Abstract

A simple procedure is described to obtain the modes of propagation in square-law lens-like media. This procedure consists of evaluating the geometrical-optics field created by a point source at the input plane of an optical system (called mode-generating system) with nonuniform losses. An expansion of the field in power series of the coordinates of the point source gives the modes of propagation. In the case of optical resonators, the mode-generating system is described by the modal matrix of the resonator round-trip ray matrix. This representation of modes by point sources allows the coupling factor between two modes with different parameters (beam radii, wave-front curvatures, and axes) to be evaluated without integration. Only matrix algebra is used. In the general three-dimensional case, the coupling factor is expressed as a product of Gauss functions and Hermite polynomials in four complex variables. The quantities introduced are generalized ray invariants.

© 1971 Optical Society of America

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References

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  1. H. Kogelnik, in Quasi-Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1964).
  2. J. A. Arnaud, Bell System Tech. J. 49, 2311 (1970). Note that a factor j is missing in Eqs. (15a), (20), (43) and that the minus sign in front of h in Eqs. (47) and (50) should be deleted.
    [Crossref]
  3. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).
  4. J. H. Van Vleck, Proc. Natl. Acad. Sci. (U.S.) 14, 178 (1928).
    [Crossref]
  5. G. S. S. Avila and J. B. Keller, Commun. Pure Appl. Math. 16, 363 (1963).
    [Crossref]
  6. J. S. Dowker, J. Phys. A.: Gen. Phys. 3, 451 (1970).
    [Crossref]
  7. P. Appel and J. Kampé de Fériet, Fonctions Hypergéomètriques et Hypersphérique/Polynomes d’Hermite (Gauthier-Villars, Paris, 1926).
  8. J. A. Arnaud, Appl. Opt. 8, 1909 (1969).
    [Crossref] [PubMed]
  9. H. Grad, Commun. Pure Appl. Math. 2, 325 (1949).
    [Crossref]

1970 (2)

J. A. Arnaud, Bell System Tech. J. 49, 2311 (1970). Note that a factor j is missing in Eqs. (15a), (20), (43) and that the minus sign in front of h in Eqs. (47) and (50) should be deleted.
[Crossref]

J. S. Dowker, J. Phys. A.: Gen. Phys. 3, 451 (1970).
[Crossref]

1969 (1)

1963 (1)

G. S. S. Avila and J. B. Keller, Commun. Pure Appl. Math. 16, 363 (1963).
[Crossref]

1949 (1)

H. Grad, Commun. Pure Appl. Math. 2, 325 (1949).
[Crossref]

1928 (1)

J. H. Van Vleck, Proc. Natl. Acad. Sci. (U.S.) 14, 178 (1928).
[Crossref]

Appel, P.

P. Appel and J. Kampé de Fériet, Fonctions Hypergéomètriques et Hypersphérique/Polynomes d’Hermite (Gauthier-Villars, Paris, 1926).

Arnaud, J. A.

J. A. Arnaud, Bell System Tech. J. 49, 2311 (1970). Note that a factor j is missing in Eqs. (15a), (20), (43) and that the minus sign in front of h in Eqs. (47) and (50) should be deleted.
[Crossref]

J. A. Arnaud, Appl. Opt. 8, 1909 (1969).
[Crossref] [PubMed]

Avila, G. S. S.

G. S. S. Avila and J. B. Keller, Commun. Pure Appl. Math. 16, 363 (1963).
[Crossref]

Dowker, J. S.

J. S. Dowker, J. Phys. A.: Gen. Phys. 3, 451 (1970).
[Crossref]

Grad, H.

H. Grad, Commun. Pure Appl. Math. 2, 325 (1949).
[Crossref]

Kampé de Fériet, J.

P. Appel and J. Kampé de Fériet, Fonctions Hypergéomètriques et Hypersphérique/Polynomes d’Hermite (Gauthier-Villars, Paris, 1926).

Keller, J. B.

G. S. S. Avila and J. B. Keller, Commun. Pure Appl. Math. 16, 363 (1963).
[Crossref]

Kogelnik, H.

H. Kogelnik, in Quasi-Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1964).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

Van Vleck, J. H.

