Abstract

The excitation of a concave optical resonator with one planar mirror by a laser source incident normally on the planar mirror is described. The problem is analyzed by describing the resonator fields by both ray tracing and normal mode analysis. The latter is more general, but it is shown that in the appropriate limit the two methods of analysis yield nearly identical solutions. Main emphasis in the analysis is placed on determination of the reflection coefficient, relative mode amplitudes, and field patterns. These are the parameters that are of interest to users of the scanning interferometer and the coupled cavity laser interferometer. The latter is commonly used to diagnose gaseous plasmas.

© 1968 Optical Society of America

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  1. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
  2. G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).
  3. A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
    [CrossRef]
  4. R. L. Fork, D. R. Herriot, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
    [CrossRef]
  5. D. E. T. F. Ashby, D. F. Jephcott, Appl. Phys. Lett. 3, 13 (1963).
    [CrossRef]
  6. J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964); see also J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 3, 121 (1963).
    [CrossRef]
  7. H. Kogelnik, Bell Syst. Tech. J. 43, 334 (1964).
  8. N. Kumagai, M. Malsuhara, IEEE J. Quantum Electron. 1, 85 (1965).
    [CrossRef]
  9. H. Kogelnik, in Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, 1965), pp. 333–347.
  10. J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 6, 185 (1965).
    [CrossRef]
  11. J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, J. Appl. Phys. 36, 3526(1965).
    [CrossRef]
  12. J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, “Theory of the Laser Interferometer and Its Use in Plasma Diagnostics,” Sci. Rep. No. 1, Grant No. DA–ARO–D–31–124–G582.
  13. M. Abramowitz, L. A. Stegun, “Handbook of Mathematical Functions,” AM555 National Bureau of Standards Applied Mathematical Series No. 55.
  14. L. B. W. Jolley, Summation of Series (Dover Publications, New York, 1961), p. 166.
  15. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, Vol. II, Bateman Manuscript Project (McGraw-Hill Book Company, Inc., New York, 1954).

1965

N. Kumagai, M. Malsuhara, IEEE J. Quantum Electron. 1, 85 (1965).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 6, 185 (1965).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, J. Appl. Phys. 36, 3526(1965).
[CrossRef]

1964

R. L. Fork, D. R. Herriot, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964); see also J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 3, 121 (1963).
[CrossRef]

H. Kogelnik, Bell Syst. Tech. J. 43, 334 (1964).

1963

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[CrossRef]

D. E. T. F. Ashby, D. F. Jephcott, Appl. Phys. Lett. 3, 13 (1963).
[CrossRef]

1962

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

1961

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

Abramowitz, M.

M. Abramowitz, L. A. Stegun, “Handbook of Mathematical Functions,” AM555 National Bureau of Standards Applied Mathematical Series No. 55.

Ashby, D. E. T. F.

D. E. T. F. Ashby, D. F. Jephcott, Appl. Phys. Lett. 3, 13 (1963).
[CrossRef]

Boyd, G. D.

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

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

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, Vol. II, Bateman Manuscript Project (McGraw-Hill Book Company, Inc., New York, 1954).

Fork, R. L.

Fox, A. G.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[CrossRef]

Gerardo, J. B.

J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 6, 185 (1965).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, J. Appl. Phys. 36, 3526(1965).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964); see also J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 3, 121 (1963).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, “Theory of the Laser Interferometer and Its Use in Plasma Diagnostics,” Sci. Rep. No. 1, Grant No. DA–ARO–D–31–124–G582.

Gordon, J. P.

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

Gusinow, M. A.

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, J. Appl. Phys. 36, 3526(1965).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, “Theory of the Laser Interferometer and Its Use in Plasma Diagnostics,” Sci. Rep. No. 1, Grant No. DA–ARO–D–31–124–G582.

Herriot, D. R.

Jephcott, D. F.

D. E. T. F. Ashby, D. F. Jephcott, Appl. Phys. Lett. 3, 13 (1963).
[CrossRef]

Jolley, L. B. W.

