Abstract

A study is made of the resonant or normal modes of optic and quasioptic interferometer cavities with plane–parallel end reflectors. The solution of the integral equation governing the relation between the normal modes and the geometry of the cavity is found by means of a series expansion of orthogonal functions. The terms of the series for the normal modes can be interpreted as Fraunhofer diffraction patterns characteristic of the geometry of the end reflectors. Various geometries, such as the infinite-strip, rectangular, and circular end reflector cavities, are considered and the results plotted and interpreted.

© 1964 Optical Society of America

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References

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  1. A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
    [Crossref]
  2. A. G. Fox and TinGye Li, Bell System Tech. J. 40, 453 (1961).
    [Crossref]
  3. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).
    [Crossref]
  4. R. F. Sooho, Proc. IEEE 51, 70 (1963).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 744–751.
  6. J. Focke, Ber. Sachs. Ges. (Akad.) Wiss. 101, No. 3 (1954).
  7. It should be pointed out that the longitudinal component of the field is small for both the TE and TM modes so that the field is predominantly transverse and the modes are nearly TEM.
  8. Kantorovich and Krylov, Approximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).
  9. C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).
  10. When cos(πH) = 0, higher order terms in the expansion of Si (α) and Ci(α) must be taken into consideration. However, these terms lead to an expression for Bjk′ that tends to zero as (1/j4) and (1/k4) with increasing j and k. Similar considerations apply to the expression for Bjk″ when sin(πH) = 0.
  11. As in Sec. 2, we say that Eqs. (41) and (43) exist if the sum Σ |βm,jk| over all j and k is finite. Now, we recognize immediately that(42*)Bm,jk=2Jm(p∼mk)∫01(πHρ)2Jm−1(πHρ)Jm(p∼mjρ)p∼2mk−(πHρ)2dρis a majorant of βm,jk Since the terms Bm,jk are obtained by solving the homogeneous integral equation(36*)γ∼mRm(ρ)=πH∫01Rm(ρ1)Jm(πHρρ1)ρ1dρ1which is symmetrical in Hermitian sense, the algebraic equation [Bm,jk−γ∼mδj,k]=0 exists and so do Eqs. (43) and (41).
  12. An expansion of Eq. (52) givesAjk=Akj=−e−i(π/4)212kj12HH12[1−iπ(k2+j2)20H+⋯]..
  13. An expansion of Eq. (69) givesAm,jk=−e−i(π/4)(2)12pjpk3π2HH12[1−ipj2+pk25πH+⋯]..

1963 (1)

R. F. Sooho, Proc. IEEE 51, 70 (1963).
[Crossref]

1961 (2)

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

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).
[Crossref]

1958 (1)

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
[Crossref]

1954 (1)

J. Focke, Ber. Sachs. Ges. (Akad.) Wiss. 101, No. 3 (1954).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 744–751.

Boyd, G. D.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).
[Crossref]

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).

Focke, J.

J. Focke, Ber. Sachs. Ges. (Akad.) Wiss. 101, No. 3 (1954).

Fox, A. G.

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

Gordon, J. P.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).
[Crossref]

Kantorovich,

Kantorovich and Krylov, Approximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).

Krylov,

Kantorovich and Krylov, Approximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).

Li, TinGye

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

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
[Crossref]

Sooho, R. F.

R. F. Sooho, Proc. IEEE 51, 70 (1963).
[Crossref]

Townes, C. H.

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 744–751.

Bell System Tech. J. (2)

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

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).
[Crossref]

Ber. Sachs. Ges. (Akad.) Wiss. 101 (1)

J. Focke, Ber. Sachs. Ges. (Akad.) Wiss. 101, No. 3 (1954).

Phys. Rev. (1)

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
[Crossref]

Proc. IEEE (1)

R. F. Sooho, Proc. IEEE 51, 70 (1963).
[Crossref]

Other (8)

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 744–751.

It should be pointed out that the longitudinal component of the field is small for both the TE and TM modes so that the field is predominantly transverse and the modes are nearly TEM.

Kantorovich and Krylov, Approximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).

When cos(πH) = 0, higher order terms in the expansion of Si (α) and Ci(α) must be taken into consideration. However, these terms lead to an expression for Bjk′ that tends to zero as (1/j4) and (1/k4) with increasing j and k. Similar considerations apply to the expression for Bjk″ when sin(πH) = 0.

