Abstract

The resonant modes of optic interferometer cavities are investigated in the angular spectrum domain. The investigation is based on an integral equation (governing the relation between the normal modes and cavity geometry) which is derived by using the self-consistent Rayleigh formulation for solving diffraction problems. For plane-parallel cavities this integral equation can be solved by means of a series expansion of orthogonal functions characteristic of the cavity geometry without making any assumption about the relative magnitudes of end reflector dimensions and separation. The solution also provides, in addition to a comparison with the solutions of the approximated Huygens’ integral equation as found by other investigators, a direct way for obtaining the angular plane-wave spectrum (or the radiation pattern) of the beam emerging from the cavity. The particular cases of plane-parallel cavities with infinite-strip and circular end reflectors are considered in this paper.

© 1966 Optical Society of America

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  1. A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
    [Crossref]
  2. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
    [Crossref]
  3. G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
    [Crossref]
  4. A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).
    [Crossref]
  5. S. R. Barone, J. Appl. Phys. 34, 831 (1963).
    [Crossref]
  6. H. Schachter and L. Bergstein, in MRI Symposium Proceedings XIII, Optical Masers (Polytechnic Institute of Brooklyn, New York, 1963), pp. 173–198.
  7. L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).
    [Crossref]
  8. H. Schachter, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June1964).
  9. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.
  10. An alternative way of obtaining Eq. (11) is to set w≈1-12(u2+v2) in the integrand of Eq. (6) and then integrate over u and v.
  11. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962).
  12. The proof of this statement is somewhat lengthy and is only outlined here. Let γ˜=1/Γ. Equation (27) then becomes, D(Γ)≡[βjkΓ−δj,k]=0. To show that Eq. (27.2) is true for all γ˜<∞ (with the possible exception of γ˜=0) it is obviously sufficient to show thatD(M)(Γ)=[βjkΓ-δj,k](M)=∑r=0MCr(M)Γrconverges uniformly to D(Γ) for all |Γ|<R<∞. To show this we use a comparison test. The real seriesDˆ(M)(x)=∑r=0NCˆrxr+Cˆ∑r=N+1M(xA)r=∑r=0N(Cˆr-CˆAr)xr+Cˆ∑r=0M(xA)r,with finite but arbitrary Ĉr and Ĉ, converges uniformly (as M→∞) toDˆ(x)=∑r=0N(Cr-CˆAr)xr+Cˆ1-x/A,for all |x|<A<R<∞. D(M)(Γ) therefore converges uniformly to D(Γ) for all |Γ|<A<R<∞ if: (1) the limit of Cr(M) exist for all r, and (2) an N<∞ exist such that |Cr|⑽A−r, for all r>N. This can be shown to be the case if the elements βjk decrease at least as fast 1/jk when j and k approach infinite, i.e., if beyond a certain J⑽N and K⑽N, βjk⑽(B/jk) for all j>J and k>K, B being a finite constant. For example, it is readily found that |CM(M)|⑾(BM/M!), and, thereforelimM→∞{∣CM(M)∣/A-M}⩽limM→∞{(eBA/M)M}⩽1.0,for all A<R<∞ (since we can always take M⑾eBA). Similarly, ∣CM-1(M)∣⩽[13MBM-1/(M-1)!], and limM→∞{∣CM-1(M)∣/A-(M-1)}⩽limM→∞{13(eBA/M(M-2)/(M-1))M-1}⩽1.0, etc.(27.2)limM→∞[γ˜n(M)]=γ˜n.
  13. For the determination of the eigenvalues and eigenfunctions of the three low-order even-symmetric modes, (1), (3), (5), and the three low-order odd-symmetric modes, (2), (4), (6), it was found in each case sufficient to use only 5×5 terms of the determinant of Eq. (27.1). Truncating the determinant after 6×6 or more terms led to essentially the same results.
  14. L. Bergstein and H. Schachter, J. Opt. Soc. Am. 55, 1226 (1965).
    [Crossref]
  15. E. Wolf and E. W. Marchand, J. Opt. Soc. Am. 54, 587 (1964)
    [Crossref]
  16. L. A. Vainshtein, Soviet Physics—JETP 17, 714 (1963).
  17. H. Ogura and Y. Yoshida, Japan, J. Appl. Phys. 3, 546 (1964).
    [Crossref]
  18. H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).
    [Crossref]
  19. E. Marom, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June1965).
  20. To a very good approximation pmk≈(12m+k-14)π.
  21. The only exception is f01′(r,ϕ,z)=(π)-12 exp{iβz}, which is a uniform plane wave.
  22. For the determination of the eigenvalues and eigenfunctions of the six low-order modes (01), (02), (03), (11), (12), (13), it was found sufficient to use only 5×5 terms of the determinant of Eq. (46).
  23. W. Grobner and H. Hofreiter, Integraltafel, Part II, Bestimmte Integrale (Springer-Verlag, Vienna, Austria, 1958), 2nd ed.

