Abstract

We have generalized the ABCD propagation law, Q2 = (AQ1 + B)/(CQ1 + D), for an optical system by introducing a generalized complex radius of curvature Q for a general optical beam. The real part of 1/Q is related to the mean radius of curvature of the wave front, while the imaginary part is related to the second moment of the amplitude of the beam.

© 1991 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. A. E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng. 1224, 2 (1990).
  3. S. Lavi, R. Prochaska, E. Keren, Appl. Opt. 27, 3696 (1988).
    [CrossRef] [PubMed]
  4. M. J. Bastiaans, Optik 82, 173 (1989).
  5. A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Defining and measuring laser beam quality: the M2 factor,” IEEE J. Quantum Electron. (to be published).

1990 (1)

A. E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng. 1224, 2 (1990).

1989 (1)

M. J. Bastiaans, Optik 82, 173 (1989).

1988 (1)

Bastiaans, M. J.

M. J. Bastiaans, Optik 82, 173 (1989).

Johnston, T. F.

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Defining and measuring laser beam quality: the M2 factor,” IEEE J. Quantum Electron. (to be published).

Keren, E.

Lavi, S.

Prochaska, R.

Sasnett, M. W.

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Defining and measuring laser beam quality: the M2 factor,” IEEE J. Quantum Electron. (to be published).

Siegman, A. E.

A. E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng. 1224, 2 (1990).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Defining and measuring laser beam quality: the M2 factor,” IEEE J. Quantum Electron. (to be published).

Appl. Opt. (1)

Optik (1)

M. J. Bastiaans, Optik 82, 173 (1989).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

A. E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng. 1224, 2 (1990).

Other (2)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Defining and measuring laser beam quality: the M2 factor,” IEEE J. Quantum Electron. (to be published).

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Equations (27)

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u 2 ( x 2 ) = ( j B λ ) 1 / 2 - u 1 ( x 1 ) × exp [ - j π λ B ( A x 1 2 - 2 x 1 x 2 + D x 2 2 ) ] d x 1 ,
W 2 = 4 - x 2 u ( x ) 2 d x ,
- u ( x ) 2 d x = 1 .
W 2 2 = A 2 W 1 2 + 2 A B V 1 + B 2 U 1 ,
U = ( λ π ) 2 - | u ( x ) x | 2 d x ,
V = 4 - x [ Φ ( x ) x ] ψ 2 ( x ) d x .
u = ψ exp ( - j 2 π λ Φ ) .
V 2 = A C W 1 2 + ( A D + B C ) V 1 + B D U 1 ,
U 2 = C 2 W 1 2 + 2 D C V 1 + D 2 U 1 .
W 2 2 U 2 - V 2 2 = W 1 2 U 1 - V 1 2 = ( λ π M 2 ) 2 .
Q 2 = A Q 1 + B C Q 1 + D ,
1 Q = V W 2 - j λ M 2 π W 2 .
1 R = V W 2 = - x ( Φ x ) ψ 2 d x - x 2 ψ 2 d x .
Θ = M 2 λ π W 0 2 ,
M 4 = W o 2 - [ ψ 0 ( x ) x ] 2 d x .
2 u x 2 - 4 π j λ u z = 0.
2 ψ x 2 - ( 2 π λ ) 2 [ ( ϕ x ) 2 + 2 ϕ z ] ψ = 0 ,
ψ 2 ϕ x 2 + 2 ϕ x ψ x + 2 ψ z = 0.
n - x n - 1 ψ 2 ( ϕ x ) d x = z x n ψ 2 d x ,             n = 0 , 1 , 2 , 3 .
Φ ( x ) = Φ 0 + ( x 2 / 2 R ) + C 4 x 4 + C 6 x 6 + ,
V = 1 2 W 2 z
U = - 8 - ( ϕ z ) ψ 2 d x .
Φ 0 = - λ M 2 4 π tan - 1 z z R ,
z R = ( π W 0 2 / λ M 2 ) .
Φ 0 = - λ 4 π [ M 2 ] tan - 1 z z R ,
u ( x ) = exp ( - j π λ x 2 Q ) H [ M 2 ] ( 2 M x W ) ( Q ) [ M 2 ] 2 .
1 Q = 1 R - j M 2 λ π W 2 .

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