Abstract

A new family of exact solutions of the scalar Helmholtz equation is presented. The 0, 0 order of this family represents a new mathematical model for the fundamental mode of a propagating Gaussian beam. The family consists of nonseparable functions in the oblate spheroidal coordinate system and can easily by transformed into a different set of solutions in the prolate spheroidal coordinate system, where the 0, 0 order is a spherical wave. This transformation consists of two substitutions in the coordinate system parameters and represents a more general method of obtaining a Gaussian beam from a spherical wave than assuming a complex point source on axis. Finally, each higher-order member of the family of solutions possesses an amplitude consisting of a finite number of higher-order terms with a zero-order term that is Gaussian.

© 1988 Optical Society of America

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  1. G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 648–685 (1971).
    [CrossRef]
  2. P. D. Einziger, S. Raz, “Wave solutions under complex space-time shifts,” J. Opt. Soc. Am. A 4, 3–10 (1987). However, Eq. (21) in that paper contains an error. The sign of the term, (ω0/ν)a cos η, in the last line of the equation should be positive. Careful insertion of Eq. (20) into Eq. (18) will reveal this to be true. Furthermore, it is obvious that if the magnitude of u(τ, r, r′) is to have a maximum on axis, η= 0, the argument of the exponential amplitude must be zero.
    [CrossRef]
  3. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Palo Alto, Calif., 1957).
  4. L. A. Vainshtein, “Open resonators with spherical mirrors,” Sov. Phy. JETP 18, 471–479 (1964).
  5. M. Ito, “Theory of ellipsoidal waves and seidel aberrations of Gaussian beams,” Jpn. J. Appl Phys. 12, 856–875 (1973).
    [CrossRef]
  6. M. Ito, “Common aspects of Gaussian beams, ellipsoidal waves, and multipole radiation,” Jpn. J. Appl. Phys. 12, 1914–1925 (1973).
    [CrossRef]
  7. J. A. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. 4, 538–543 (1985).
    [CrossRef]
  8. B. Tehan Landesman, “Geometrical representation of the fundamental mode of a Gaussian beam in oblate spheroidal coordinates,” submitted to J. Opt. Soc. Am. A.
  9. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
  10. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
  11. G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
  12. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]

1987 (1)

1985 (1)

J. A. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. 4, 538–543 (1985).
[CrossRef]

1973 (2)

M. Ito, “Theory of ellipsoidal waves and seidel aberrations of Gaussian beams,” Jpn. J. Appl Phys. 12, 856–875 (1973).
[CrossRef]

M. Ito, “Common aspects of Gaussian beams, ellipsoidal waves, and multipole radiation,” Jpn. J. Appl. Phys. 12, 1914–1925 (1973).
[CrossRef]

1971 (1)

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 648–685 (1971).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1964 (1)

L. A. Vainshtein, “Open resonators with spherical mirrors,” Sov. Phy. JETP 18, 471–479 (1964).

1962 (1)

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).

1961 (2)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Arnaud, J. A.

J. A. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. 4, 538–543 (1985).
[CrossRef]

Boyd, G. D.

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Deschamps, G. A.

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 648–685 (1971).
[CrossRef]

Einziger, P. D.

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Palo Alto, Calif., 1957).

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Ito, M.

M. Ito, “Theory of ellipsoidal waves and seidel aberrations of Gaussian beams,” Jpn. J. Appl Phys. 12, 856–875 (1973).
[CrossRef]

M. Ito, “Common aspects of Gaussian beams, ellipsoidal waves, and multipole radiation,” Jpn. J. Appl. Phys. 12, 1914–1925 (1973).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Raz, S.

Tehan Landesman, B.

B. Tehan Landesman, “Geometrical representation of the fundamental mode of a Gaussian beam in oblate spheroidal coordinates,” submitted to J. Opt. Soc. Am. A.

Vainshtein, L. A.