1965 (2)

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 55, 1226 (1965).
[Crossref]

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).
[Crossref]

1964 (3)

1963 (3)

L. A. Vainshtein, Soviet Physics—JETP 17, 714 (1963).

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

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

1962 (1)

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
[Crossref]

1961 (2)

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

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

Barone, S. R.

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

Bergstein, L.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 55, 1226 (1965).
[Crossref]

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).
[Crossref]

H. Schachter and L. Bergstein, in MRI Symposium Proceedings XIII, Optical Masers (Polytechnic Institute of Brooklyn, New York, 1963), pp. 173–198.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

Boyd, G. D.

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
[Crossref]

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

Fox, A. G.

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

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

Gordon, J. P.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

Grobner, W.

W. Grobner and H. Hofreiter, Integraltafel, Part II, Bestimmte Integrale (Springer-Verlag, Vienna, Austria, 1958), 2nd ed.

Hofreiter, H.

W. Grobner and H. Hofreiter, Integraltafel, Part II, Bestimmte Integrale (Springer-Verlag, Vienna, Austria, 1958), 2nd ed.

Ikenoue, J.

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).
[Crossref]

Kogelnik, H.

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
[Crossref]

Li, T.

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

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

Marchand, E. W.

Marom, E.

E. Marom, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June1965).

Ogura, H.

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).
[Crossref]

H. Ogura and Y. Yoshida, Japan, J. Appl. Phys. 3, 546 (1964).
[Crossref]

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962).

Schachter, H.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 55, 1226 (1965).
[Crossref]

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).
[Crossref]

H. Schachter, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June1964).

H. Schachter and L. Bergstein, in MRI Symposium Proceedings XIII, Optical Masers (Polytechnic Institute of Brooklyn, New York, 1963), pp. 173–198.

Vainshtein, L. A.

L. A. Vainshtein, Soviet Physics—JETP 17, 714 (1963).

Wolf, E.

E. Wolf and E. W. Marchand, J. Opt. Soc. Am. 54, 587 (1964)
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

Yoshida, Y.

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).
[Crossref]

H. Ogura and Y. Yoshida, Japan, J. Appl. Phys. 3, 546 (1964).
[Crossref]

Bell System Tech. J. (3)

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

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
[Crossref]

J. Appl. Phys. (1)

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

J. Opt. Soc. Am. (3)

Japan, J. Appl. Phys. (2)

H. Ogura and Y. Yoshida, Japan, J. Appl. Phys. 3, 546 (1964).
[Crossref]

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).
[Crossref]

Proc. IEEE (1)

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

Soviet Physics—JETP (1)

L. A. Vainshtein, Soviet Physics—JETP 17, 714 (1963).

Other (12)

H. Schachter and L. Bergstein, in MRI Symposium Proceedings XIII, Optical Masers (Polytechnic Institute of Brooklyn, New York, 1963), pp. 173–198.

E. Marom, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June1965).

To a very good approximation pmk≈(12m+k-14)π.

The only exception is f01′(r,ϕ,z)=(π)-12 exp{iβz}, which is a uniform plane wave.

For the determination of the eigenvalues and eigenfunctions of the six low-order modes (01), (02), (03), (11), (12), (13), it was found sufficient to use only 5×5 terms of the determinant of Eq. (46).

W. Grobner and H. Hofreiter, Integraltafel, Part II, Bestimmte Integrale (Springer-Verlag, Vienna, Austria, 1958), 2nd ed.

H. Schachter, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June1964).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

An alternative way of obtaining Eq. (11) is to set w≈1-12(u2+v2) in the integrand of Eq. (6) and then integrate over u and v.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962).

The proof of this statement is somewhat lengthy and is only outlined here. Let γ˜=1/Γ. Equation (27) then becomes, D(Γ)≡[βjkΓ−δj,k]=0. To show that Eq. (27.2) is true for all γ˜<∞ (with the possible exception of γ˜=0) it is obviously sufficient to show thatD(M)(Γ)=[βjkΓ-δj,k](M)=∑r=0MCr(M)Γrconverges uniformly to D(Γ) for all |Γ|<R<∞. To show this we use a comparison test. The real seriesDˆ(M)(x)=∑r=0NCˆrxr+Cˆ∑r=N+1M(xA)r=∑r=0N(Cˆr-CˆAr)xr+Cˆ∑r=0M(xA)r,with finite but arbitrary Ĉr and Ĉ, converges uniformly (as M→∞) toDˆ(x)=∑r=0N(Cr-CˆAr)xr+Cˆ1-x/A,for all |x|<A<R<∞. D(M)(Γ) therefore converges uniformly to D(Γ) for all |Γ|<A<R<∞ if: (1) the limit of Cr(M) exist for all r, and (2) an N<∞ exist such that |Cr|⑽A−r, for all r>N. This can be shown to be the case if the elements βjk decrease at least as fast 1/jk when j and k approach infinite, i.e., if beyond a certain J⑽N and K⑽N, βjk⑽(B/jk) for all j>J and k>K, B being a finite constant. For example, it is readily found that |CM(M)|⑾(BM/M!), and, thereforelimM→∞{∣CM(M)∣/A-M}⩽limM→∞{(eBA/M)M}⩽1.0,for all A<R<∞ (since we can always take M⑾eBA). Similarly, ∣CM-1(M)∣⩽[13MBM-1/(M-1)!], and limM→∞{∣CM-1(M)∣/A-(M-1)}⩽limM→∞{13(eBA/M(M-2)/(M-1))M-1}⩽1.0, etc.(27.2)limM→∞[γ˜n(M)]=γ˜n.

