Abstract

The integral equation that describes the mode structure of unstable resonators of rectangular aperture is solved by a new method that makes use of an asymptotic expansion of the integral for large Fresnel number. This expansion is carried out to terms of order 1/F, where F is the Fresnel number, and includes effects of diffraction at the edges of the feedback mirror, as well as a term corresponding to the geometric-optics approximation. Results obtained by this method are in good agreement with those obtained by gaussian integration, even at an effective Fresnel number of unity. The method has the advantage that it involves finding the roots of a polynomial, rather than the eigenvalues of a matrix, and therefore requires less computer time and storage. The results of the present work are at variance with a class of theories based on geometric optics, as regards the higher-loss modes even for very large Fresnel number, although the lowest-loss symmetric mode is correctly given by geometric optics in this limit.

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  1. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  2. A. E. Siegman and H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  3. A. E. Siegman and R. Arrathoon, IEEE J. Quantum Electron. 3, 156 (1966).
    [CrossRef]
  4. R. L. Sanderson and W. Streifer, Appl. Opt. 8, 2129 (1969).
    [CrossRef] [PubMed]
  5. W. Streifer, IEEE J. Quantum Electron. 4, 229 (1967).
    [CrossRef]
  6. L. Bergstein, Appl. Opt. 7, 495 (1968).
    [CrossRef] [PubMed]
  7. S. R. Barone, Appl. Opt. 10, 935 (1970).
    [CrossRef]
  8. A. E. Siegman, Proc. IEEE 53, 277 (1965).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  10. Mathematically, the reason for the continuous spectrum of eigenvalues is that taking only the first term in the expansion of I(x) corresponds to neglecting contributions from the end points of the integration, which contribute at the next order. The end-point contributions do not contribute if the integral is taken over the range (- ∞, ∞), provided it exists. Their neglect, therefore, is equivalent to considering the small mirror to be infinitely big, an approximation the dangers of which we have touched on above. It requires, of course, the introduction of a convergence factor to insure the finiteness of the integral if Reξ> 0. It also makes Eq. (8) a singular integral equation, in the sense that the range of integration is infinite, and the appearance of a continuum of eigenvalues is a common pathology of such equations. See, e.g. Ref. 11, p. 273.
  11. F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J. 1965).
  12. M. A. Jenkins and J. F. Traub, Numer. Math. 14, 252 (1970).
    [CrossRef]
  13. A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).
    [CrossRef]
  14. L. Sirovich, Techniques of Asymptotic Analysis (Springer, New York, 1971).
    [CrossRef]
  15. N. Bleistein and A. Handelsman, SIAM J. Appl. Math. (Soc. Ind. Appl. Math.) 15, 422 (1967).

1970

A. E. Siegman and H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

S. R. Barone, Appl. Opt. 10, 935 (1970).
[CrossRef]

M. A. Jenkins and J. F. Traub, Numer. Math. 14, 252 (1970).
[CrossRef]

1969

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 2129 (1969).
[CrossRef] [PubMed]

1968

L. Bergstein, Appl. Opt. 7, 495 (1968).
[CrossRef] [PubMed]

1967

W. Streifer, IEEE J. Quantum Electron. 4, 229 (1967).
[CrossRef]

N. Bleistein and A. Handelsman, SIAM J. Appl. Math. (Soc. Ind. Appl. Math.) 15, 422 (1967).

1966

A. E. Siegman and R. Arrathoon, IEEE J. Quantum Electron. 3, 156 (1966).
[CrossRef]

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

1965

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

1961

A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).
[CrossRef]

Arrathoon, R.

A. E. Siegman and R. Arrathoon, IEEE J. Quantum Electron. 3, 156 (1966).
[CrossRef]

Barone, S. R.

S. R. Barone, Appl. Opt. 10, 935 (1970).
[CrossRef]

Bergstein, L.

L. Bergstein, Appl. Opt. 7, 495 (1968).
[CrossRef] [PubMed]

Bleistein, N.

N. Bleistein and A. Handelsman, SIAM J. Appl. Math. (Soc. Ind. Appl. Math.) 15, 422 (1967).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Fox, A. G.