J. H. Van Vleck, Proc. Natl. Acad. Sci. (U.S.) 14, 178 (1928).
[Crossref]

Appl. Opt. (1)

Bell System Tech. J. (1)

J. A. Arnaud, Bell System Tech. J. 49, 2311 (1970). Note that a factor j is missing in Eqs. (15a), (20), (43) and that the minus sign in front of h in Eqs. (47) and (50) should be deleted.
[Crossref]

Commun. Pure Appl. Math. (2)

H. Grad, Commun. Pure Appl. Math. 2, 325 (1949).
[Crossref]

G. S. S. Avila and J. B. Keller, Commun. Pure Appl. Math. 16, 363 (1963).
[Crossref]

J. Phys. A.: Gen. Phys. (1)

J. S. Dowker, J. Phys. A.: Gen. Phys. 3, 451 (1970).
[Crossref]

Proc. Natl. Acad. Sci. (U.S.) (1)

J. H. Van Vleck, Proc. Natl. Acad. Sci. (U.S.) 14, 178 (1928).
[Crossref]

Other (3)

H. Kogelnik, in Quasi-Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1964).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

P. Appel and J. Kampé de Fériet, Fonctions Hypergéomètriques et Hypersphérique/Polynomes d’Hermite (Gauthier-Villars, Paris, 1926).

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Figures (3)

F. 1
F. 1

In this figure ( S ) denotes an optical system having a ray matrix M s. The modes of propagation in this system are obtained by considering the geometrical-optics field created by a point source S through a mode-generating system ( G ) whose ray matrix M is generally complex.

F. 2
F. 2

This figure represents schematically a ring-type resonator having a round-trip ray matrix M s, defined from a plane B. With respect to the plane B , the round-trip ray matrix is the diagonal matrix M 1 M s M, where M is the modal matrix of the reso nator. The modes of resonance can be viewed as resulting from a multipole expansion of a point source S at B .

F. 3
F. 3

This figure indicates how to evaluate the coupling factor at plane (x) between a mode generated by a point source Sα through the mode-generating system ( G α ), and a mode generated by a point sink Sβ through the mode-generating system ( G β ). It is obtained by evaluating the field at plane (y) created by a point source at plane (y), or vice versa.

Equations (73)