As in Sec. 2, we say that Eqs. (41) and (43) exist if the sum Σ |βm,jk| over all j and k is finite. Now, we recognize immediately that(42*)Bm,jk=2Jm(p∼mk)∫01(πHρ)2Jm−1(πHρ)Jm(p∼mjρ)p∼2mk−(πHρ)2dρis a majorant of βm,jk Since the terms Bm,jk are obtained by solving the homogeneous integral equation(36*)γ∼mRm(ρ)=πH∫01Rm(ρ1)Jm(πHρρ1)ρ1dρ1which is symmetrical in Hermitian sense, the algebraic equation [Bm,jk−γ∼mδj,k]=0 exists and so do Eqs. (43) and (41).

An expansion of Eq. (52) givesAjk=Akj=−e−i(π/4)212kj12HH12[1−iπ(k2+j2)20H+⋯]..

An expansion of Eq. (69) givesAm,jk=−e−i(π/4)(2)12pjpk3π2HH12[1−ipj2+pk25πH+⋯]..

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Figures (14)

F. 1
F. 1

A general Fabry-Perot interferometer.

F. 2
F. 2

Analog of the Fabry–Perot cavity of finite lateral extent. Surfaces “ R ” have the same geometry as the end reflectors of the cavity and are of perfect reflectivity; Surfaces “ B ” extend to infinity and are perfectly absorbing or “black.”

F. 3
F. 3

Geometry of the infinite-strip plane reflector cavity.

F. 4
F. 4

Terms, fn,k(ξ) = αn,k sin[π(k)]/[π(k)], of the series expansion, fn(ξ) = exp[(/2)2]{Σkfn,k(ξ)}, of the resonant modes of the infinite-strip plane reflector cavity; (a) shows the expansion terms for the case when H>1.0, and (b) shows the terms for the case when H<1.0. The coefficients αn,k are constants, and H=2a2l is the Fresnel number of the cavity. (a is the strip width, l is the reflector separation, and λ is the resonant wavelength.)

F. 5
F. 5

Relative power loss, P n ( H ) = 1 | γ n | 2, of the six low-order modes, (1), (2), (3), (4), (5), (6), of the infinite-strip plane reflector cavity as a function of the Fresnel number H. ( γ n is the eigenvalue associated with the nth mode.)

F. 6
F. 6

Departure, δ ν n ( H ) = ν n ν = ( c / 2 r 1 2 l ) [ arg ( γ n * ) / π ], of the resonant frequency νn of the six low-order modes, (1), (2), (3), (4), (5), (6), of the infinite-strip plane reflector cavity from the resonant frequency ν = N c / 2 ( r ) 1 2 l of a Fabry–Perot interferometer of infinite lateral extent as a function of the Fresnel number H. ( γ n * is the complex conjugate of the eigenvalue γ n associated with the nth mode, c is the speed of light in vacuum, l is the reflector separation, r is the relative dielectric constant of the cavity host medium, and N is a positive integer which gives the axial mode order of the cavity.)

F. 7
F. 7

Amplitude, |fn(ξ;H)|, and phase, arg{fn(ξ; H)}, of the relative field distribution fn(ξ; H) over the reflecting surfaces of the lowest-order even-symmetric and odd-symmetric modes of the infinite-strip plane reflector cavity for various values of the Fresnel number H. (a) Relative amplitude distribution of the dominant (even-symmetric) mode, (1); (b) Relative amplitude distribution of the lowest-order odd-symmetric mode, (2); (c) Relative plase distribution of the dominant (even-symmetric) mode, (1); (d) Relative phase distribution of the lowest-order odd-symmetric mode, (2).

F. 8
F. 8

Amplitude, |fn(ξ)|, and phase, arg{fn(ξ)}, of the relative field distribution fn(ξ) over the reflecting surfaces of the six low-order modes, (1), (2), (3), (4), (5), (6), of the infinite-strip plane reflector cavity for a Fresnel number H = 2.0. (a) Relative amplitude distribution of the three even-symmetric modes, (1), (3), (5); (b) Relative amplitude distribution of the three odd-symmetric modes, (2), (4), (6); (c) Relative phase distribution of the three even-symmetric modes, (1), (3), (5); (d) Relative phase distribution of the three odd-symmetric modes, (2), (4), (6).

F. 9
F. 9

Geometry of the rectangular plane reflector cavity.

F. 10
F. 10

Geometry of the circular plane reflector cavity.

F. 11
F. 11

Relative power loss, P m n ( H ) = 1 | γ m n | 2, of the six low-order modes, (01), (02), (03), (11), (12), (13), of the circular plane reflector cavity as a function of the Fresnel number H. ( γ m n is the eigenvalue associated with the mnth mode.)