For the determination of the eigenvalues and eigenfunctions of the three low-order even-symmetric modes, (1), (3), (5), and the three low-order odd-symmetric modes, (2), (4), (6), it was found in each case sufficient to use only 5×5 terms of the determinant of Eq. (27.1). Truncating the determinant after 6×6 or more terms led to essentially the same results.

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Figures (25)

Fig. 1
Fig. 1

Fabry–Perot cavity.

Fig. 2
Fig. 2

Geometry of the infinite-strip plane reflector cavity.

Fig. 3
Fig. 3

Relative losses, P1(H; κ)=1−| γ ˜ 1|2, of the first-order mode of the infinite-strip plane reflector cavity as a function of the Fresnel number H=2a2 for various values of κ=2a/λ. ( γ ˜ n is the eigenvalue associated with the nth mode, 2a is the width of the end reflectors, is the reflector separation, and λ is the resonant wavelength.)

Fig. 4
Fig. 4

Relative power losses, P2(H; κ)=1−| γ ˜ 2|2, of the second-order mode of the infinite-strip plane reflector cavity as a function of the Fresnel number H=2a2 for various values of κ=2a.

Fig. 5
Fig. 5

Relative power losses, P3(H; κ)=1−| γ ˜ 3|2, of the third-order mode of the infinite-strip plane reflector cavity as a function of the Fresnel number H=2a2 for various values of κ=2a/λ.

Fig. 6
Fig. 6

Relative power losses, P4(H; κ)=1−| γ ˜ 4|2, of the fourth-order mode of the infinite-strip plane reflector cavity as a function of the Fresnel number H=2a2 for various values of κ=2a.

Fig. 7
Fig. 7

Departure, δ ν n ( H ; κ ) = ν n - ν = ( c / 2 r 1 2 ) [ arg ( γ ˜ n * ) / π, of the resonant frequencies νn of the four low-order modes, (1), (2), (3), (4), of the infinite-strip plane reflector cavity from the resonant frequency ν = N c / 2 r 1 2 of a plane-parallel Fabry–Perot cavity of length and infinite lateral extent as a function of the Fresnel number H=2a2 for various values of κ=2a/λ. ( γ ˜ n * is the complex conjugate of the eigenvalue γ ˜ n associated with the nth mode, N=20 is a positive integer which specifies the axial mode order of the cavity, r is the relative dielectric constant of the cavity host medium, and c is the speed of light in vacuum.)

Fig. 8
Fig. 8

Relative power loss P1(H; ∊) of the dominant mode of the infinite-strip plane reflector cavity as a function of the Fresnel number H=2a2 for various values of =a/.

Fig. 9
Fig. 9

Comparison of the relative power losses P1(H) of the first-order mode of the infinite-strip plane reflector cavity with the results of Fox and Li1 and Bergstein and Schachter.7

Fig. 10
Fig. 10

Comparison of the relative power losses P2(H) of the second-order mode of the infinite-strip plane reflector cavity with the results of Fox and Li1 and Bergstein and Schachter.7

Fig. 11
Fig. 11

The angular plane-wave spectra |Fn(τ; 0.2)| of the first-and second-order modes of the infinite-strip plane reflector cavity with a Fresnel number H=0.2 and κ=2a/λ≥ 2.0. For comparison, the angular plane-wave spectra |Fn(τ; 0)| for the limiting case H→0 are shown in dashed lines. The angular spectra are normalized such that the integral of |Fn(τ)|2 over-all −∞<τ<+∞ is unity. [τ=κu=(2a/λ)u.]

Fig. 12
Fig. 12

The angular plane-wave spectra |Fn(τ; 0.2)| of the third- and fourth-order modes of the infinite-strip plane reflector cavity with a Fresnel number H=0.2 and κ=2a/λ≥2.0. The comparisons and normalizations are the same as in Fig. 11.

Fig. 13
Fig. 13

The angular plane-wave spectra |Fn(τ; 100)| of the first- and second-order modes of the infinite-strip plane reflector cavity with a Fresnel number H=100 and κ=2a/λ≥2.0. The angular spectra are normalized such that the integral of |Fn(τ)|2 over-all −∞<τ<+∞ is unity. [τ=κu=(2a/λ)u.]

Fig. 14
Fig. 14

The angular plane-wave spectra |Fn(τ; 100)| of the third- and fourth-order modes of the infinite-strip plane reflector cavity with a Fresnel number H=100 and κ=2a/λ≥2.0. The normalizations are the same as in Fig. 13.