A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).
[CrossRef]

Handelsman, A.

N. Bleistein and A. Handelsman, SIAM J. Appl. Math. (Soc. Ind. Appl. Math.) 15, 422 (1967).

Hildebrand, F. B.

F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J. 1965).

Jenkins, M. A.

M. A. Jenkins and J. F. Traub, Numer. Math. 14, 252 (1970).
[CrossRef]

Kogelnik, H.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Li, T.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).
[CrossRef]

Miller, H. Y.

A. E. Siegman and H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

Sanderson, R. L.

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 2129 (1969).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman and H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

A. E. Siegman and R. Arrathoon, IEEE J. Quantum Electron. 3, 156 (1966).
[CrossRef]

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

Sirovich, L.

L. Sirovich, Techniques of Asymptotic Analysis (Springer, New York, 1971).
[CrossRef]

Streifer, W.

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 2129 (1969).
[CrossRef] [PubMed]

W. Streifer, IEEE J. Quantum Electron. 4, 229 (1967).
[CrossRef]

Traub, J. F.

M. A. Jenkins and J. F. Traub, Numer. Math. 14, 252 (1970).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Other

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

A. E. Siegman and H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

A. E. Siegman and R. Arrathoon, IEEE J. Quantum Electron. 3, 156 (1966).
[CrossRef]

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 2129 (1969).
[CrossRef] [PubMed]

W. Streifer, IEEE J. Quantum Electron. 4, 229 (1967).
[CrossRef]

L. Bergstein, Appl. Opt. 7, 495 (1968).
[CrossRef] [PubMed]

S. R. Barone, Appl. Opt. 10, 935 (1970).
[CrossRef]

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Mathematically, the reason for the continuous spectrum of eigenvalues is that taking only the first term in the expansion of I(x) corresponds to neglecting contributions from the end points of the integration, which contribute at the next order. The end-point contributions do not contribute if the integral is taken over the range (- ∞, ∞), provided it exists. Their neglect, therefore, is equivalent to considering the small mirror to be infinitely big, an approximation the dangers of which we have touched on above. It requires, of course, the introduction of a convergence factor to insure the finiteness of the integral if Reξ> 0. It also makes Eq. (8) a singular integral equation, in the sense that the range of integration is infinite, and the appearance of a continuum of eigenvalues is a common pathology of such equations. See, e.g. Ref. 11, p. 273.

F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J. 1965).

M. A. Jenkins and J. F. Traub, Numer. Math. 14, 252 (1970).
[CrossRef]

A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).
[CrossRef]

L. Sirovich, Techniques of Asymptotic Analysis (Springer, New York, 1971).
[CrossRef]

N. Bleistein and A. Handelsman, SIAM J. Appl. Math. (Soc. Ind. Appl. Math.) 15, 422 (1967).

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Figures (14)

Fig. 1
Fig. 1

Modulus of λ vs effective Fresnel number for M = 2.9, antisymmetric roots.

Fig. 2
Fig. 2

Modulus of λ vs effective Fresnel number for M = 2.9, symmetric roots.

Fig. 3
Fig. 3

Modulus of λ vs effective Fresnel number for M = 1.9, antisymmetric roots.

Fig. 4
Fig. 4

Modulus of λ vs effective Fresnel number for M = 1.9, symmetric roots.

Fig. 5
Fig. 5

Plot of paths in the complex plane traced out by the symmetric roots as Feff varies from 10 to 11, M = 1.9. Arrows denote positions of the roots for Feff = 10. The α and β roots cross near Feff = 11.

Fig. 6
Fig. 6

Plot of paths in the complex plane traced out by the symmetric roots as Feff varies from 56 to 57, M = 1.9. Arrows denote positions of the roots for Feff = 56. The α and β roots cross near Feff = 57.

Fig. 7
Fig. 7

Plot of paths in the complex plane traced out by the symmetric roots as Feff varies from 57 to 58. Arrows denote positions of the roots for Feff = 57. The α and β roots approach one another very closely, but do not cross, near Feff = 58.