L. B. W. Jolley, Summation of Series (Dover Publications, New York, 1961), p. 166.

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 43, 334 (1964).

R. L. Fork, D. R. Herriot, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
[CrossRef]

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

H. Kogelnik, in Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, 1965), pp. 333–347.

Kumagai, N.

N. Kumagai, M. Malsuhara, IEEE J. Quantum Electron. 1, 85 (1965).
[CrossRef]

Li, T.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, Vol. II, Bateman Manuscript Project (McGraw-Hill Book Company, Inc., New York, 1954).

Malsuhara, M.

N. Kumagai, M. Malsuhara, IEEE J. Quantum Electron. 1, 85 (1965).
[CrossRef]

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, Vol. II, Bateman Manuscript Project (McGraw-Hill Book Company, Inc., New York, 1954).

Stegun, L. A.

M. Abramowitz, L. A. Stegun, “Handbook of Mathematical Functions,” AM555 National Bureau of Standards Applied Mathematical Series No. 55.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, Vol. II, Bateman Manuscript Project (McGraw-Hill Book Company, Inc., New York, 1954).

Verdeyen, J. T.

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, J. Appl. Phys. 36, 3526(1965).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 6, 185 (1965).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964); see also J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 3, 121 (1963).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, “Theory of the Laser Interferometer and Its Use in Plasma Diagnostics,” Sci. Rep. No. 1, Grant No. DA–ARO–D–31–124–G582.

Appl. Opt.

Appl. Phys. Lett.

D. E. T. F. Ashby, D. F. Jephcott, Appl. Phys. Lett. 3, 13 (1963).
[CrossRef]

J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 6, 185 (1965).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, Bell Syst. Tech. J. 43, 334 (1964).

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

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

IEEE J. Quantum Electron.

N. Kumagai, M. Malsuhara, IEEE J. Quantum Electron. 1, 85 (1965).
[CrossRef]

J. Appl. Phys.

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, J. Appl. Phys. 36, 3526(1965).
[CrossRef]

Proc. IEEE

J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964); see also J. B. Gerardo, J. T. Verdeyen, Appl. Phys. Lett. 3, 121 (1963).
[CrossRef]

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[CrossRef]

Other

H. Kogelnik, in Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, 1965), pp. 333–347.

J. B. Gerardo, J. T. Verdeyen, M. A. Gusinow, “Theory of the Laser Interferometer and Its Use in Plasma Diagnostics,” Sci. Rep. No. 1, Grant No. DA–ARO–D–31–124–G582.

M. Abramowitz, L. A. Stegun, “Handbook of Mathematical Functions,” AM555 National Bureau of Standards Applied Mathematical Series No. 55.

L. B. W. Jolley, Summation of Series (Dover Publications, New York, 1961), p. 166.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, Vol. II, Bateman Manuscript Project (McGraw-Hill Book Company, Inc., New York, 1954).

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Figures (9)

Fig. 1
Fig. 1

(a) The laser interferometer using planar mirrors for the reference cavity, (b) off-axis excitation of a semiconfocal external cavity.

Fig. 2
Fig. 2

Graphs showing the dependence of the partition coefficients on the displacement of the input beam for s = 4. The parameter s equals the number of discrete beams observed in the reference cavity for large relative displacements [(x0/w) > s] between the input beam and the axis of the passive cavity (see Fig. 1 for an example, or Ref. 6).

Fig. 3
Fig. 3

Relative cavity response for three different displacements.

Fig. 4
Fig. 4

Graph of the partition coefficients as a function of the normalized displacement of the input beam for s = 7. The parameter s equals the number of discrete beams observed in the reference cavity for large relative displacements [(x0/w) > s] between the input beam and the axis of the passive cavity (see Fig. 1 for an example, or Ref. 6).

Fig. 5
Fig. 5

Relative cavity response for (x0/w0)2 = 2 for r/s = 1/7 (top) and r/s = 2/7 (bottom).