L. A. Vainshtein, “Open resonators with spherical mirrors,” Sov. Phy. JETP 18, 471–479 (1964).

Appl. Opt. (1)

J. A. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. 4, 538–543 (1985).
[CrossRef]

Bell Syst. Tech. J. (3)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 648–685 (1971).
[CrossRef]

J. Opt. Soc. Am. A (1)

Jpn. J. Appl Phys. (1)

M. Ito, “Theory of ellipsoidal waves and seidel aberrations of Gaussian beams,” Jpn. J. Appl Phys. 12, 856–875 (1973).
[CrossRef]

Jpn. J. Appl. Phys. (1)

M. Ito, “Common aspects of Gaussian beams, ellipsoidal waves, and multipole radiation,” Jpn. J. Appl. Phys. 12, 1914–1925 (1973).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Sov. Phy. JETP (1)

L. A. Vainshtein, “Open resonators with spherical mirrors,” Sov. Phy. JETP 18, 471–479 (1964).

Other (2)

B. Tehan Landesman, “Geometrical representation of the fundamental mode of a Gaussian beam in oblate spheroidal coordinates,” submitted to J. Opt. Soc. Am. A.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Palo Alto, Calif., 1957).

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Figures (3)

Fig. 1
Fig. 1

(a) Prolate spheroidal coordinate system, (b) oblate spheroidal coordinate system.

Fig. 2
Fig. 2

(a) Focal points for the prolate spheroidal coordinate system, (b) focal ring for the oblate spheroidal coordinate system.

Fig. 3
Fig. 3

Geometry for a polar coordinate system whose origin has been shifted to one of the foci of the prolate spheroidal coordinate system.

Tables (3)

Tables Icon

Table 1 Legendre Polynomials

Tables Icon

Table 2 Associated Legendre Functions

Tables Icon

Table 3 Higher-Order Values of the Function ekdhn(1)[kd(ξ − iη)]

Equations (84)