Fig. 8
Fig. 8

Plot of paths in the complex plane traced out by the symmetric roots as Feff varies from 99 to 100. Arrows denote positions of the roots for the Feff = 99. The α root has split off from the others completely, and is describing a small circle around λ = 1.

Fig. 9
Fig. 9

Irradiance and phase of g0(x) on the feed-back mirror, for Feff = 49.4, M = 1.9.

Fig. 10
Fig. 10

Irradiance and phase of g1(x) on the feed-back mirror, for Feff = 49.4, M = 1.9.

Fig. 11
Fig. 11

Irradiance and phase of g2(x) on the feed-back mirror, for Feff = 49.4, M = 1.9.

Fig. 12
Fig. 12

Irradiance of go and g2 for Feff = 16.4, M = 2.9. This is a point of maximum separation.

Fig. 13
Fig. 13

Irradiance of g0 and g2 for Feff = 16.874, M = 2.9. This is a crossing point.

Fig. 14
Fig. 14

Irradiance of g0 and g2 for Feff = 15.863, M = 2.9. This is a cusping point.

Tables (2)

Tables Icon

Table I Polynomial sizes and computation times. Degree of polynomial and times required to compute roots on the IBM 360/50, in some typical cases. N is the degree of the polynomial used, T1 is the time (in s) for the first run, using the unmodified CPOLY routine, and T2 is the time for subsequent runs in which previously found roots were used as starting points in the program. All previous roots were used, except as noted. Times given are for complete runs; i.e., symmetric and antisymmetric cases.

Tables Icon

Table II Values of critical Feff for various values of M.

Equations (93)