Fig. 6
Fig. 6

The amplitude of electric field on the right side of M2 of Fig. 1 predicted from the mode coupling approach. The square of this field should be compared to the dot pattern illustrated in Fig. 1.

Fig. 7
Fig. 7

An elementary approach to the prediction of the spatial fringes in the central spot of Fig. 6.

Fig. 8
Fig. 8

Temporal fringes observed in the reference cavity for the case of r/s = ¼ with various values of the displacement between the axis of the input beam and the axis of the reference cavity. Relative amplitudes between cases are not significant inasmuch as there is a sensitivity change in the detector.

Fig. 9
Fig. 9

Observed temporal fringes for the case r/s = 2/7.

Equations (58)

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ρ 23 = ( ρ 2 - ρ 3 e - j θ ) / ( 1 - ρ 2 ρ 3 e - j θ ) ,
E 2 = E 0 ( 1 + ρ 2 s - 1 ρ 3 s e - j s θ 0 ) ,
H 2 = ( E 0 / η 0 ) ( 1 - ρ 2 s - 1 ρ 3 s e - j s θ 0 ) .
[ E 1 H a ] = [ A B C D ] [ E 2 H 2 ] ,
E 1 = E inc ( 1 + ρ 23 ) , H 1 = ( E inc / η 0 ) ( 1 - ρ 23 ) .
ρ 23 = ρ 2 ( 1 - ρ 2 s - 2 ρ 3 s e - j s θ 0 1 - ρ 2 s ρ 3 s e - j s θ 0 ) .
E 1 ( r , z = 0 ) = E 0 ψ 0 ( r ) ( 1 + ρ ) + t 0 E t ψ t ( r ) ,
H 1 ( r , z = 0 ) = ( E 0 / η 0 ) ψ 0 ( r ) ( 1 - ρ ) - t 0 E t η 0 ψ t ( r ) .
E 2 = l = 0 A l ϕ l ( r ) ( 1 + ρ 3 e - j θ l ) ,
H 2 = l = 0 A l η l ϕ l ( r ) ( 1 - ρ 3 e - j θ l ) ,
[ 2 π λ 0 - ( 1 + l ) 2 d r cos - 1 ( 1 - 2 d r b 3 ) ] d r = q π β l d r ,
E 0 ψ 0 ( r ) ( 1 + ρ ) + t 0 E t ψ t ( r ) = B η 0 l = 0 A l ϕ l ( r ) ( 1 - ρ 3 e - j θ l ) ,
E 0 η 0 ψ 0 ( r ) ( 1 - ρ ) - t 0 E t η 0 ψ t ( r ) = C l = 0 A l ϕ l ( r ) ( 1 + ρ 3 e - j θ l ) .
2 E 0 ψ 0 ( r ) = B + C η 0 2 η 0 l = 0 A l ϕ l ( r ) ( 1 - ρ 2 ρ 3 e - j θ l ) ,
A l = 2 η 0 E 0 ( B + C η 0 2 ) [ + ψ 0 ( r ) ϕ l * ( r ) d A 1 - ρ 2 ρ 3 e - j θ l ]
A l = 2 η 0 E 0 ( B + C η 0 2 ) C 0 ¯ l 1 - ρ 2 ρ 3 e - j θ l ,