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r + = [ ( z + d ) 2 + x 2 + y 2 ] 1 / 2 , r = [ ( z d ) 2 + x 2 + y 2 ] 1 / 2 .
r + = d ( 1 + cosh 2 μ cos 2 θ + 2 cosh μ cos θ + sinh 2 μ sin 2 θ ) 1 / 2 , r = d ( 1 + cosh 2 μ cos 2 θ 2 cosh μ cos θ + sinh 2 μ sin 2 θ ) 1 / 2 .
cosh 2 μ sinh 2 μ = 1 ,
cos 2 θ + sin 2 ϕ = 1 ,
r + = d ( cosh μ + cos θ ) = d ( ξ + η ) , r = d ( cosh μ cos θ ) = d ( ξ η ) .
r + = [ ( ρ + d ) 2 + z 2 ] 1 / 2 , r = [ ( ρ d ) 2 + z 2 ] 1 / 2 ,
r + = d [ ( 1 + ξ 2 ) 1 / 2 + ( 1 η 2 ) 1 / 2 ] , r = d [ ( 1 + ξ 2 ) 1 / 2 ( 1 η 2 ) 1 / 2 ] .
( 2 + k 2 ) ψ = 0
[ ξ ( ξ 2 1 ) ξ + η ( 1 η 2 ) η + ξ 2 η 2 ( ξ 2 1 ) ( 1 η 2 ) 2 ϕ 2 + k 2 d 2 ( ξ 2 η 2 ) ] ψ = 0.
ξ P / S ± i ξ O / S , i k d P / S ± k d O / S
[ ξ ( ξ 2 + 1 ) ξ + η ( 1 η 2 ) η + ξ 2 + η 2 ( ξ 2 + 1 ) ( 1 η 2 ) 2 ϕ 2 + k 2 d 2 ( ξ 2 + η 2 ) ] ψ = 0.
ψ P / S = exp ( i k d ) exp ( i k r + ) i k r + ,
exp ( i k d ) exp [ i k d ( η + ξ ) ] i k d ( η + ξ ) exp ( k d ) exp [ k d ( η + ξ ) ] k d ( η + i ξ ) .
( 2 + k 2 ) ψ P / S ( 2 + k 2 ) ψ O / S , ξ P / S ± i ξ O / S , i k d P / S ± k d O / S ,
ψ O / S = exp ( i k d ξ ) exp [ k d ( 1 η ) ] k d ( η 2 + ξ 2 ) 1 / 2 exp [ i tan 1 ξ / η ] ,
x = d O / S ( 1 + ξ O / S 2 ) 1 / 2 ( 1 η 2 ) 1 / 2 cos ϕ , y = d O / S ( 1 + ξ O / S 2 ) 1 / 2 ( 1 η 2 ) 1 / 2 sin ϕ , z = d O / S ξ O / S η i d O / S .
ξ ( O / S ) i ξ P / S , i d ( O / S ) d P / S ,
x = d P / S ( ξ P / S 2 1 ) 1 / 2 ( 1 η 2 ) 1 / 2 cos ϕ , x = d P / S ( ξ P / S 2 1 ) 1 / 2 ( 1 η 2 ) 1 / 2 sin ϕ , z = d P / S ( ξ P / S η 1 ) .
u ( x , y , z ) = w 0 w ( z ) exp { i [ k z Φ + k ( x 2 + y 2 ) 2 R ( z ) ] x 2 + y 2 w 2 ( z ) } ,
Φ = tan 1 z z 0 ,
R ( z ) = z 2 + z 0 2 z ,
w ( z ) = w 0 [ 1 + ( z z 0 ) 2 ] 1 / 2 .
z 0 = k w 0 2 / 2 ,
u ( x , y , z ) = ψ ( x , y , z ) exp ( i k z ) ,
ψ ( x , y , z ) = exp [ i ( P + k ρ 2 2 q ) ] ,
w 0 2 = x 2 + y 2 = d 2 cosh 2 μ sin 2 δ = d 2 sin 2 δ .
w 2 ( z ) w 0 2 = d 2 cosh 2 μ sin 2 δ d 2 sin 2 δ = 1 + z 2 z 0 2 .
cosh 2 μ = 1 + z 2 z 0 2 .
sinh μ = z / z 0 .
Φ = tan 1 ξ .
sin 2 θ = x 2 + y 2 d 2 cosh 2 μ .
k d ( 1 cos θ ) = k d [ 1 ( 1 x 2 + y 2 d 2 cosh 2 μ ) 1 / 2 ] .
k d ( 1 cos θ ) k d [ 1 ( 1 x 2 + y 2 2 d 2 cosh 2 μ ) ] k ( x 2 + y 2 ) 2 d 2 cosh 2 μ .
