ ω ) / ( υ d 2 + υ ω 2 ) ,
ω 0 = ( y w υ d υ ω y d ) / ( υ d 2 + υ ω 2 ) 1 / 2 .
ω 2 = ( λ / π ω 1 ) z t ,
ω 0 = [ 1 + ( π ω 1 2 / λ z t ) 2 ] 1 / 2 ω 1 ,
z 1 = ( π ω 1 ω 0 / λ ) 2 / z t ,
F 1 1 = R 1 1 + z t 1 .

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