C 0 ¯ , l = - + ψ 0 ( r ) ϕ l * ( r ) d A .
2 ρ E 0 ψ 0 ( r ) + 2 l 0 E t ψ t ( r ) = B + C η 0 2 η 0 l = 0 A l ϕ l ( r ) ( ρ 2 - ρ 3 e - j θ l ) .
2 ρ 0 E 0 = B + C η 0 2 η 0 l = 0 A l C * 0 ¯ , l ( ρ 2 - ρ 3 e - j θ l ) .
ρ = l = 0 C 0 ¯ , l 2 { ρ 2 - ρ 3 e - j θ l 1 - ρ 2 ρ 3 e - j θ l } ,
C * 0 ¯ , l = - + ψ 0 * ( r ) ϕ l ( r ) d A .
( 1 / 2 π ) cos - 1 [ 1 - ( 2 d r / b 3 ) ] = r / s < 1 2 .
θ l = θ 0 + l ( 2 π r / s ) .
θ l = θ 0 + l π / 2 ,
ρ = ρ 2 - ρ 3 e - j θ o 1 - ρ 2 ρ 3 e - j θ o ( C 2 0 ¯ , 0 ¯ , 0 , 0 + C 2 0 ¯ , 0 ¯ , 2 , 2 + C 2 0 ¯ , 0 ¯ , 4 , 0 + = K 0 ) + ρ 2 + j ρ 3 e j θ o 1 + j ρ 2 ρ 3 e - j θ o ( C 2 0 ¯ , 0 ¯ , 0 , 1 + C 2 0 ¯ , 0 ¯ , 1 , 0 + C 2 0 ¯ , 0 ¯ , 2 , 3 + = K 1 ) + ρ 2 + ρ 3 e - j θ o 1 + ρ 2 ρ 3 e - j θ o ( C 2 0 ¯ , 0 ¯ , 2 , 0 + C 2 0 ¯ , 0 ¯ , 1 , 1 + C 2 0 ¯ , 0 ¯ , 0 , 2 + = K 2 ) + ρ 2 - j ρ 3 e - j θ o 1 - j ρ 2 ρ 3 e - j θ o ( C 2 0 ¯ , 0 ¯ , 0 , 3 + C 2 0 , 0 , 2 , 1 + C 2 0 ¯ , 0 ¯ , 1 , 2 + = K 3 ) .
ρ { 1 - ρ 0 6 e - j 4 θ 0 1 - ρ 0 8 e - j 4 θ 0 - ρ 0 3 ( 1 - ρ 0 2 ) ( K 0 + K 1 - K 2 - K 3 ) e - j 2 θ o 1 - ρ 0 e - j 4 θ 0 } .
C 2 0 ¯ , 0 ¯ , m , 0 = [ ( x 0 / w 0 ) 2 m / m ! ] exp - ( x 0 / w 0 ) 2 ,
K n = 1 2 ( 1 + [ ( - 1 ) n exp - 2 ( x 0 / w 0 ) 2 if s is even 0 if s is odd ] + 2 × { exp [ - ( x 0 w 0 ) 2 ( 1 - cos 2 π s ) ] } cos [ 2 π n s - ( x 0 w 0 ) 2 sin 2 π s ] + 2 { exp [ - ( x 0 w 0 ) 2 ( 1 - cos 2 π k s ) ] } cos [ 2 π k n s - ( x 0 w 0 ) 2 sin 2 π k s ] + + etc . ,
E scat = K 0 A l ϕ l ρ 3 e - j θ l ,
E scat = 2 K η 0 E 0 B + C η 0 2 l = 0 C 0 ¯ , l ϕ l ( r ) ρ 3 e - j θ l 1 - ρ 2 ρ 3 e - j θ l ,
E scat K C 0 ¯ , 0 = 1 1 - ρ 2 ρ 3 e - j θ o p = 0 C 0 ¯ , 4 p C 0 ¯ , 0 ϕ 4 p + 1 1 + j ρ 2 ρ 3 e - j θ o × p = 0 C 0 ¯ , 4 p + 1 C 0 ¯ , 0 ϕ 4 p + 1 + 1 1 + ρ 2 ρ 3 e - j θ o p = 0 C 0 ¯ , 4 p + 2 C 0 ¯ , 0 ϕ 4 p + 2 + 1 1 - j ρ 2 ρ 3 e - j θ o p = 0 C 0 ¯ , 4 p + 3 C 0 ¯ , 0 ϕ 4 p + 3 .