F. 12
F. 12

Departure, δ ν m n ( H ) = ν m n ν = ( c / 2 r 1 2 l ) [ arg ( γ m n * ) / π ], of the resonant frequency νmn of the six low-order modes, (01), (02), (03), (11), (12), (13), of the circular plane reflector cavity from the resonant frequency ν = N c / 2 ( r ) 1 2 l of a Fabry–Perot interferometer of infinite lateral extent as a function of the Fresnel number H. ( γ m n * is the complex conjugate of the eigenvalue γ m n associated with the mnth mode.)

F. 13
F. 13

Amplitude, |Rmn(ρ; H)|, and phase, arg{Rmn(ρ; H)}, of the relative radial field distribution Rmn(ρ; H) over the reflecting surfaces of the lowest-order modes (01) and (11) of the circular plane reflector cavity for various values of the Fresnel number H. (a) Relative amplitude distribution of the dominant mode, (01); (b) Relative amplitude distribution of the lowest-order 1-mode, (11); (c) Relative phase distribution of the dominant mode, (01); (d) Relative phase distribution of the lowest-order 1-mode, (11).

F. 14
F. 14

Amplitude, |Rmn(ρ)|, and phase, arg{Rmn(ρ)}, of the relative radial field distribution Rmn(ρ) over the reflecting surfaces of the six low-order modes, (01), (02), (03), (11), (12), (13), of the circular plane reflector cavity for a Fresnel number H = 2.0. (a) Relative amplitude distribution of the three 0-modes, (01), (02), (03); (b) Relative amplitude distribution of the three 1-modes, (11), (12), (13); (c) Relative phase distribution of the three 0-modes, (01), (02), (03); (d) Relative phase distribution of the three 1-modes, (11), (12), (13).

Tables (2)

Tables Icon

Table I Coefficients αn,k of the series expansion of the resonant modes fn(ξ) of the infinite strip plane reflector cavity for various values of the Fresnel number H.

Tables Icon

Table II Coefficients αmn,k of the series expansion of the resonant modes fmn(ρ,ϕ) = Rmn(ρ) exp(−imϕ) of the circular plane reflector cavity for various values of the Fresnel number H.

Equations (136)