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ψ ( x , y ) = u ( x ) υ ( y ) ,
γ x ( 1 ) u ( 1 ) ( x 1 ) = a 2 a 2 K ( x 1 , x 2 ) u ( 2 ) ( x 2 ) d x 2 ,
γ x ( 2 ) u ( 2 ) ( x 2 ) = a 1 a 1 K ( x 1 , x 2 ) u ( 1 ) ( x 1 ) d x 1 ,
K ( x , y ) = [ i / ( λ L ) ] 1 2 exp [ i ( k / 2 L ) ( g 1 x 2 + g 2 y 2 2 x y ) ] .
γ x ( 1 ) γ x ( 2 ) u ( 1 ) ( x 1 ) = a 1 a 1 d x 1 a 2 a 2 d x 2 K ( x 1 , x 2 ) × K ( x 1 , x 2 ) u ( 1 ) ( x i ) .
γ x u ( x ) = a 1 a 1 K ( 2 ) ( x , x ) u ( x ) d x ,
K ( 2 ) ( x , x ) = K ( x , x ) K ( x , x ) d x = [ i / ( 2 λ L g 2 ) ] 1 2 exp { i π λ L 1 2 g 2 × [ ( x 2 + x 2 ) ( 2 g 1 g 2 1 ) 2 x x ] } .
γ x u ( x ) = ( i F ) 1 2 1 1 e i π F [ g ( x 2 + y 2 ) 2 x y ] u ( y ) d y ,
F eff = F ( g 2 1 ) 1 2 = F 2 ( M 1 M ) ,
M = ( g + 1 ) 1 2 + ( g 1 ) 1 2 ( g + 1 ) 1 2 ( g 1 ) 1 2 = ( g 1 g 2 ) 1 2 + ( g 1 g 2 1 ) 1 2 ( g 1 g 2 ) 1 2 ( g 1 g 2 1 ) 1 2 .
g ( x ) = e i π F eff x 2 u ( x ) .
γ g ( x ) = ( i F ) 1 2 1 1 e i π M F ( y x / M ) 2 g ( y ) d y .
t = π M F
λ g ( x ) = ( i t / π ) 1 2 1 1 e i t ( y x / M ) 2 g ( y ) d y ,
λ = γ M 1 2
( Loss ) i j = 1 | λ i λ j | M 2 ,
( Frequency ) i j = c 2 L [ k + 1 π arg ( λ i λ j ) ] ,
I ( x ) = 1 1 e i t ( y x / M ) 2 g ( y ) d y ,
I ( x ) ( π / i t ) 1 2 g ( x / M ) .
λ g ( x ) = g ( x / M ) .
g ( x ) = x ξ
λ = M ξ .
Re ξ 0
I ( x ) = π 1 2 ( i t ) 1 2 g ( x / M ) 1 2 ( i t ) 1 [ e i t ( 1 + x / M ) 2 1 + x / M g ( 1 ) + e i t ( 1 x / M ) 2 1 x / M g ( 1 ) ] + 1 4 π 1 2 ( i t ) 3 2 g ( x / M ) + 1 4 ( i t ) 2 { e i t ( 1 + x / M ) 2 [ g ( 1 ) ( 1 + x / M ) 3 + g ( 1 ) ( 1 + x / M ) 2 ] + e i t ( 1 x / M ) 2 [ g ( 1 ) ( 1 x / M ) 3 + g ( 1 ) ( 1 x / M ) 2 ] } + O ( t 3 2 ) ,
λ g ( x ) = g ( x / M ) + g ( 1 ) G ± ( x ) ,
= 1 2 ( i π t ) 1 2 , G ± ( x ) = e i t ( 1 x / M ) 2 1 x / M ± e i t ( 1 + x / M ) 2 1 + x / M ,
g ( x ) = ± g ( x ) .
λ g ( x / M ) = g ( x / M 2 ) + g ( 1 ) G ± ( x / M ) .
λ 2 g ( x ) = g ( x M 2 ) + g ( 1 ) [ G ± ( x M ) + λ G ± ( x ) ] .
λ n g ( x ) = g ( x M n ) + g ( 1 ) [ G ± ( x M n 1 ) + λ G ± ( x M n 2 ) + + λ n 1 G ± ( x ) ] .
g ( x M n ) g ( 0 ) .
g ( 0 ) = G ± ( 0 ) g ( 1 ) λ 1 .
g ( m ) ( x ) O ( ( t / M ) m ) .
E n ( x ) = exp { i t [ ( 1 + x / M n ) 2 M n 1 ] } , n = 1 , 2 , ,
M n = m = 0 n M 2 m , n = 0 , 1 , .
G ± ( x ) = E 1 ( x ) 1 x / M ± E 1 ( x ) 1 + x / M .
I n ( x ) = 1 1 e i t ( y x / M ) 2 E n ( y ) f n ( y ) d y ,
I n ( x ) = 1 1 e i t ( a n y 2 2 b n y + c n ) f n ( y ) d y ,
a n = M n / M n 1 , b n = x / M M n / M n 1 , c n = x 2 / M 2 + M n 1 1 .
