Abstract

Systematic procedures are presented for determining the the optical components needed to obtain an arbitrary transformation of a propagating light ray or Gaussian beam.

© 1981 Optical Society of America

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References

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  1. For an early review of this subject see H. W. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  2. H. W. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  3. H. W. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  4. L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
    [CrossRef]
  5. L. W. Casperson, S. D. Lunnam, Appl. Opt. 14, 1193 (1975) and references therein.
    [CrossRef] [PubMed]
  6. L. W. Casperson, J. Opt. Soc. Am. 66, 1373 (1976).
    [CrossRef]
  7. L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
    [CrossRef]
  8. A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
    [CrossRef]
  9. J. J. Degnan, Appl. Phys. 11, 1 (1976) and references therein.
    [CrossRef]

1976 (3)

L. W. Casperson, J. Opt. Soc. Am. 66, 1373 (1976).
[CrossRef]

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

J. J. Degnan, Appl. Phys. 11, 1 (1976) and references therein.
[CrossRef]

1975 (1)

1974 (1)

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

1968 (1)

L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

1966 (1)

1965 (2)

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

H. W. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

Casperson, L. W.

L. W. Casperson, J. Opt. Soc. Am. 66, 1373 (1976).
[CrossRef]

L. W. Casperson, S. D. Lunnam, Appl. Opt. 14, 1193 (1975) and references therein.
[CrossRef] [PubMed]

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Degnan, J. J.

J. J. Degnan, Appl. Phys. 11, 1 (1976) and references therein.
[CrossRef]

Kogelnik, H. W.

Li, T.

Lunnam, S. D.

Siegman, A. E.

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

Yariv, A.

L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

J. J. Degnan, Appl. Phys. 11, 1 (1976) and references therein.
[CrossRef]

Appl. Phys. Lett. (1)

L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Bell Syst. Tech. J. (1)

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

IEEE J. Quantum Electron. (2)

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

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Figures (3)

Fig. 1
Fig. 1

Matrices for a Gaussian beam incident from the left.

Fig. 2
Fig. 2

Possible representation of a complex α type element.

Fig. 3
Fig. 3

Identity systems for (A) transmission and (B) reflection of rays or beams.

Equations (43)