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E ( x , y , z ; t ) = E ( x , y , z ) e i 2 π ν t ,
H ( x , y , z ; t ) = H ( x , y , z ) e i 2 π ν t ;
E ( x , y , l ) = γ E ( x , y , 0 ) ,
H ( x , y , l ) = γ H ( x , y , 0 ) ,
ν = c λ = N c 2 r 1 2 l [ 1 arg ( γ e i β l ) N π ] ,
P = 1 | γ | 2 .
γ f ( x , y ) = ( β 2 π ) 2 + R f ( x 1 , y 1 ) e + i β [ u ( x x 1 ) + υ ( y y 1 ) + w l ] × d u d υ d x 1 d y 1 ,
f ( x , y ) f ( x , y , 0 ) = { either E T ( x , y , 0 ) or H n ( x , y , 0 ) ,
w = { [ 1 ( u 2 + υ 2 ) ] 1 2 when u 2 + υ 2 1.0 , i ( u 2 + υ 2 1 ) 1 2 when u 2 + υ 2 1.0 ,
γ f ( x , y ) = β i 2 π R f ( x 1 , y 1 ) ( l r ) e + i β r r d x 1 d y 1 ,
r = [ ( x x 1 ) 2 + ( y y 1 ) 2 + l 2 ] 1 2 ,
r l + { [ ( x x 1 ) 2 + ( y y 1 ) 2 ] / 2 l }
γ f ( x , y ) = 1 i λ l R f ( x 1 , y 1 ) × exp { + i π λ l [ ( x x 1 ) 2 + ( y y 1 ) 2 ] } d x 1 d y 1 ,
γ = γ e i β l .
γ f ( x ) = ( β i 2 π ) 1 2 a + a f ( x 1 ) ( l r ) e + i β r r 1 2 d x 1 ,
γ f ( x ) = ( 1 i λ l ) 1 2 a + a f ( x 1 ) exp { + i π λ l ( x x 1 ) 2 } d x 1 ,
H = 2 a 2 / λ l ,
γ f ( ξ ) = ( H i 2 ) 1 2 1 + 1 f ( ξ 1 ) e + i ( π / 2 ) H ( ξ ξ 1 ) 2 d ξ 1 .
γ f ( ξ ) = A K ( ξ , ξ 1 ) f ( ξ 1 ) d ξ 1 ,
K ( ξ , ξ 1 ) = K ( ξ 1 , ξ ) = e + i ( π / 2 ) H ( ξ ξ 1 ) 2 = e + i ( π / 2 ) H ( ξ 2 + ξ 1 2 ) e i π H ξ ξ 1 ,
K ( ξ , ξ 1 ) = g 1 ( ξ ) g 2 ( ξ 1 ) j a j ( ξ ) β j ( ξ 1 ) .
e i π H ξ ξ 1 = k = + [ sin [ π ( H ξ k ) ] π ( H ξ k ) e i π k ξ 1 ] ,
γ f ( ξ ) = ( H i 2 ) 1 2 1 + 1 f ( ξ 1 ) e + i ( π / 2 ) H ( ξ 1 + ξ 1 2 ) × k = + { sin [ π ( H ξ k ) ] π ( H ξ k ) × [ cos ( π k ξ 1 ) i sin ( π k ξ 1 ) ] } d ξ 1 .
f ( ξ ) = e + i ( π / 2 ) H ξ 2 k = + a k sin [ π ( H ξ k ) ] π ( H ξ k ) ,
α k = a k = [ ( H / i 2 ) 1 2 γ ] 1 + 1 f ( ξ 1 ) cos ( π k ξ 1 ) × e + i ( π / 2 ) H ξ 1 2 d ξ 1 ,
α k = a k = [ ( H / i 2 ) 1 2 γ ] 1 + 1 f ( ξ 1 ) sin ( π k ξ 1 ) × e + i ( π / 2 ) H ξ 1 2 d ξ 1 ,
f ( ξ ) = e + i ( π / 2 ) H ξ 2 [ α 0 sin π H ξ π H ξ + k = 1 α k ( sin [ π ( H ξ k ) ] π ( H ξ k ) + sin [ π ( H ξ + k ) ] π ( H ξ + k ) ) ] = e + i ( π / 2 ) H ξ 2 [ α 0 sin ( π H ξ ) ( π H ξ ) + k = 1 ( ( 1 ) k α k 2 H ξ sin ( π H ξ ) π [ ( H ξ ) 2 k 2 ] ) ] ,
f ( ξ ) = e + i ( π / 2 ) H ξ 2 [ k = 1 α k ( sin [ π ( H ξ k ) ] π ( H ξ k ) sin [ π ( H ξ + k ) ] π ( H ξ + k ) ) ] = e + i ( π / 2 ) H ξ 2 [ k = 1 ( 1 ) k α k 2 k sin ( π H ξ ) π [ ( H ξ ) 2 k 2 ] ] .