( i t / π ) 1 2 I n ( x ) = a n 1 2 E n + 1 ( x ) f n ( r n ( x ) ) [ E n ( 1 ) f n ( 1 ) E 1 ( x ) d n + x / M + E n ( 1 ) f n ( 1 ) E 1 ( x ) d n + x / M ] ,
g ( x ) = n = 1 [ E n ( x ) f n ( x ) ± E n ( x ) f n ( x ) ] .
g ( x ) = h ( x ) + n = 1 N [ E n ( x ) f n ( x ) ± E n ( x ) f n ( x ) ] ,
λ h ( x ) = h ( x / M ) + a N 1 2 [ E N + 1 ( x ) f N ( r N ( x ) ) ± E N + 1 ( x ) f N ( r N ( x ) ) ] ,
λ f 1 ( x ) = { h ( 1 ) 1 + x / M + [ E n ( 1 ) f n ( 1 ) 1 d n + x / M ± E n ( 1 ) f n ( 1 ) 1 d n + + x / M ] } ,
λ f n ( x ) = a n 1 1 2 f n 1 ( r n 1 ( x ) ) , n = 2 , 3 , , N .
λ n f n + 1 ( x ) = M n 1 2 f 1 ( s n ( x ) ) , n = 1 , 2 , ,
s n ( x ) = r 1 ( r 2 ( ( r n ( x ) ) ) ) .
λ h ( x ) = h ( x / M ) + λ N M N 1 2 [ E N + 1 ( x ) f 1 ( s N ( x ) ) ± E N + 1 ( x ) f 1 ( s N ( x ) ) ] ,
f 1 ( x ) = [ λ 1 h ( 1 ) 1 + x / M + n = 1 N λ n M n 1 1 2 ( E n ( 1 ) f 1 ( s n 1 ( 1 ) ) d n + x / M ± E n ( 1 ) f 1 ( s n 1 ( 1 ) ) d n + + x / M ) ] .
f 1 ( x ) = f 1 ( 0 ) 1 + x / M ,
f 1 ( 0 ) = [ λ 1 h ( 1 ) + n = 1 N λ n A n ± f 1 ( 0 ) ] ,
A n ± = M n 1 1 2 [ E n ( 1 ) 1 + s n 1 ( 1 ) / M ± E n ( 1 ) 1 + s n 1 ( 1 ) / M ] .
f 1 ( 0 ) = λ 1 h ( 1 ) [ 1 + n = 1 N A n ± λ n ] 1 .
λ h ( x ) = h ( x / M ) + h ( 1 ) H ± ( x ) ,
H ± ( x ) = λ N M N 1 2 / [ 1 + A n ± λ n ] × [ E N + 1 ( x ) 1 + s n ( x ) / M ± E N + 1 ( x ) 1 + s n ( x ) / M ] .
λ N h ( x ) / h ( 1 ) = [ H ± ( 0 ) λ 1 + H ± ( x / M N 1 ) + λ H ± ( x / M N 2 ) + + λ N 1 H ± ( x ) ] ,
h ( x / M N ) h ( 0 ) .
( λ 1 ) P N + ( λ ) + A N + 1 + = 0 ( symmetric case ) ,
P N ± ( λ ) = λ N [ 1 + n = 1 N A n ± λ n ]
A N + 1 + = 2 M N 1 2 E N ( 0 ) 1 + s N ( 0 ) / M .
λ N + 1 + ( A 1 + + 1 ) λ N + ( A 2 + A 1 + ) λ N 1 + + ( A N + 1 + A N + ) = 0.
P N ( λ ) = 0 ( antisymmetric case ) ,
λ N + A 1 λ N 1 + + A N = 0.
lim n A n = 2 i lim n e 2 π i F eff sin 2 π F eff M = 4 π i F eff M e 2 π i F eff .
1 + n = 1 N A n μ n = 0.
λ > 1 / M ( antisymmetric ) .
λ > 1 / M 2 ( symmetric ) .
t / M N [ ( 1 + 1 / M N ) 2 ( 1 1 / M N ) 2 ] 2 π .
lim N t / M N = 2 π F eff
M N 4 F eff .
f 1 ( 0 ) ( x ) = f 1 ( 0 ) ( 0 ) 1 + x / M .
f 1 ( 1 ) ( x ) = { λ j 1 h ( 0 ) 1 + x / M + n = 1 N λ j 1 M n 1 1 2 ( E n ( 1 ) f 1 ( 0 ) ( s n 1 ( 1 ) ) d n + x / M ± E n ( 1 ) f 1 ( 0 ) ( s n 1 ( 1 ) ) d n + + x / M ) } ,
λ j 1 h ( 0 ) = f 1 ( 0 ) [ 1 + n = 1 N A n + λ j n ]
I = α β exp { i t ( a y 2 2 b y + c ) } g ( y ) d y .
ν a b ( a ) = 1 , ν a b ( b ) = 0 ,
d n ν a b ( y ) d y n | y = a , b = 0 , n = 1 , 2 , ,