Fig. 15
Fig. 15

Smoothing of a discontinuous function fn(x/a) by its Fourier-type expansion.

Fig. 16
Fig. 16

Geometry of the circular plane-reflector cavity.

Fig. 17
Fig. 17

Relative power losses, Pmn(H; κ)=1−| γ ˜ m n|2 of the (01) mode of the circular plane reflector cavity as a function of the Fresnel number H=2a2 for various values of κ=2a/λ.

Fig. 18
Fig. 18

Relative power losses, Pmn(H; κ)=1−| γ ˜ m n|2, of the (11) mode of the circular plane reflector cavity as a function of the Fresnel number H=2a2 for various values of κ=2a/λ.

Fig. 19
Fig. 19

Relative power losses, Pmn(H; κ)=1−| γ ˜ m n|2, of the (02) mode of the circular plane reflector cavity as a function of the Fresnel number H=2a2 for various values of κ=2a/λ.

Fig. 20
Fig. 20

Relative power losses, Pmn(H; κ)=1−|γmn|2, of the (12) mode of the circular plane reflector cavity as a function of the Fresnel number H=2a2 for various values of κ=2a/λ.

Fig. 21
Fig. 21

Departure δ ν m n ( H ; κ ) = ν m n - ν = ( c / 2 r 1 2 ) [ arg ( γ ˜ m n * ) / π ], of the resonant frequencies νmn of the four low-order modes (01), (11), (02), (12), of the circular plane reflector cavity from the resonant frequency ν = ( N c / 2 r 1 2 ) of a plane-parallel Fabry–Perot cavity of length and infinite lateral extent as a function of the Fresnel number H=2a2 for various values of κ=2a/λ.

Fig. 22
Fig. 22

The angular spectra |Fmn(τ; 0.2)| of the (01) and (11) modes of the circular plane reflector cavity with a Fresnel number H=2a2=0.2 and κ=2a/λ≥2.0. For comparison, the angular spectra |Fmn(τ; 0)| for the limiting case H→0 are shown in dashed lines. The angular spectra are normalized such that the integral of |Fmn(τ)|2 over-all 0<τ<+∞ is unity. [τ=κs.]

Fig. 23
Fig. 23

The angular spectra |Fmn(τ; 0.2)| of the (02) and (12) modes of the circular plane reflector cavity with a Fresnel number H=2a2=0.2 and κ=2a/λ≥2.0. The comparisons and normalizations are the same as in Fig. 22.

Fig. 24
Fig. 24

The angular spectra |Fmn(τ; 100)| of the (01) and (11) modes of the circular plane reflector cavity with a Fresnel number H=2a2=100 and κ=2a/λ≥2.0. The angular spectra are normalized such that the integral of |Fmn(τ)|2 over-all 0<τ<+∞ is unity. [τ=κs.]

Fig. 25
Fig. 25

The angular spectra |Fmn(τ; 100)| of the (02) and (12) modes of the circular plane reflector cavity with a Fresnel number H=2a2=100 and κ=2a/λ≥2.0. The normalizations are the same as in Fig. 24.

Equations (113)