γ α j = k = 0 ( β j k α k ) ,
k = 0 ( β j k γ δ j , k ) α k = 0 , j = 0 , 1 , 2 , 3 , ,
β j 0 = β j 0 = e i ( π / 4 ) ( H 2 ) 1 2 1 + 1 e + i π H ξ 2 ( sin ( π H ξ ) π H ξ ) × cos ( π j ξ ) d ξ ,
β j k = β j k = e i ( π / 4 ) ( 2 H ) 1 2 1 + 1 e + i π H ξ 2 ( sin [ π ( H ξ k ) ] π ( H ξ k ) ) × cos ( π j ξ ) d ξ ,
β j k = β j k = e i ( 3 π / 4 ) ( 2 H ) 1 2 1 + 1 e + i π H ξ 2 ( sin [ π ( H ξ k ) ] π ( H ξ k ) ) × sin ( π j ξ ) d ξ ,
[ β j k γ δ j , k ] = 0 .
f n ( ξ ) = e + i ( π / 2 ) H ξ 2 k = + ( α n , k sin [ π ( H ξ k ) ] π ( H ξ k ) ) .
1 + 1 f m ( ξ ) f n ( ξ ) d ξ = 0 , if m n .
δ ν n = ν n ν = ( c 2 r 1 2 l ) [ arg ( γ n * ) π ] ,
P n = 1 | γ n | 2
H y = 0 , E x = 0 , E z = 0 , H x = ( 1 i ω μ ) E y z , H z = + ( 1 i ω μ ) E y x ,
E y = 0 , H x = 0 , H z = 0 , E x = + ( 1 i ω ) H y z , E z = ( 1 i ω ) H y x ,
[ B j k γ δ j , k ] = 0 ,
B j k = ( 2 H ) 1 2 1 + 1 ( sin [ π ( H ξ k ) ] π ( H ξ k ) ) cos ( π j ξ ) d ξ
γ ψ ( ξ ) = ( H 2 ) 1 2 1 1 ψ ( ξ 1 ) cos ( π H ξ ξ 1 ) d ξ 1 ,
B j k = 1 π ( 2 H ) 1 2 ( cos ( π H j k ) { S i [ π H ( j + H ) ( k + H ) ] S i [ π H ( j + H ) ( k H ) ] + S i [ π H ( j H ) ( k H ) ] S i [ π H ( j H ) ( k + H ) ] } sin ( π H j k ) { C i [ π H ( j + H ) ( k + H ) ] C i [ π H ( j + H ) ( k H ) ] + C i [ π H ( j H ) ( k H ) ] C i [ π H ( j H ) ( k + H ) ] } ) ,
S i ( α ) = 0 α sin t t d t and C i ( α ) = α cos t t d t .
S i ( α ) ( π / 2 ) ( cos α / α ) ,
C i ( α ) sin α / α .
B j k 2 H 2 cos ( π H ) π 2 ( 2 H ) 1 2 ( j 2 H 2 ) ( k 2 H 2 ) ,
B j k = ( 2 H ) 1 2 1 + 1 ( sin [ π ( H ξ k ) ] π ( H ξ k ) ) sin ( π j ξ ) d ξ ,
B j k + 2 j k sin ( π H ) π 2 ( 2 H ) 1 2 ( j 2 k 2 ) ( k 2 H 2 ) .
lim N γ n ( N ) = γ n .
γ n = e i π n 2 / 8 H [ ( 1 n 2 12 H H 1 2 π n 4 480 H 2 H 1 2 + ) + i ( n 2 12 H H 1 2 π n 4 480 H 2 H 1 2 + ) ] .
f n ( ξ ) sin [ n ( π / 2 ) ( ξ + 1 ) ] ,
ν n = ν + ( c 2 r 1 2 l ) ( n 2 / 8 H ) ,
P n = n 2 / [ 6 H H 1 2 ] .
f ( x , y ) = X ( x ) Y ( y )
γ x X ( x ) = ( 1 i λ l ) 1 2 a + a X ( x 1 ) exp { + i π λ l ( x x 1 ) 2 } d x 1 , γ y Y ( y ) = ( 1 i λ l ) 1 2 b + b Y ( y 1 ) exp { + i π λ l ( y y 1 ) 2 } d y 1 ,
γ x X ( ξ ) = ( H x i 2 ) 1 2 1 + 1 X ( ξ 1 ) e + i ( π / 2 ) H x ( ξ ξ 1 ) 2 d ξ 1 , γ y Y ( η ) = ( H y i 2 ) 1 2 1 + 1 Y ( η 1 ) e + i ( π / 2 ) H y ( ξ ξ 1 ) 2 d η 1 ,
ξ x / a , η y / b , H x = ( 2 a 2 / λ l ) , H y = ( 2 b 2 / λ l ) ,
γ = γ x γ y = γ e i β l .