1 / w 0 2 = k / 2 d .
k d ( 1 cos θ ) x 2 + y 2 w 0 2 ( z ) ( 1 + z 2 / z 0 2 ) = ( x 2 + y 2 ) w 2 ( z ) .
( η 2 + ξ 2 ) 1 / 2 ( 1 + ξ 2 ) 1 / 2 = cosh 1 μ .
lim η 1 ψ O / S = exp ( i k d ξ ) exp ( i tan 1 ξ ) exp [ k ( x 2 + y 2 ) 2 d ( 1 + ξ 2 ) ] k d ( 1 + ξ 2 ) 1 / 2 .
r 2 2 R ( r ) r 2 + 2 r R ( r ) r + [ k 2 r 2 n ( n + 1 ) ] R ( r ) = 0.
1 sin θ θ [ sin θ Ө ( θ ) ] + [ n ( n + 1 ) m 2 sin 2 θ ] Ө ( θ ) = 0 ,
Ө ( θ ) = P n m ( cos θ ) .
1 Φ ( ϕ ) 2 Φ ( ϕ ) ϕ 2 = m 2 ,
Φ ( ϕ ) = exp ( ± i m ϕ ) .
ψ ( r , θ , ϕ ) = R ( r ) Ө ( θ ) Φ ( ϕ ) = h n ( 1 ) ( r ) P n m ( cos θ ) exp ( ± i m ϕ ) ,
ψ P / S = exp ( i k d ) h n ( 1 ) [ k d ( ξ + η ) ] ,
ψ P / S = exp ( i k d ) h n ( 2 ) [ k d ( ξ + η ) ] ,
ψ m n ( ξ , η , ϕ ) P / S = exp ( i k d ) h n ( 1 ) [ k d ( ξ + η ) ] P n m ( cos θ ) exp ( i m ϕ ) ,
cos θ = z + d r + = 1 + ξ η ξ + η .
ξ P / S ± i ξ O / S k d P / S i d k O / S r = k d P / S ( ξ P / S + η ) t = i k d O / S ( η + i ξ O / S ) = k d O / S [ ξ O / S i η ] ,
cos θ = 1 + ξ P / S η ξ P / S + η s = 1 + i ξ O / S η η + i ξ O / S .
ψ m n ( ξ , η , ϕ ) O / S = e k d h n ( 1 ) ( t ) P n m ( s ) exp ( i m ϕ ) ,
P n ( m ) ( s ) = 1 2 n n ! ( 1 s 2 ) m / 2 d m + n d s m + n ( s 2 1 ) n
e k d h n ( 1 ) ( t ) = e k d ( i ) n e i t i t j = 0 n ( 1 i t ) j ( n + j ) ! 2 j j ! ( n j ) ! .
s 2 1 = ( 1 η 2 ) ( 1 + ξ 2 ) ( η + i ξ ) 2 .
P n m ( s ) = i m 2 n n ! ( 1 η 2 ) m / 2 ( ξ 2 + 1 ) m / 2 ( η 2 + ξ 2 ) 3 / 2 × exp ( i m tan 1 ξ η ) d m + n d s m + n [ ( 1 η 2 ) ( ξ 2 + 1 ) ( η + i ξ ) 2 ] n ,
h n ( 1 ) ( t ) = ( i ) n exp [ k d ( 1 η ) exp ( i k d ξ ) exp ( i tan 1 ξ / η ) ] k d ( η 2 + ξ 2 ) 1 / 2 × j = 0 n exp ( i j tan 1 ξ / η ) [ k d ( η 2 + ξ 2 ) 1 / 2 ] j ( n + j ) ! 2 j j ! ( n j ) ! .
x = d sinh μ sin θ cos ϕ , y = d sinh μ sin θ sin ϕ , z = d cosh μ cos θ ,
0 θ π , 0 μ < , 0 ϕ 2 π .
x = d ( ξ 2 1 ) 1 / 2 ( 1 η 2 ) 1 / 2 cos ϕ , y = d ( ξ 2 1 ) 1 / 2 ( 1 η 2 ) 1 / 2 sin ϕ , z = d ξ η ,
1 η 1 , 1 ξ < , 0 ϕ 2 π .
x = d cosh μ sin θ cos ϕ , y = d cosh μ sin θ sin ϕ , z = d sinh μ cos θ ,
0 θ π , 0 μ < , 0 ϕ 2 π
0 θ 2 π , < μ < , 0 ϕ 2 π .
x = d ( 1 + ξ 2 ) 1 / 2 ( 1 η 2 ) 1 / 2 cos ϕ , y = d ( 1 + ξ 2 ) 1 / 2 ( 1 η 2 ) 1 / 2 sin ϕ , z = d ξ η ,