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f + ( x , y , z ; t ) = f ( x , y , z ) e - i 2 π ν t
f ( x , y , - ) = γ f ( x , y , 0 + ) ,
f ( x , y , z ) = - + - + F ( u , v ) e i β ( u x + v y + w z ) d u d v ,
w = { + [ 1 - ( u 2 + v 2 ] 1 2 , when ( u 2 + v 2 ) 1 . 0 , + i [ ( u 2 + v 2 ) - 1 ] 1 2 , when ( u 2 + v 2 ) 1.0.
F ( u , v ) = ( β / 2 π ) 2 f ( x , y , 0 + ) e - i β ( u x + v y ) d x d y ,
γ ˜ f ( x , y ) = ( β 2 π ) 2 - + - + f ( x 1 , y 1 ) × e i β [ u ( x - x 1 ) + v ( y - y 1 ) + ( w - 1 ) ] d u d v d x 1 d y 1 ,
γ ˜ F ( u , v ) = ( β 2 π ) 2 - + - + F ( u 1 , v 1 ) × e i β [ x ( u 1 - u ) + y ( v 1 - v ) + ( w 1 - 1 ) ] d x d y d u 1 d v 1 ,
γ ˜ = γ e - i β .
γ ˜ f ( x , y ) = i β 2 π f ( x 1 , y 1 ) r ( 1 - 1 / i β r ) × ( e + i β r / r ) d x 1 d y 1 ,
r = [ ( x - x 1 ) 2 + ( y - y 1 ) 2 + 2 ] 1 2 .
r + [ ( x - x 1 ) 2 + ( y - y 1 ) 2 ] / 2
γ ˜ f ( x , y ) = 1 i λ f ( x 1 , y 1 ) × exp { i ( π / λ ) [ ( x - x 1 ) 2 + ( y - y 1 ) 2 ] } d x 1 d y 1 .
γ ˜ F ( u ) = β 2 π - + - a + a F ( u 1 ) e + i β [ x ( u 1 - u ) + ( w 1 - 1 ) ] d x d u ,
F ( u ) = β 2 π - a + a f ( x ) e - i β u x d x
w = { + ( 1 - u 2 ) 1 2 , when u 2 1.0 , + i ( u 2 - 1 ) 1 2 , when u 2 1.0.
γ ˜ F ( τ ) = - F ( τ 1 ) sin [ π ( τ 1 - τ ) ] π ( τ 1 - τ ) × exp { - i π ( κ 2 / H ) ( 1 - w 1 ) } d τ 1 ,
κ = 2 a / λ ,
H = 2 a 2 / λ ,
τ = ( 2 a / λ ) u = κ u .
K ( τ , τ 1 ) = K ˜ ( τ , τ 1 ) exp { - i π ( κ 2 / H ) ( 1 - w 1 ) } = { sin [ π ( τ 1 - τ ) ] / π ( τ 1 - τ ) } × exp { - i π ( κ 2 / H ) ( 1 - w 1 ) } .
K ˜ ( τ , τ 1 ) = k g k ( τ ) h k ( τ 1 ) .
γ ˜ F ( τ ) = - + F ( τ 1 ) sin [ π ( τ 1 - τ ) ] π ( τ 1 - τ ) d τ 1 .
- + f ( t ) sin [ ω 0 ( t - τ ) ] π ( t - τ ) d t = f ( τ ) .
{ g k + c ( κ u ) } = { sin [ π κ u - ( k + c ) π / 2 ] / π [ κ u - 1 2 ( k + c ) ] } ,
G k ( τ ) = ( e k / 2 ) [ g ( k - 1 ) ( τ ) + ( - 1 ) ( k - 1 ) g - ( k - 1 ) ( τ ) ] = e k 2 ( π τ ) sin [ π τ - ( k - 1 ) π / 2 ] π 2 { τ 2 - [ ( k - 1 ) / 2 ] 2 } ,
G k ( τ ) = ( 1 / 2 ) [ g k ( τ ) - ( - 1 ) k g - k ( τ ) ] = 2 ( k / 2 ) sin [ π τ - k π / 2 ] / π [ τ 2 - ( k / 2 ) 2 ] ,
- + G k ( τ ) G j ( τ ) d τ = δ k , j ,
K ˜ ( τ , τ 1 ) = sin [ π ( τ 1 - τ ) ] π ( τ 1 - τ ) = k = 1 G k ( τ ) G k ( τ 1 ) .
F ( τ ) = k = 1 A k G k ( τ ) ,
A k = 1 γ ˜ - + F ( τ ) G k ( τ ) exp { - i π ( κ 2 / H ) ( 1 - w ) } d τ .
k = 1 ( β j k - γ ˜ δ j , k ) A k = 0 ,             j = 1 , 2 , 3 , 4 , ,
β j k = - + G k ( τ ) G j ( τ ) exp [ - i π ( κ 2 / H ) ( 1 - w ) ] d τ .
[ β j k - γ ˜ δ j , k ] = 0.
[ β j k - γ ˜ δ j , k ] ( M ) = 0 ,
F n ( k u ) = k A n , k G k ( κ u ) ,
