Abstract

Approximate solutions of the wave equation are investigated, which can be considered as Gaussian beams affected by third-order spherical aberration. In practical cases, such beams are good approximations to the beams emerging from media whose square refractive index is described by a fourth-degree polynomial of the transverse coordinate. The measure of their spherical aberration could be a means for the determination of the coefficient of the fourth-order term in the expression of the square refractive index of the graded-index medium.

© 1978 Optical Society of America

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References

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  1. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
  2. G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
    [CrossRef]
  3. R. Pratesi, L. Ronchi, J. Opt. Soc. Am. 67, 1274 (1977).
    [CrossRef]
  4. A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).
    [CrossRef]
  5. Y. W. M. Antar, W. M. Boerner, IEEE Trans. Antennas Propag. AP-22, 837 (1974).
    [CrossRef]
  6. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  7. R. Pratesi, L. Ronchi, “Wave Propagation in a Non-Exactly-Quadratic Graded-Index Medium,” in press, IEEE Trans. Microwave Theory Techn. (1978).
    [CrossRef]
  8. L. B. Felsen, J. Opt. Soc. Am. 66, 751 (1976).
    [CrossRef]
  9. S. Choudhari, L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730 (1973).
  10. G. Toraldo di Francia, “Geometrical and Interferential Aspects of the Ronchi Test,” in Optical Image Evaluation (NBS, Washington, D.C., 1951), p. 161.
  11. T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
    [CrossRef]

1977

1976

1974

Y. W. M. Antar, W. M. Boerner, IEEE Trans. Antennas Propag. AP-22, 837 (1974).
[CrossRef]

1973

S. Choudhari, L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730 (1973).

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).
[CrossRef]

1961

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Antar, Y. W. M.

Y. W. M. Antar, W. M. Boerner, IEEE Trans. Antennas Propag. AP-22, 837 (1974).
[CrossRef]

Boerner, W. M.

Y. W. M. Antar, W. M. Boerner, IEEE Trans. Antennas Propag. AP-22, 837 (1974).
[CrossRef]

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Choudhari, S.

S. Choudhari, L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730 (1973).

Felsen, L. B.

L. B. Felsen, J. Opt. Soc. Am. 66, 751 (1976).
[CrossRef]

S. Choudhari, L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730 (1973).

Furukawa, M.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Goubau, G.

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Kitano, I.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Kitano, T.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

Matsumura, H.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Pratesi, R.

R. Pratesi, L. Ronchi, J. Opt. Soc. Am. 67, 1274 (1977).
[CrossRef]

R. Pratesi, L. Ronchi, “Wave Propagation in a Non-Exactly-Quadratic Graded-Index Medium,” in press, IEEE Trans. Microwave Theory Techn. (1978).
[CrossRef]

Ronchi, L.

R. Pratesi, L. Ronchi, J. Opt. Soc. Am. 67, 1274 (1977).
[CrossRef]

R. Pratesi, L. Ronchi, “Wave Propagation in a Non-Exactly-Quadratic Graded-Index Medium,” in press, IEEE Trans. Microwave Theory Techn. (1978).
[CrossRef]

Schwering, F.

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Siegman, A. E.

Toraldo di Francia, G.

G. Toraldo di Francia, “Geometrical and Interferential Aspects of the Ronchi Test,” in Optical Image Evaluation (NBS, Washington, D.C., 1951), p. 161.

Bell Syst. Tech. J.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

IEEE J. Quantum Electron.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

IEEE Trans. Antennas Propag.

Y. W. M. Antar, W. M. Boerner, IEEE Trans. Antennas Propag. AP-22, 837 (1974).
[CrossRef]

S. Choudhari, L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730 (1973).

IRE Trans. Antennas Propag.

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

J. Opt. Soc. Am.

Other

G. Toraldo di Francia, “Geometrical and Interferential Aspects of the Ronchi Test,” in Optical Image Evaluation (NBS, Washington, D.C., 1951), p. 161.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

R. Pratesi, L. Ronchi, “Wave Propagation in a Non-Exactly-Quadratic Graded-Index Medium,” in press, IEEE Trans. Microwave Theory Techn. (1978).
[CrossRef]

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Equations (38)