ν a b ( y ) = ( y b exp { 1 u a + 1 u b } d u ) / ( a b exp { 1 u a + 1 u b } d u ) .
I = I α + I y 0 + I y 0 + I β ,
I α = α y 0 exp { i t ( a y 2 2 b y + c ) } g ( y ) ν α y 0 ( y ) d y ,
I y 0 = α y 0 exp { i t ( a y 2 2 b y + c ) } × g ( y ) [ 1 ν α y 0 ( y ) ] d y ,
I y 0 = y 0 β exp { i t ( a y 2 2 b y + c ) } g ( y ) ν y 0 β ( y ) d y ,
I β = y 0 β exp { i t ( a y 2 2 b y + c ) } × g ( y ) [ 1 ν y 0 β ( y ) ] d y .
g ˜ ( y ) = g ( y ) ν α y 0 ( y ) , a y y 0 = 0 , y > y 0
I α = exp { i t ( α a 2 2 b α + c ) } α exp { 2 i t ( a + b ) ( y α ) } × exp { i a t ( y α ) 2 } g ˜ ( y ) d y .
I = exp { i t ( a α 2 2 b α + c ) } α exp { 2 i t ( a + b ) s } × n = 0 m = 0 ( i a t ) n n ! g ˜ ( m ) ( α ) m ! s 2 n + m d s .
0 e z s s p f ( s ) d s = Γ ( p + 1 ) z p + 1 f ( 0 ) + O ( z )
I α = exp { i t ( a α 2 2 b α + c ) } n = 0 m = 0 ( 2 n + m ) ! n ! m ! × g ( m ) ( α ) ( i a ) n t n m 1 [ 2 i ( a + b ) ] 2 n + m + 1 .
g ˜ ( y ) = g ( y ) [ 1 ν α y 0 ( y ) ] , α y y 0 g ˜ ( y ) = 0 , y < α
I y 0 = exp { i t ( c b 2 / a ) } y 0 exp { i a t ( y y 0 ) 2 } g ˜ ( y ) d t = exp { i t ( c b 2 / a ) } 0 exp { i a t s 2 } m = 0 g ( m ) ( x ) m ! s m d s = 1 2 exp { i t ( c b 2 / a ) } m = 0 g ( m ) ( y 0 ) m ! × 0 exp ( i a t u ) u 1 2 ( m 1 ) d u .
I y 0 = 1 2 exp { i t ( c b 2 / a ) } × m = 0 Γ ( 1 2 ( m + 1 ) ) m ! g ( m ) ( y 0 ) ( i a t ) 1 2 ( m + 1 ) .
I = exp { i t ( c b 2 / a ) } m = 0 1 + ( 1 ) m 2 Γ ( 1 2 ( m + 1 ) ) m ! g ( m ) ( y 0 ) ( i a t ) 1 2 ( m + 1 ) + exp { i t ( a α 2 2 b α + c ) } n = 0 m = 0 ( 2 n + m ) ! n ! m ! g ( m ) ( α ) ( i a ) n t n m 1 [ 2 i ( a + b ) ] 2 n + m + 1 + exp { i t ( a β 2 2 b β + c ) } n = 0 m = 0 ( 2 n + m ) ! n ! m ! g ( m ) ( β ) ( i a ) n t n m 1 [ 2 i ( a + b ) ] 2 n + m + 1 .
( i t / π ) 1 2 1 1 exp { i t ( y x / M ) 2 } h ( y ) d y = h ( x / M ) + ( 4 i t ) 1 h ( x / M ) 1 2 ( i π t ) 1 2 { E 1 ( x ) [ h ( 1 ) 1 + x / M ( 2 i t ) 1 ( h ( 1 ) ( 1 + x / M ) 3 + h ( 1 ) ( 1 + x / M ) 2 ) ] + E 1 ( x ) [ h ( 1 ) 1 + x / M ( 2 i t ) 1 ( h ( 1 ) ( 1 + x / M ) 3 + h ( 1 ) ( 1 + x / M ) 2 ) ] } ,
( i t / π ) 1 2 1 1 exp { i t ( y x / M ) 2 } E n ( x ) f n ( x ) d x = a n 1 2 E n + 1 ( x ) [ f n ( r n ( x ) ) + ( 4 i t a n ) 1 f n ( r n ( x ) ) ] 1 2 ( i π t ) 1 2 { E 1 ( x ) E n ( 1 ) [ f n ( 1 ) d n + x / M ( 2 i t ) 1 ( a n f n ( 1 ) ( d n + x / M ) 3 + f n ( 1 ) ( d n + x / M ) 2 ) ] + E 1 ( x ) E n ( 1 ) [ f n ( 1 ) d n + x / M ( 2 i t ) 1 ( a n f n ( 1 ) ( d n + x / M ) 3 + f n ( 1 ) ( d n + x / M ) 2 ) ] } ,