- + F n ( τ ) F m ( τ ) W ( τ ) d τ = C n δ n , m ,
P n = 1 - γ ˜ n 2
δ ν n = ν n - ν = ( ν / N ) [ arg ( γ ˜ n * ) / π ] ,
w 1 - 1 2 u 2 ,
β j k - + G k ( τ ) G j ( τ ) exp { - i ( π / 2 H ) τ 2 } d τ = β ˆ j k ,
F n ( κ u ) = A n , n 2 k ( A n , k A n , n ) e k × ( π κ u ) sin [ π κ u - ( k - 1 ) π / 2 ] π 2 { ( κ u ) 2 - [ ( k - 1 ) / 2 ] 2 } ,
F n ( κ u ) = A n , n 2 k ( A n , k A n , n ) × ( k / 2 ) sin [ π κ u - k π / 2 ] π [ ( κ u ) 2 - ( k / 2 ) 2 ] ,
F n ( κ u ) G n ( κ u ) = 2 e n ( π κ u ) sin [ π κ u - ( n - 1 ) π / 2 ] π 2 { ( κ u ) 2 - [ ( n - 1 ) / 2 ] 2 } ,
f n ( x / a ) e n sin [ ( n - 1 ) ( π / 2 ) ( x / a ) + n π / 2 ] .
F n ( κ u ) G n ( κ u ) = 2 ( n / 2 ) sin ( π κ u - n π / 2 ) / π [ ( κ u ) 2 - ( n / 2 ) 2 ] ,
f n ( x / a ) sin [ n ( π / 2 ) ( x / a ) + n π / 2 ] .
γ ˜ n β ˆ n n e - i π ( n 2 / 8 H ) { 1 - [ ( 1 - i ) n 2 / 12 H ( H ) 1 2 ] + } ,
P n [ n 2 / 6 H ( H ) 1 2 ] { 1 + [ π n 2 / 40 H ] + } ,
ν n - ν ( ( n 2 / 8 H ) { 1 - [ 2 / 3 π ( H ) 1 2 ] + } ) ( ν / N ) .
s = ( u 2 + v 2 ) 1 2 ,             ϑ = t g - 1 ( v / u ) .
γ ˜ F ( s , ϑ ) = ( β 2 π ) 2 s 1 = 0 ϑ 1 = 0 2 π r = 0 ϕ = 0 2 π F ( s 1 , ϑ 1 ) × exp { + i β r [ s 1 cos ( ϑ 1 - ϕ ) - s cos ( ϑ - ϕ ) ] + i β ( w 1 - 1 ) } r d r d ϕ s 1 d s 1 d ϑ 1 ,
w = { + ( 1 - s 2 ) 1 2 , when s 1.0 , + i ( s 2 - 1 ) 1 2 , when s 1.0.
F ( s , ϑ ) = F ( s ) Θ ( ϑ )
Θ m ( ϑ ) = ( 2 π ) - 1 2 e + i m ϑ ,
γ ˜ m F ˜ m ( τ ) = 0 F ˜ m ( τ 1 ) K ˜ m ( τ , τ 1 ) × exp { - i π ( κ 2 / H ) ( 1 - w 1 ) } τ 1 d τ 1 ,
K ˜ m ( τ , τ 1 ) = π 2 0 1 J m ( π τ ρ ) J m ( π τ 1 ρ ) ρ d ρ = π τ 1 J m - 1 ( π τ 1 ) J m ( π τ ) - τ J m - 1 ( π τ ) J m ( π τ 1 ) τ 2 - τ 1 2 ,
K m ( τ , τ 1 ) = K ˜ m ( τ , τ 1 ) exp { - i π ( κ 2 / H ) ( 1 - w 1 ) }
D m k ( τ ) = 2 π ( π τ ) J m - 1 ( π τ ) / [ ( π τ ) 2 - p ˜ m k 2 ] ,
D m k ( τ ) = 2 π p m k J m ( π τ ) / [ ( π τ ) 2 - p m k 2 ] ,
J m ( p m k ) = 0 ,             k = 0 , 1 , 2 , 3 , ,
- + D m k ( τ ) D j ( τ ) τ d τ = δ m , δ m , j ,
K ˜ m ( τ , τ 1 ) = k = 1 D m k ( τ ) D m k ( τ 1 ) ,
F ˜ m ( τ ) = k = 1 A m , k D m k ( τ ) .
k = 1 ( β m , j k - γ ˜ m δ j , k ) A m , k = 0 ,             j = 1 , 2 , 3 , 4 , ,
β m , j k = 0 D m k ( τ ) D m j ( τ ) × exp [ - i π ( κ 2 / H ) ( 1 - w ) ] τ d τ .
( β m , j k - γ ˜ m δ j , k ) = 0.
F m n ( s , ϑ ) = e i m ϑ k = 1 A m n , k D m k ( κ s ) ,
f m k ( r a , ϕ , z ) = [ - ( i ) m ( π ) 1 2 J m ( p ˜ m k ) ] J m ( p m k r a ) × exp ( i { m ϕ + [ 1 - ( p ˜ m k / π κ ) 2 ] 1 2 β z } ) ,
f m k ( r a , ϕ , z ) = [ - ( i ) m ( π ) 1 2 J m + 1 ( p m k ) ] J m ( p m k r a ) × exp ( i { m ϕ + [ 1 - ( p m k / π κ ) 2 ] 1 2 β z } ) ,