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u m ( x , z ) = H m [ ( 2 ) 1 / 2 x w ] exp ( i k z + i k x 2 2 q + i P ) ,
q = q 0 + z , ( 1 / w 2 ) = Re [ ( i k ) / ( 2 q ) ] , i P = ( m + 1 2 ) ( 1 / q ) + m ( w / w ) ,
( 2 / w 2 ) + ( i k / q ) i k ( w / w ) = 0
w 2 = ( 4 / i k ) q ( 1 + c 0 q ) ,
n 2 ( x ) = n 0 2 n 2 x 2 + n 4 x 4 n 6 x 6 +
u m ( x , z ) = K m ( x , z ) exp [ i k z + i k Ω j ( x , z ) + i P ( z ) ] ,
ψ ( x , z ) = K m ( x , z ) exp [ i k Ω i ( x , z ) + i P ( z ) ]
2 ψ z 2 + 2 i k ψ z = 0.
2 K m x 2 + 2 i k [ K m x Ω j x + K m z ] + [ ( i k ) 2 ( Ω j x ) 2 + 2 ( i k ) 2 Ω j z + i k 2 Ω j x 2 + 2 i k · i P ] K m = 0.
Ω 4 = x 2 2 q 1 + x 4 4 Q 1 3 ,
K m ( x , z ) = c n [ 1 + Δ n ( z ) ] X n = a n X n ,
c n = ( 1 ) 1 / 2 n 2 n ( m 2 ) ! n ! ( m 2 n 2 ) ! for n even , 0 n m c n = 0 for n odd , n < 0 , n > m .
Δ n ( z ) = G ( z ) Δ n ,
Δ m = 0 ,
2 q 1 Q 1 3 + 1 2 d d z 1 Q 1 3 = 0 i k [ 1 q 1 2 + d d z 1 q 1 ] + 2 m + 3 Q 1 3 = 0 ,
( 1 / q 1 ) = ( 1 / q ) + ( 1 ) / ( i k q 2 ) ,
( 1 / Q 1 3 ) = ( 4 s 0 / q 4 ) ,
( 1 / q 2 ) = 4 s 0 ( 2 m + 3 ) / ( q 3 ) + ( s 1 / q 2 ) ,
exp ( i k S ) = exp [ i k ( z + x 2 2 q + x 4 4 Q 1 3 ) ] ,
exp ( i k S ) = exp [ i k ( z + x 2 2 q ) + i k s 0 x 4 q 4 ] .
2 w 1 2 ( n + 2 ) ( n + 1 ) a n + 2 + i k ( 2 n + 1 q 1 2 n w 1 w 1 + 2 i P ) a n + 2 i k a n 4 s 0 i k q 4 w 1 2 ( m n + 2 ) a n 2 = 0.
c m ( 2 m + 1 q 1 2 m w 1 w 1 + 2 i P ) = 8 s 0 q 4 w 1 2 c m 2 [ 1 + G ( z ) Δ m 2 ] ,
i P = ( m + 1 2 ) 1 q 1 + m w 1 w 1 m ( m 1 ) s 0 q 4 w 1 2 ,
4 w 1 2 ( m n ) ( 1 + G Δ n + 2 ) + 2 i k × [ ( m n ) ( w 1 w 1 1 q 1 ) m ( m 1 ) s 0 q 4 w 1 2 ] × ( 1 + G Δ n ) + 2 i k G Δ n + 2 n ( n 1 ) i k q 4 s 0 w 1 2 = 0 ,
w 1 w 1 1 q 1 = 2 i k w 1 2 + α ( z ) s 0 ,
2 w 1 2 G ( m n ) ( Δ n + 2 Δ n ) + i k s 0 ( m n ) α ( z ) + i k G Δ n i k q 4 ( m n ) ( m + n 1 ) s 0 w 1 2 = 0.
Δ n = ( m n ) ( 3 m + n 4 ) ,
( 3 m + n 4 ) ( i k G + 8 w 1 2 G i k q 4 s 0 w 1 2 ) + i k s 0 α ( z ) ( 2 m 3 ) ( 8 G w 1 2 i k q 4 w 1 2 ) = 0.
G + 8 i k w 1 2 G s 0 q 4 w 1 2 = 0 s 0 α ( z ) = ( 2 m 3 ) G = ( 2 m 3 ) ( 8 i k w 1 2 G s 0 q 4 w 1 2 ) .
G ( z ) = s 0 i k [ F 2 ( z ) + g 0 F 2 ( z ) ] .
F ( z ) = ( 1 / q ) ( 1 + c 0 q ) .
α ( z ) = 2 i k q 2 ( 2 m 3 ) [ F ( z ) g 0 F 3 ( z ) ] .
w 1 2 = w 2 + 4 s 1 ( i k ) 2 ( 1 + 2 c 0 q ) + 16 s 0 ( i k ) 2 { 2 m + 3 3 q ( 2 + 3 c 0 q ) + ( 2 m 3 ) [ ( 1 + c 0 q 3 ) 3 q g 0 q 2 2 c 0 1 c 0 q 1 + c 0 q ] } .
exp ( i P ) = 1 q m + 1 / 2 w 1 m exp { ( m + 1 2 ) s 1 i k q + 2 s 0 i k q 2 [ ( m + 1 2 ) ( 2 m + 3 ) m ( m 1 ) ( 1 + c 0 q 2 ) ] } .
u m ( x , z ) = K m ( x , z ) exp [ i k ( z + x 2 2 q + s 0 x 4 q 4 ) ] · exp { [ 4 q ( 2 m + 3 ) s 0 + s 1 ] x 2 q 2 + i P } ,
K m ( x , z ) = Σ c n [ 1 + ( m n ) ( 3 m + n 4 ) G ( z ) ] [ ( 2 ) 1 / 2 x w 1 ] n ,
n 2 ( x ) = n 0 2 n 2 x 2 + n 4 x 4 .
U m ( x , z ) = K m [ ( 2 ) 1 / 2 x W ] exp ( i β z x 2 w 2 p 2 x 4 w 4 ) ,

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