1 η 1 , 0 ξ < , 0 ϕ 2 π
0 η 1 , < ξ < , 0 ϕ 2 π .
( 2 + k 2 ) ψ O / S = ξ ( ξ 2 + 1 ) ξ ψ O / S + η ( 1 η 2 ) η ψ O / S + ξ 2 + η 2 ( 1 + ξ 2 ) ( 1 η 2 ) 2 ψ O / S ϕ 2 + k 2 d 2 ( ξ 2 + η 2 ) ψ O / S = 0.
t ξ = k d , t η = i k d
s ξ = i ( 1 η 2 ) ( η + i ξ ) 2 , s η = ξ 2 + 1 ( η + i ξ ) 2 .
ξ ψ O / S = exp ( i m ϕ ) exp ( k d ) { d d t [ h n ( 1 ) ( t ) ] [ P n m ( s ) ] t ξ + h n ( 1 ) ( t ) d d s [ P n m ( s ) ] s ξ } , η ψ O / S = exp ( i m ϕ ) exp ( k d ) { d d t [ h n ( 1 ) ( t ) ] P n m ( s ) t η + h n ( 1 ) ( t ) d d s [ P n m ( s ) ] s η } .
ξ ( ξ 2 + 1 ) ξ ψ O / S = 2 ξ [ h h k d P P i ( 1 η 2 ) ( η + i ξ ) 2 ] ψ O / S + ( 1 + ξ 2 ) { ( h h ) 2 k 2 d 2 ( P P ) 2 ( 1 η 2 ) 2 ( η + i ξ ) 4 2 i k d h h P P ( 1 η 2 ) ( η + i ξ ) 2 + k 2 d 2 [ h h ( h h ) 2 ] + ( 1 η ) 2 ( η i ξ ) 4 [ P P ( P P ) 2 ] } ψ O / S ,
η ( 1 η 2 ) η ψ O / S = 2 η [ i k d h h P P ξ 2 + 1 ( η + i ξ ) 2 ] ψ O / S + ( 1 η 2 ) { k 2 d 2 ( h h ) 2 + ( P P ) 2 ( ξ 2 + 1 ) 2 ( η + i ξ ) 4 + 2 i k d h h P P ξ 2 + 1 ( η + i ξ ) 2 k 2 d 2 [ h h ( h h ) 2 ] + ( ξ 2 + 1 ) 2 ( η + i ξ ) 4 [ P P ( P P ) 2 ] } ψ O / S ,
ξ 2 + η 2 ( ξ 2 + 1 ) ( 1 η 2 ) 2 ψ O / S ϕ 2 = m 2 ( ξ 2 + η 2 ) ( ξ 2 + 1 ) ( 1 η 2 ) ψ O / S .
k 2 d 2 ( η + i ξ ) 2 ( 1 + h h ) ψ O / S + 2 i k d ( η + i ξ ) h h ψ O / S + P P ( 1 η 2 ) ( ξ 2 + 1 ) ( η + i ξ ) 2 ψ O / S + 2 P P 1 + i ξ η η + i ξ ψ O / S m 2 ( η + i ξ ) 2 ( 1 η 2 ) ( ξ 2 + 1 ) ψ O / S = 0.
[ t 2 ( 1 + h h ) 2 t h h ] ψ O / S + [ ( 1 s 2 ) P P + 2 s P P + m 2 1 s 2 ] ψ O / S = 0.
t 2 h + 2 t h + [ t 2 n ( n + 1 ) ] h = 0 ,
( 1 s 2 ) P 2 s P + [ n ( n + 1 ) m 2 1 s 2 ] P = 0.
P n m ( s ) = 1 2 n n ! ( 1 s 2 ) m / 2 d m + n d s m + n ( s 2 1 ) n ,
P n o ( s ) = P n ( s ) = 1 2 n n ! d n d s n ( s 2 1 ) n ,
s = 1 + i ξ η n + i ξ = [ 1 + ( ξ η ) 2 η 2 + ξ 2 ] 1 / 2 exp ( i tan 1 ξ η ) exp ( i tan 1 ξ / η )
s 2 1 = ( 1 η 2 ) ( ξ 2 + 1 ) ( η + i ξ ) 2 = ( 1 η 2 ) ( ξ 2 + 1 ) η 2 + ξ 2 exp ( i 2 tan ξ / η ) .
h n ( 1 ) ( t ) = ( i ) n e i t i t j = 0 n ( 1 i t ) j ( n + j ) ! 2 j j ! ( n j ) ! .
h n ( 2 ) ( t ) = ( i ) n e i t i t j = 0 n ( 1 i t ) j ( n + j ) ! 2 j j ! ( n j ) ! .
i t = k d ( η + i ξ )
t = k d ( ξ i η ) .

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