s = 0 ϑ = 0 2 π F m n ( s , ϑ ) F p ( s , ϑ ) W ( s ) s d s d ϑ = C m n δ m , δ n , p ,
P m n = 1 - γ ˜ m n 2
δ ν m n = ν m n - ν = ( ν / N ) [ arg ( γ ˜ m n * ) / π ] ,
w 1 - 1 2 s 2
β m , j k = 0 D m k ( τ ) D m j ( τ ) e - i ( π / 2 H ) r 2 τ d τ = β ˆ m , j k ,
F m n ( s , ϑ ) ( 2 π ) - 1 2 e i m ϑ D m n ( κ s ) = ( π ) - 1 2 { ( κ s ) J m - 1 ( π κ s ) ( κ s ) 2 - [ p ( m - 1 ) n / π ] 2 } e i m ϑ
f m n ( r / a , ϕ ) { - ( i ) m / ( π ) 1 2 J m [ p ( m - 1 ) n ] } × J m [ p ( m - 1 ) n r / a ] e i m ϕ .
F m n ( s , ϑ ) ( 2 π ) - 1 2 e i m ϑ D m n ( κ s ) = ( π ) - 1 2 [ ( p m n / π ) J m ( π κ s ) ( κ s ) 2 - ( p m n / π ) 2 ] e i m ϑ ,
f m n ( r / a , ϕ ) [ - ( i ) m / ( π ) 1 2 J m + 1 ( p m n ) ] J m ( p m n r / a ) e i m ϕ .
γ ˜ m n β ˆ m , n n [ 1 - ( p m n / π ) 2 3 H ( H ) 1 2 + ] × exp [ - i π ( p m n / π ) 2 2 H ] ,
P m n ( p m n / π ) 2 / 1.5 H ( H ) 1 2 ( n + 1 2 m - 1 4 ) 2 / 1.5 H ( H ) 1 2
ν - ν ( p m n / π ) 2 2 H ( ν N ) ( n + 1 2 m - 1 4 ) 2 2 H ( ν N ) .
ϑ ˆ m n = sin - 1 [ p m n / π κ ] sin - 1 [ ( m + 2 n - 1 2 ) / 2 κ ]
β j k β ˆ j k = 2 e k e j - + τ 2 sin [ π τ - ( k - 1 ) π / 2 ] sin [ π τ - ( j - 1 ) π / 2 ] π 2 { τ 2 - [ ( k - 1 ) / 2 ] 2 } { τ 2 - [ ( j - 1 ) / 2 ] 2 } e - i ( π / 2 H ) τ 2 d τ ,
β j k β ˆ j k = 2 - + ( k / 2 ) ( j / 2 ) sin [ π τ - k π / 2 ] sin [ π τ - j π / 2 ] π 2 [ τ 2 - ( k / 2 ) 2 ] [ τ 2 - ( j / 2 ) 2 ] e - i ( π / 2 H ) τ 2 d τ .
β ˆ k j = ( - 1 ) 1 2 ( k - j ) e k e j { ( 2 i ) 1 2 / π [ ( k - 1 ) 2 - ( j - 1 ) 2 ] } [ ( k - 1 ) B ( k - 1 , H ) - ( j - 1 ) B ( j - 1 , H ) ]
β ˆ 11 = ( 2 i ) - 1 2 { 2 N [ 2 ( H ) 1 2 ] - [ i / π ( H ) 1 2 ] ( 1 - e i 2 π H ) } ,
β ˆ k k = ( 2 i ) - 1 2 { D ( k - 1 , H ) + ( k - 1 4 H + i π ( k - 1 ) ) B ( k - 1 , H ) - [ i / π ( H ) 1 2 ] [ 1 - ( - 1 ) k - 1 e i 2 π H ] }
β ˆ k j = ( - 1 ) 1 2 ( k - j ) [ ( 2 i ) 1 2 / π ( k 2 - j 2 ) ] [ j B ( k , H ) - k B ( j , H ) ]
β ˆ k k = ( 2 i ) - 1 2 { D ( k , H ) + [ ( k / 4 H ) - ( i / π k ) ] B ( k , H ) - [ i / π ( H ) 1 2 ] [ 1 - ( - 1 ) k e i 2 π H ] } .
N ( ξ ) = 0 ξ e i ( π / 2 ) τ 2 d τ = C ( ξ ) + i S ( ξ ) ,
B ( m , H ) = e - i π m 2 / 8 H { N [ m 2 ( H ) 1 2 + 2 ( H ) 1 2 ] + N [ m 2 ( H ) 1 2 - 2 ( H ) 1 2 ] - 2 N [ m 2 ( H ) 1 2 ] } ,
D ( m , H ) = e - i ( π m 2 / 8 H ) { N [ m 2 ( H ) 1 2 + 2 ( H ) 1 2 ] - N [ m 2 ( H ) 1 2 - 2 ( H ) 1 2 ] } .
β ˆ 11 = e - i ( π / 4 ) ( 2 H ) 1 2 [ 1 + i 1 3 ( π H ) - ( 2 / 15 ) ( π H ) 2 - i ( 1 / 21 ) ( π H ) 3 + ] ,