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( r 2 r 2 ) = ( A B C D ) ( r 1 r 1 ) ,
( A B C D ) = ( 2 0 0 0.5 ) ,
Q k 0 = 1 q = 1 R i λ π w 2 ,
1 q 2 = C + D / q 1 A + B / q 1 ,
1 R 2 i λ π w 2 2 = ( C r + D r R 1 + D i λ π w 1 2 ) + i ( C i + D i R 1 + D r λ π w 1 2 ) ( A r + B r R 1 + B i λ π w 1 2 ) + i ( A i + B i R 1 + B r λ π w 1 2 ) .
1 R 2 = ( C r + D r R 1 + D i λ π w 1 2 ) ( A r + B r R 1 + B i λ π w 1 2 ) + ( C i + D i R 1 D r λ π w 1 2 ) ( A i + B i R 1 B r λ π w 1 2 ) ( A r + B r R 1 + B i λ π w 1 2 ) 2 + ( A i + B i R 1 B r λ π w 1 2 ) 2 ,
λ π w 2 2 = ( C i + D i R 1 D r λ π w 1 2 ) ( A r + B r R 1 + B i λ π w 1 2 ) + ( C r + D r R 1 + D i λ π w 1 2 ) ( A i + B i R 1 B r λ π w 1 2 ) ( A r + B r R 1 + B i λ π w 1 2 ) 2 + ( A i + B i R 1 B r λ π w 1 2 ) 2 .
E ( x , y , z ) = E 0 H m ( 2 ½ x w x ) H n ( 2 ½ y w y ) × exp { i [ k 0 z + Q x ( z ) x 2 2 + Q y ( z ) y 2 2 + S x ( z ) x + S y ( z ) y + P ( z ) ] } .
S x 2 = S x 1 A x + B x / q x 1
P 2 P 1 = i 2 Re ln ( A x + B x q x 1 ) + ( m + 1 2 ) Im ln ( A x + B x q x 1 ) i 2 Re ln ( A y + B y q y 1 ) + ( n + 1 2 ) Im ln ( A y + B y q y 1 ) S x 1 2 2 k 0 1 B x A x + B x / q x 1 S y 1 2 2 k 0 1 B y A y + B y / q y 1 ,
AD BC = n 1 / n 2 .
1 | A + D 2 ± [ ( A + D 2 ) 2 1 ] 1 / 2 | ,
α = ( 1 α 0 1 ) ;
β = ( 1 0 β 1 ) ;
γ = ( 1 0 0 γ ) .
( A B C D ) = ( 1 α 0 1 ) ( 1 0 β 1 ) ( 1 0 0 γ ) ( 1 α 0 1 ) = ( 1 ( A 1 ) / C 0 1 ) ( 1 0 C 1 ) ( 1 0 0 AD BC ) ( 1 B + ( 1 A ) D / C 0 1 ) .
( 1 0 0 γ ) ( 1 α 0 1 ) = ( 1 α 0 γ ) = ( 1 α / γ 0 1 ) ( 1 0 0 γ ) ,
( 1 0 0 γ ) ( 1 0 β 1 ) = ( 1 0 β γ γ ) = ( 1 0 β γ 1 ) ( 1 0 0 γ ) .
( 1 α 0 1 ) ( 1 0 β 1 ) = ( 1 0 β 1 ) ( 1 α 0 1 ) .
( 1 + α β α β 1 ) = ( 1 α β 1 + α β )
α 1 β 1 γ 1 α 1 = α 2 β 2 γ 2 α 2 .
β 1 γ α 1 ( γ α 2 ) 1 = α 1 1 α 2 β 2 .
β 1 α 3 = α 4 β 2 .
( A B C D ) = ( 1 0 0 AD BC ) ( 1 ( A 1 ) ( AD BC ) / C 0 1 ) ( 1 0 C / ( AD BC ) 1 ) ( 1 B + ( 1 A ) D / C 0 1 )
= ( 1 ( A 1 ) / C 0 1 ) ( 1 0 0 AD BC ) ( 1 0 C / ( AD BC ) 1 ) ( 1 B + ( 1 A ) D / C 0 1 )
= ( 1 ( A 1 ) / C 0 1 ) ( 1 0 C 1 ) ( 1 0 0 AD BC ) ( 1 B + ( 1 A ) D / C 0 1 )
= ( 1 ( A 1 ) / C 0 1 ) ( 1 0 C 1 ) ( 1 [ B + ( 1 A ) D / C ] / ( AD BC ) 0 1 ) ( 1 0 0 AD BC ) .
( A B C D ) = ( 1 0 0 AD BC ) ( 1 0 [ C + ( 1 A ) D / B ] / ( AD BC ) 1 ) ( 1 B 0 1 ) ( 1 0 ( A 1 ) / B 1 )
= ( 1 0 C + ( 1 A ) D / B 1 ) ( 1 0 0 AD BC ) ( 1 B 0 1 ) ( 1 0 ( A 1 ) / B 1 )
= ( 1 0 C + ( 1 A ) D / B 1 ) ( 1 B / ( AD BC ) 0 1 ) ( 1 0 0 AD BC ) ( 1 0 ( A 1 ) / B 1 )
= ( 1 0 C + ( 1 A ) D / B 1 ) ( 1 B / ( AD BC ) 0 1 ) ( 1 0 ( A 1 ) ( AD BC ) / B 1 ) ( 1 0 0 AD BC ) .
( A 0 0 D ) = ( 1 α 0 1 ) ( A α D 0 D )
= ( A α A 0 D ) ( 1 α 0 1 )
= ( 1 0 β 1 ) ( A 0 β A D )
= ( A 0 β D D ) ( 1 0 β 1 ) .
det ( A B C D ) = j γ j = n 1 n 2 n 2 n 3 n 3 n 4 n j 1 n j ,
( A B C D ) = ( 1 ( A 1 ) / C 0 1 ) ( 1 0 C 1 ) ( 1 ( D 1 ) / C 0 1 ) ,
( A B C D ) = ( 1 0 ( D 1 ) / B 1 ) ( 1 B 0 1 ) ( 1 0 ( A 1 ) / B 1 ) .
( 1 0 1 / f 1 ) ( 1 0 i λ / π w a 2 1 ) = ( 1 0 1 / f i λ / π w a 2 1 ) = ( 1 0 β r + i β i 1 ) .
( 1 α 0 1 ) = ( 1 0 2 ( α 1 l 1 ) 1 ) ( 1 l / 2 0 1 ) ( 1 0 4 ( α l 2 l 1 ) 1 ) × ( 1 l / 2 0 1 ) ( 1 0 2 ( α 1 l 1 ) 1 ) .
( 1 0 0 1 ) = ( 1 l 1 0 1 ) ( 1 l 1 0 1 ) ,
( 1 0 0 1 ) = ( 1 0 2 ( l 1 1 + l 2 1 ) 1 ) ( 1 l 2 / 2 0 1 ) ( 1 0 4 ( l 1 l 2 2 + l 2 1 ) 1 ) ( 1 l 2 / 2 0 1 ) × ( 1 0 2 ( l 1 1 + l 2 1 ) 1 ) ( 1 l 1 0 1 ) .
( 1 0 0 1 ) = ( 1 0 3 / l 1 ) ( 1 l 0 1 ) ( 1 0 3 / l 1 ) ( 1 l 0 1 ) ( 1 0 3 / l 1 ) ( 1 l 0 1 ) .

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