f m n ( x , y ) = f m ( x ; H x ) f n ( y ; H y ) ,
γ m n = γ m ( H x ) γ n ( H y ) .
ν m n ν = ( c 2 r 1 2 l ) [ arg γ m * ( H x ) + arg γ n * ( H y ) π ] ,
P m n = 1 | γ m n | 2 = 1 | γ m ( H x ) γ n ( H y ) | 2 ,
γ f ( r , ϕ ) = 1 i λ l r = 0 a ϕ = 0 2 π f ( r 1 , ϕ 1 ) exp { + i π λ l [ r 2 + r 1 2 2 r r 1 cos ( ϕ ϕ 1 ) ] } r 1 d r 1 d ϕ 1 .
f ( r , ϕ ) = R ( r ) Φ ( ϕ )
f ( r , ϕ ) = R m ( r ) e i m ϕ .
γ m R m ( r ) = e i ( m + 1 ) ( π / 2 ) ( 2 π λ l ) 0 a R m ( r 1 ) J m ( 2 π r r 1 λ l ) × exp { + i π λ l ( r 2 + r 1 2 ) } r 1 d r 1 ,
ρ r / a
H = ( 2 a 2 / λ l )
γ m R m ( ρ ) = e i ( m + 1 ) ( π / 2 ) ( π H ) 0 1 R m ( ρ 1 ) J m ( π H ρ ρ 1 ) × e + i ( π / 2 ) H ( ρ 2 + ρ 1 2 ) ρ 1 d ρ 1 ,
J m ( π H ρ ρ 1 ) = k = 0 2 ( π H ρ ) J m 1 ( π H ρ ) J m ( p m k ) [ p m k 2 ( π H ρ ) 2 ] J m ( p m k ρ 1 ) ,
R m ( ρ ) = e + i ( π / 2 ) H ρ 2 × k = 0 α m , k 2 ( π H ρ ) J m 1 ( π H ρ ) J m ( p m k ) [ p m k 2 ( π H ρ ) 2 ] ,
α m , k = e i ( m + 1 ) ( π / 2 ) ( π H γ m ) × 0 1 R m ( ρ 1 ) J m ( p m k ρ 1 ) e + i ( π / 2 ) H ρ 1 2 ρ 1 d ρ 1 .
k = 0 ( β m , j k γ m δ j , k ) α m , k = 0 j = 0 , 1 , 2 , 3 , ,
β m , j k = 2 e i ( m + 1 ) ( π / 2 ) J m ( p m k ) 0 1 e + i π H ρ 2 ( π H ρ ) 2 J m 1 ( π H ρ ) [ p m k 2 ( π H ρ ) 2 ] × J m ( p m j ρ ) d ρ .
[ β m , j k γ m δ j , k ] = 0 .
{ 2 ( π H ρ ) J m 1 ( π H ρ ) / J m ( p m k ) [ p m k 2 ( π H ρ ) 2 ] } .
f m n ( ρ , ϕ ) = R m n ( ρ ) e i m ϕ = e + i ( π / 2 ) H ρ 2 ( k = 0 α m n , k 2 ( π H ρ ) J m 1 ( π H ρ ) J m ( p m k ) [ p m k 2 ( π H ρ ) 2 ] ) e i m ϕ .
ϕ = 0 2 π ρ = 0 1 f m n ( ρ , ϕ ) f k l ( ρ , ϕ ) ρ d ρ d ϕ = C δ m , k δ n , l ,
γ m n = exp ( i π ( p m n / π ) 2 2 H ) × [ ( 1 ( p m n / π ) 2 3 H H 1 2 π ( p m n / π ) 2 30 H 2 H 1 2 + ) + i ( ( p m n / π ) 2 3 H H 1 2 π ( p m n / π ) 4 30 H 2 H 1 2 + ) ] ,
f m n ( r , ϕ ) 2 1 2 / [ J m ( p m n ) ] J m ( p m n ρ ) e i m ϕ .
ν m n = ν + p m n 2 2 π 2 H ( c 2 r 1 2 l ) ν + ( n + 1 2 m 1 4 ) 2 2 H ( c 2 r 1 2 l ) ,
P m n = p m n 2 1.5 π 2 H H 1 2 ( n + 1 2 m 1 4 ) 2 1.5 H H 1 2 .
f n ( ξ ) = exp { + i ( π 2 H ξ 2 ψ n ( 0 ) ) } ( k = + α n , k sin [ π ( H ξ k ) ] π ( H ξ k ) )
{ α n , k = ( 1 ) n 1 α n , k ψ n ( 0 ) = arg { f n ( 0 ) } + ψ n ( 0 )
f m n ( ρ , ϕ ) = R m n ( ρ ) exp ( i m ϕ ) = exp { i [ π 2 H ρ 2 ψ m n ( 0 ) ] } ( k = 0 α m n , k 2 ( π H ρ ) J m 1 ( π H ρ ) J m ( p m k ) [ p m k 2 ( π H ρ ) 2 ] ) e i m ϕ { p m k is the k th zero of the Bessel function J m 1 , ie . , J m 1 ( p m k ) = 0 ψ m n ( 0 ) = arg { R m n ( 0 ) } + ψ m n ( 0 )