β ˆ 1 k = β ˆ k 1 = ( - 1 ) 1 2 ( k - j ) e i π / 4 2 2 π ( k - 1 ) 2 ( 2 H ) 3 2 [ 1 + i ( 1 - 12 π 2 ( k - 1 ) 2 ) ( π H ) - 2 3 ( 1 - 30 π 2 ( k - 1 ) 2 + 360 π 4 ( k - 1 ) 4 ) ( π H ) 2 + ]
β ˆ k j = - ( - 1 ) 1 2 ( k - j ) e - i π / 4 [ 24 / π 2 ( k - 1 ) 2 ( j - 1 ) 2 ] ( 2 H ) 5 2 { 1 + i 5 3 [ 1 - 12 ( k - 1 ) 2 + ( j - 1 ) 2 π 2 ( k - 1 ) 2 ( j - 1 ) 2 ] ( π H ) + }
β ˆ k j = - ( - 1 ) 1 2 ( k - j ) e i ( π / 4 ) [ 16 / π 3 ( k - 1 ) 2 ( j - 1 ) 2 ] ( 2 H ) 3 2 { 1 + i 3 [ 1 - 4 ( k - 1 ) 2 + ( j - 1 ) 2 π 2 ( k - 1 ) 2 ( j - 1 ) 2 ] ( π H ) - 5 [ 1 - 12 ( k - 1 ) 2 + ( j - 1 ) 2 π 2 ( k - 1 ) 2 ( j - 1 ) 2 + 48 ( k - 1 ) 4 + ( k - 1 ) 2 ( j - 1 ) 2 + ( j - 1 ) 4 π 4 ( k - 1 ) 4 ( j - 1 ) 4 ] ( π H ) 2 + } .
β ˆ k j = - e i π / 4 2 k j 12 H ( H ) 1 2 [ 1 - i π ( k 2 + j 2 ) / 20 H + ]
β ˆ k k = e - i π k 2 / 8 H [ 1 - ( 1 - i ) k 2 12 H ( H ) 1 2 ( 1 + i ) π k 4 480 H 2 ( H ) 1 2 + ] .
J m ( x ρ ) J m ( y ρ ) ρ d ρ = ρ y J m - 1 ( ρ y ) J m ( ρ x ) - x J m - 1 ( ρ x ) J m ( ρ y ) x 2 - y 2 ,
π 2 0 J m ( π ρ τ ) J m ( π q τ ) e - i ( π / 2 H ) τ 2 τ d τ = ( - i ) m + 1 π H J m ( π H ρ q ) e i ( π / 2 ) H ( ρ 2 + q 2 ) ,
β ˆ m , k j = ( - i ) m + 1 2 π H J c + 1 ( p c k ) J c + 1 ( p c j ) 0 1 0 1 J m ( p c k ρ ) J m ( p c j q ) J m ( π H ρ q ) e i ( π / 2 ) H ( ρ 2 + q 2 ) ρ q d ρ d q ,
0 1 ρ 2 n J m + 1 ( p m k ρ ) ρ d ρ = 2 ( m + 1 + n ) J m + 1 ( p m k ) p m k 2 r = 0 n [ ( - 1 ) r ( n + 1 - r ) r ( m + 1 + n - r ) r ( 2 / p m k ) 2 r ] ,
β ˆ 0 , 11 = - i [ ( π / 2 ) H ] [ 1 + i 1 2 ( π H ) - ( 5 / 24 ) ( π H ) 2 - i ( 7 / 96 ) ( π H ) 3 + ] ,
β ˆ 0 , k 1 = β ˆ 0 , 1 k = ( 4 p 1 k 2 ) ( π 2 H ) 2 [ 1 + i ( 1 - 4 p 1 k 2 ) ( π H ) - 5 8 [ 1 - ( 12 / 5 ) ( 4 / p 1 k 2 ) + ( 12 / 5 ) ( 16 / p 1 k 4 ] ( π H ) 2 + ]
β ˆ 0 , k j = i ( 32 p 1 k 2 p 1 j 2 ) ( π 2 H ) 3 [ 1 + i 3 2 ( 1 - 4 p 1 k 2 + p 1 j 2 p 1 k 2 p 1 j 2 ) ( π H ) + ]
β ˆ m , k j = ( - i ) m + 1 ( 16 m 2 m ! p ˜ m k 2 p ˜ m j 2 ) ( π 2 H ) m + 1 { 1 + i m + 1 m [ 1 - 2 m ( p ˜ m k 2 + p ˜ m j 2 p ˜ m k 2 p ˜ m j 2 ) ] ( π H ) - ( m + 2 ) ( 2 m + 1 ) 4 m 2 [ 1 - 8 m ( m + 1 ) 2 m + 1 ( p ˜ m k 2 + p ˜ m j 2 p ˜ m k 2 p ˜ m j 2 ) + 16 m 2 ( m + 1 ) 2 m + 1 ( p ˜ m k 4 + p ˜ m k 2 p ˜ m j 2 + p ˜ m j 4 p ˜ m k 4 p ˜ m j 4 ) ] ( π H ) 2 + }
β ˆ m , k j = - e - i ( π / 4 ) [ 2 p m k p m j 3 π 2 H ( H ) 1 2 ] [ 1 - i p m k 2 + p m j 2 5 π H + ]
β ˆ m , k k = exp { - i p m k 2 π H } [ 1 - ( 1 - i ) p m k 2 3 π 2 H ( H ) 1 2 - ( 1 + i ) p m k 4 30 π 3 H 2 ( H ) 1 2 + ] .
D(M)(Γ)=[βjkΓ-δj,k](M)=r=0MCr(M)Γr
Dˆ(M)(x)=r=0NCˆrxr+Cˆr=N+1M(xA)r=r=0N(Cˆr-CˆAr)xr+Cˆr=0M(xA)r,
Dˆ(x)=r=0N(Cr-CˆAr)xr+Cˆ1-x/A,
limM{CM(M)/A-M}limM{(eBA/M)M}1.0,
limM[γ˜n(M)]=γ˜n.