E scat K = 1 ( 1 - ρ 2 ρ 3 e - j θ o ) { cos ( 2 x 0 w 0 x w 0 ) + exp [ - ( x 0 w 0 ) 2 ] cosh ( 2 x 0 w 0 x w 0 ) } exp [ - ( x w 0 ) 2 ] + 1 ( 1 + j ρ 2 ρ 3 e - j θ o ) { sin ( 2 x 0 w 0 x w 0 ) + exp [ - ( x 0 w 0 ) 2 ] sinh ( 2 x 0 w 0 x w 0 ) } exp [ - ( x w 0 ) 2 ] + 1 ( 1 + ρ 2 ρ 3 e - j θ o ) { - cos ( 2 x 0 w 0 x w 0 ) + exp [ - ( x 0 w 0 ) 2 ] cosh ( 2 x 0 w 0 x w ) } exp [ - ( x w 0 ) 2 ] + 1 ( 1 - j ρ 2 ρ 3 e - j θ o ) { - sin ( 2 x 0 w 0 x w 0 ) + exp [ - ( x 0 w 0 ) 2 ] sinh ( 2 x 0 w 0 x w ) } exp [ - ( x w 0 ) 2 ] .
E scat = ± K 2 exp - [ ( x w 0 ) 2 2 ( x 0 w 0 ) ( x w 0 ) + ( x 0 w 0 ) 2 ] .
E ( z = 0 ) = exp [ - ( x w 0 ) 2 ] { exp [ + j ( ω / c ) sin δ x ] + exp [ - j ( ω / c ) sin δ x ] } = 2 exp [ - ( x w 0 ) 2 ] cos ( ω c sin δ x ) 2 exp [ - ( x w 0 ) 2 ] cos ( ω c δ ) x .
δ x 0 / d r .
w 0 4 = ( λ 0 / π ) 2 d r b [ 1 - ( d r / b ) ] for b = 2 d r , ( 2 π / λ 0 ) = ( 2 d r / w 0 2 ) .
E ~ 2 E 0 exp [ - ( x w 0 ) 2 ] cos ( 2 x 0 w 0 x w 0 ) .
ψ 0 ( x , y ) = ( 2 / π ) 1 2 1 w 0 exp - [ ( x - x 0 ) 2 / w 0 2 ] exp - ( y 2 / w 0 2 ) .
ϕ m , n ( x , y ) = 1 w 0 { 2 π 2 n + m m ! n ! } H m ( ( 2 ) 1 2 x w 0 ) H n ( ( 2 ) 1 2 y w 0 ) × exp - [ ( x 2 + y 2 ) / w 0 2 ] .
C 0 ¯ , m = w 0 ( π 2 ) 1 2 - + ψ 0 ( x ) ϕ m ( x ) d x .
C 0 ¯ , m = { 1 π 2 m m ! } 1 2 ( - 1 ) m - + exp [ ( 2 ) 1 2 x 0 w 0 ξ - x 0 2 w 0 2 ] d m d ξ m e - ξ 2 d ξ ,
ξ = ( 2 ) 1 2 x / w 0 .
C 2 0 ¯ , m = ( x 0 / w 0 ) 2 m m !             exp - ( x 0 w 0 ) 2 .
K 0 = C 0 ¯ , 0 2 + C 0 ¯ , s 2 + C 0 ¯ , 2 s 2 + = e - u { 1 + u s / s ! + u 2 s / ( 2 s ) ! + } ,
K n ( u ) = e - u 0 u e + u K n - 1 ( u ) d u ,
u = ( x 0 / w 0 ) 2 .
K 0 e + u = 1 s { e u + exp [ ( j ) 4 / s u ] + exp [ ( j ) 4 k s u ] + } = 1 + u S s ! + u 2 S ( 2 s ) ! + , k s - 1.