f ( ξ ) = k c k sin [ k ( π / 2 ) ( ξ + 1 ) ] ,
γ k c k sin [ k π 2 ( ξ + 1 ) ] = ( H i 2 ) 1 2 1 + 1 e + i ( π / 2 ) H ( ξ ξ 1 ) 2 × ( k c k sin [ k π 2 ( ξ 1 + 1 ) ] ) d ξ 1 .
γ j | c j | 2 = j , k ( A j k c j * c k ) ,
A j k = A k j = ( 1 ) 1 2 ( k j ) ( H i 2 ) 1 2 1 + 1 1 + 1 e + i ( π / 2 ) H ( ξ ξ 1 ) 2 × sin [ k ( π / 2 ) ( ξ + 1 ) ] sin [ j ( π / 2 ) ( ξ 1 + 1 ) ] d ξ d ξ 1 .
A j k = A k j = ( i 2 ) 1 2 π ( k 2 j 2 ) { k e i ( π j 2 / 8 H ) [ N ( 2 H 1 2 j 2 H 1 2 ) N ( 2 H 1 2 + j 2 H 1 2 ) + 2 N ( j 2 H 1 2 ) ] j e i ( π k 2 / 8 H ) [ N ( 2 H 1 2 k 2 H 1 2 ) N ( 2 H 1 2 + k 2 H 1 2 ) + 2 N ( k 2 H 1 2 ) ] } ,
A k k = e i ( π k 2 / 8 H ) { 1 + 1 i 2 ( [ N ( 2 H 1 2 k 2 H 1 2 ) + N ( 2 H 1 2 + k 2 H 1 2 ) ( 1 + i ) ] + ( i π k k 4 H ) × [ N ( 2 H 1 2 k 2 H 1 2 ) N ( 2 H 1 2 + k 2 H 1 2 ) + 2 N ( k 2 H 1 2 ) ] i π H 1 2 exp ( + i π k 2 8 H ) [ 1 e i k π e + i 2 π H ] ] } .
N ( α ) = 0 α e + i ( π / 2 ) t 2 d t = 0 α cos ( π 2 t 2 ) d t + i 0 α sin ( π 2 t 2 ) d t = C ( α ) + i S ( α ) .
j [ k ( γ δ j , k A j k ) c k ] c j * γ maximum ,
k ( A j k γ δ j , k ) c k = 0 .
[ A j k γ δ j , k ] = 0 .
γ n A n n = e i π ( n 2 / 8 H ) { 1 + 1 i 2 ( [ N ( 2 H 1 2 n 2 H 1 2 ) + N ( 2 H 1 2 + n 2 H 1 2 ) ( 1 + i ) ] + ( i π n n 4 H ) × [ N ( 2 H 1 2 n 2 H 1 2 ) N ( 2 H 1 2 + n 2 H 1 2 ) + 2 N ( n 2 H 1 2 ) ] i π H 1 2 e + i ( π n 2 / 8 H ) [ 1 ( 1 ) n e + i 2 π H ] ) } ,
f n ( ξ ) sin [ n ( π / 2 ) ( ξ + 1 ) ] .
C ( α ) = 1 2 + ( 1 π α 3 π 3 α 5 + ) sin ( π 2 α 2 ) ( 1 π 2 α 3 15 π 4 α 7 + ) cos ( π 2 α 2 )
S ( α ) = 1 2 ( 1 π α 5 π 3 α a + ) cos ( π 2 α 2 ) ( 1 π 2 α 3 15 π 4 α 7 + ) sin ( π 2 α 2 ) ,
C ( α ) = α 1 2 ! 5 ( π 2 ) 2 α 5 + 1 4 ! 9 ( π 2 ) 4 α 9 ,
S ( α ) = 1 1 ! 3 ( π 2 ) α 3 1 3 ! 7 ( π 2 ) 3 α 7 + 1 5 ! 11 ( π 2 ) 11 α 11 .
γ n A n n = e i π ( n 2 / 8 H ) [ ( 1 n 2 12 H H 1 2 π n 4 480 H 2 H 1 2 + ) + i ( n 2 12 H H 1 2 π n 4 480 H 2 H 1 2 + ) ] ,
| γ n | = 1 n 2 12 H H 1 2 ( 1 + π n 2 40 H n 2 24 H H 1 2 + ) ,
arg ( γ n ) = π n 2 8 H ( 1 2 3 π H 1 2 + n 2 60 H H 1 2 n 2 18 π H 2 + ) .
γ n ( 1 n 2 12 H H 1 2 ) e i π ( n 2 / 8 H ) ,
ν n = ν + ( n 2 / 8 H ) ( c / 2 r 1 2 l ) ,
P n = n 2 / 6 H H 1 2 ,
Q n = 2 π P n ( l λ ) = 12 π H H 1 2 n 2 ( l λ ) .
R m ( ρ ) = k = 1 c m , k 2 1 2 J m ( p m k ) J m ( p m k ρ ) ,
J m ( p m k ) = 0 , k = 1 , 2 , 3 , .
γ m k = 1 2 1 2 c m , k J m ( p m k ) J m ( p m k ρ ) = e i ( m + 1 ) ( π / 2 ) ( π H ) 0 1 e + i ( π / 2 ) ( ρ 2 + ρ 1 2 ) J m ( π H ρ ρ 1 ) × ( k = 1 2 1 2 c m , k J m ( p m k ) J m ( p m k ρ ) ) ρ d ρ .
γ m j | c m , j | 2 = j , k ( A m , j k c m , j * c m , k ) ,
A m , j k = e i ( m + 1 ) ( π / 2 ) ( 2 π H J m ( p m k ) J m ( p m j ) ) × 0 1 0 1 e + i ( π / 2 ) H ( ρ 2 + ρ 1 2 ) J m ( π H ρ ρ 1 ) × J m ( p m k ρ ) J m ( p m j ρ 1 ) ρ ρ 1 d ρ d ρ 1 .
k ( A m , j k γ m δ j , k ) c m , k = 0 .
[ A m , j k γ δ j , k ] = 0 .
0 1 0 1 f ( ρ , ρ 1 ) d ρ d ρ 1 = I ( 0 , 0 ) I ( 0 , 1 ) I ( 1 , 0 ) + I ( 1 , 1 ) ,
I ( b , c ) = ρ = b ρ 1 = c f ( ρ , ρ 1 ) d ρ d ρ 1 .
J m ( α ) = ( 2 / π α ) 1 2 cos ( α 1 4 π m 1 2 π ) .
A m , j k = ( i 2 ) 1 2 2 ( p m k 2 p m j 2 ) { p m k exp ( i p m j 2 2 π H ) [ N ( 2 H 1 2 p m j π H 1 2 ) ( 1 + i ) + 2 N ( p m j π H 1 2 ) ] p m j exp ( i p m k 2 2 π H ) [ N ( 2 H 1 2 p m k π H 1 2 ) + N ( 2 H 1 2 + p m k π H 1 2 ) ( 1 + i ) + 2 N ( p m k π H 1 2 ) ] } ,
A m , k k = exp ( i p m k 2 2 π H ) { 1 + 1 i 2 ( [ N ( 2 H 1 2 p m k π H 1 2 ) N ( 2 H 1 2 + p m k π H 1 2 ) ] + ( i 2 p m k p m k 2 π H ) [ N ( 2 H 1 2 p m k π H 1 2 ) + N ( 2 H 1 2 + p m k π H 1 2 ) ( 1 + i ) + 2 N ( p m k π H 1 2 ) ] i π H 1 2 exp ( + i p m k 2 2 π H ) [ 1 e i 2 p m k e + i 2 π H ] ) } ,
γ m n A m , n n = exp ( i p m n 2 2 π H ) { 1 + 1 i 2 ( [ N ( 2 H 1 2 p m n π H 1 2 ) N ( 2 H 1 2 + p m n π H 1 2 ) ] + ( i 2 p m n p m n 2 π H ) × [ N ( 2 H 1 2 p m n π H 1 2 ) + N ( 2 H 1 2 + p m n π H 1 2 ) ( 1 + i ) + 2 N ( p m n π H 1 2 ) ] + 1 π H 1 2 exp ( + i p m n 2 2 π H ) [ 1 e i 2 p m n e + i 2 π H ] ) } ,
f m ( ρ , ϕ ) [ π 1 2 J m ( p m n ) ] 1 J m ( p m n ρ ) e i m ϕ .
p m n ( 1 2 m + n 1 4 ) π .
γ m + 2 k , n γ m , n + k .
γ 2 k , n γ 0 , n + k
γ 2 k + 1 , n γ 1 , n + k
γ m n A m , n n = exp ( i p m n 2 2 π H ) [ ( 1 p 2 3 π 2 H H 1 2 p 4 30 π 3 H 2 H 1 2 + ) + i ( p 2 3 π 2 H H 1 2 p 4 30 π 3 H 2 H 1 2 + ) ] .
γ m n ( 1 ( p m n / π ) 2 3 H H 1 2 ) exp ( i π ( p m n / π ) 2 2 H ) ,
ν m n = ν + ( n + 1 2 m 1 4 ) 2 2 H ( c 2 r 1 2 l ) ,
P m n = ( n + 1 2 m 1 4 ) 2 1.5 H H 1 2 .
Bm,jk=2Jm(pmk)01(πHρ)2Jm1(πHρ)Jm(pmjρ)p2mk(πHρ)2dρ
γmRm(ρ)=πH01Rm(ρ1)Jm(πHρρ1)ρ1dρ1
Ajk=Akj=ei(π/4)212kj12HH12[1iπ(k2+j2)20H+].
Am,jk=ei(π/4)(2)12pjpk3π2HH12[1ipj2+pk25πH+].