s e u K 0 = e u + [ exp ( j ) 4 / s u + exp ( j ) 4 ( s - 1 ) s u ] + + [ exp ( j ) 4 / s u + exp ( j ) 4 ( s - p ) s u ] + + [ e - u if s is even o if s is odd ] .
[ exp ( j ) 4 p s u + exp ( j ) 4 ( s - p ) s u ] = 2 exp [ u cos ( 2 p π / s ) ] cos [ u sin ( 2 p π / s ) ] .
t 2 ( 1 - ρ 2 ρ 3 e - j θ o ) 4 C 0 ¯ , 0 K E scat = l = 0 C 0 ¯ , 4 l C 0 ¯ , 0 ϕ 4 l ( x ) ,
C 0 ¯ , m / C 0 ¯ , 0 = ( x 0 / w 0 ) m [ 1 / ( m ! ) 1 2 ] ,
ϕ m ( ξ ) = 1 ( 2 m m ! ) 1 2 m ! k = 0 [ m / 2 ] ( - 1 ) k ( 2 ξ ) m - 2 k k ! ( m - 2 k ) ! ,
l = 0 : R . H . S . = 1 l = 1 , 4 l = 4 : + 1 2 2 ( x 0 w 0 ) 4 [ ( 2 ξ ) 4 4 ! - ( 2 ξ ) 2 1 ! 2 ! + 1 2 ! ] l = 2 , 4 l = 8 + 1 2 4 ( x 0 w 0 ) 8 [ 2 ( ξ ) 6 8 ! - ( 2 ξ ) 6 1 ! 6 ! + ( 2 ξ ) 4 2 ! 4 ! - 2 ( ξ ) 2 3 ! 2 ! + 1 4 ! ]
Coefficient of ( 2 ξ ) 4 p = ( x 0 ( 2 ) 1 2 w 0 ) 4 p ( 4 ρ ) ! { j = 0 ( x 0 ( 2 ) 1 2 w 0 ) 4 j ( 2 j ) ! = cosh ( x 0 ( 2 ) 1 2 w 0 ) 2 } , Coefficient of - ( 2 ξ ) 4 p + 2 = ( x 0 ( 2 ) 1 2 w 0 ) 4 p + 2 ( 4 p + 2 ) ! { ( x 0 ( 2 ) 1 2 w 0 ) 4 j + 2 ( 2 j + 1 ) ! = sinh ( x 0 ( 2 ) 1 2 w 0 ) 2 } .
= { cosh ( x 0 ( 2 ) 1 2 w 0 ) 2 0 { [ ( 2 ) 1 2 x 0 / w 0 ] ξ } 4 k 4 k ! - sinh ( x 0 ( 2 ) 1 2 w 0 ) 2 0 { [ ( 2 ) 1 2 x 0 / w 0 ] ξ } 4 k + 2 ( 4 k + 2 ) ! } .
1 - [ ( 2 ) 1 2 θ ] 4 4 ! + [ ( 2 ) 1 2 θ ] 8 8 ! + = cosh θ cos θ ,             ( a ) [ ( 2 ) 1 2 θ ] 2 2 ! - [ ( 2 ) 1 2 θ ] 6 6 ! + [ ( 2 ) 1 2 θ ] 10 10 ! + = sinh θ sin θ .             ( b )
t 2 ( 1 - ρ 2 ρ 3 e - j θ a ) 4 C 0 ¯ , 0 K = 1 2 ( cos [ ( 2 ) 1 2 x 0 ξ w 0 ] { cosh [ x 0 ( 2 ) 1 2 w 0 ] 2 + sinh [ x 0 ( 2 ) 1 2 w 0 ] 2 } + cosh ( ( 2 ) 1 2 x 0 ξ w 0 ) [ cosh ( x 0 ( 2 ) 1 2 w 0 ) 2 - sinh ( x 0 ( 2 ) 1 2 w 0 ) 2 ] ) exp - [ x w 0 ] 2 .
C 2 0 ¯ , 0 = exp - ( x 0 / w 0 ) 2 E scat = 2 K t 2 ( 1 - ρ 2 ρ 3 e - j θ o ) { cos ( 2 x 0 x w 0 2 ) + [ exp - ( x 0 w 0 ) 2 ] cosh ( 2 x 0 x w 0 2 ) } exp - ( x w 0 ) 2 .

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