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p = x V ( x , x ) ,
p = x V ( x , x ) ,
E ( x ; x ) = ± j λ 1 | 2 V / x i x j | 1 2 exp ( j k V ) ,
V ( x , x ) = L + 1 2 x Ux + x V x + 1 2 x W x ,
p = Uq + V q
p = V q + W q .
E ( x ; x ) = ± j λ 1 | V | 1 2 × exp [ j k ( L + 1 2 x Ux + x V x + 1 2 x W x ) ] .
E ( x ) = + E ( x ; x ) E ( x ) d x 1 d x 2 .
[ q p ] = [ A B C D ] [ q p ] M [ q p ] .
U = B 1 A ,
V = B 1 ,
V = C D B 1 A ,
W = D B 1 .
B D D B = 0 ,
C A Ã C = 0 ,
D A B C = 1 .
M 1 = [ D B C Ã ] .
Q [ q 1 q 2 ] ,
P [ p 1 p 2 ] ,
[ Q P ] = [ A B C D ] [ Q P ] .
q ¯ α p ¯ β p ¯ α q ¯ β
q ¯ α p β p ¯ α q ¯ β 4 j k 1 ( α ¯ ; β ¯ ) .
Q α P β P α Q β 4 j k 1 ( α ; β ) .
[ q p 1 ] = [ A B a C D c 0 0 1 ] [ q p 1 ] M [ q p 1 ] ,
M 1 = [ D B B c D a C à C a à c 0 0 1 ] .
u = B 1 a
u = c D B 1 a .
( q ¯ α q ¯ γ ) ( p ¯ β p ¯ γ ) ( p ¯ α p ¯ γ ) ( q ¯ β q ¯ γ ) 4 j k 1 ( α ¯ γ ¯ ; β ¯ γ ¯ ) .
Q α ( p ¯ β p ¯ γ ) P α ( q ¯ β q ¯ γ ) 4 j k 1 ( α ; β ¯ γ ¯ ) ,
M 1 2 [ j k Q Q j k P P ] .
QP PQ = 0 ,
Q P P Q = 0 ,
Q P P Q = 4 j k 1 1 .
E ( x ; y ) = ± ( π / 2 ) 1 2 | Q | 1 2 exp ( j k 2 x P Q 1 x ) × exp ( 2 j k Q 1 x 1 2 j k Q 1 Q j k y ) .
E ( x ; 0 ) = ± ( π / 2 ) 1 2 | Q | 1 2 exp ( j k 2 x P Q 1 x )
exp ( x Q 1 Q * 1 x ) ,
E ( x ; y ) = m 1 , m 2 = 0 ( m 1 ! m 2 ! ) 1 2 × ( j k y 1 ) m 1 ( j k y 2 ) m 2 E m 1 m 2 ( x ) ,
E m 1 m 2 ( x ) = ± ( π / 2 ) 1 2 ( m 1 ! m 2 ! ) 1 2 | Q | 1 2 × exp ( j k 2 x P Q 1 x ) ) H e m 1 m 2 ( 2 Q 1 x ; Q 1 Q ) .
M = M s M .
M s [ A B C D ] ,
M = 1 2 [ j k Q Q j k P P ] ,
[ Q P ] = [ A B C D ] [ Q P ] ,
[ Q P ] = [ A B C D ] [ Q P ] .
M P Q 1 = ( C + DM ) ( A + BM ) 1 .
M 1 2 [ j k R 1 j k R 2 R 1 R 2 ] .
± exp ( j k L ) λ 1 m 1 + 1 2 λ 2 m 2 + 1 2 = exp ( 2 j l π ) ,
M s = M 1 M s M = D ,
C α β = + E α ( x , z ) E β ( x , z ) d x 1 d x 2 .
M α = 1 2 [ j k Q α Q α j k P α P α ] ,
M β = 1 2 [ j k Q β Q β j k P β P β ] .
E α m 1 m 2 ( x ) = ( π / 2 ) 1 2 ( m 1 ! m 2 ! ) 1 2 | Q α | 1 2 × exp ( j k 2 x P α Q α 1 x ) × H e m 1 m 2 ( 2 Q α 1 x ; Q α 1 Q α ) .
E β m 1 m 2 ( x ) = ( π / 2 ) 1 2 ( m 1 ! m 2 ! ) 1 2 | Q β | 1 2 × exp ( j k 2 x P β Q β 1 x ) × H e m 1 m 2 ( 2 Q β 1 x ; Q β 1 Q β ) .
M = M β 1 M α = 1 4 [ P β Q β j k P β j k Q β ] [ j k Q α Q α j k P α P α ] = 1 4 [ j k ( P β Q α Q β P α ) P β Q α Q β P α ( j k ) 2 ( P β Q α Q β P α ) j k ( P β Q α Q β P α ) ] = [ ( β ; α ) ( β ; α ) / j k j k ( β ; α ) ( β ; α ) ] .
E ( y ; y ) = | ( α ; β ) | 1 2 exp [ 1 2 j k ( β ; α ) 1 ( β ; α ) j k y + j k ( β ; α ) 1 j k y 1 2 j k ( β ; α ) ( β ; α ) 1 j k y ] ,
C α β m 1 m 2 m 1 m 2 = | ( α ; β ) | 1 2 ( m 1 ! m 2 ! m 1 ! m 2 ! ) 1 2 × H e m 1 m 2 m 1 m 2 ( 0 ; N ) ,
N [ ( β ; α ) 1 ( β ; α ) ( β ; α ) 1 ( α ; β ) 1 ( β ; α ) ( β ; α ) 1 ] .
N = [ 0 1 1 0 ] .
C α β 0 m = 2 m / 2 ( m ! ) 1 2 [ ( m / 2 ) ! ] 1 × ( β * ; α * ) m / 2 ( α ; β * ) ( m + 1 ) / 2 .
( β * ; α * ) ( β ; α ) = ( α ; β * ) ( α * ; β ) + ( α ; α * ) ( β ; β * ) ,
κ 1 ( α * ; β ) ( β * ; α ) = ( w α / w β + w β / w α ) 2 / 4 + k 2 w α 2 w β 2 ( 1 / R α 1 / R β ) 2 / 16 ,
C α β 0 m C α β 0 m * = 2 m m ! [ ( m / 2 ) ! ] 2 κ 1 2 ( 1 κ ) m / 2 ,
M α = [ j k 2 Q α 1 2 Q α q ¯ α j k 2 P α 1 2 P α p ¯ α 0 0 1 ] ,
M β = [ j k 2 Q β 1 2 Q β q ¯ β j k 2 P β 1 2 P β p ¯ β 0 0 1 ] ,
C α β m 1 m 2 m 1 m 2 = | ( α ; β ) | 1 2 exp [ 2 ( β ; α ¯ β ¯ ) ( α ; β ) 1 ( α ; α ¯ β ¯ ) ] × exp [ 1 2 ( α ¯ γ ¯ ; β ¯ γ ¯ ) ] ( m 1 ! m 2 ! m 1 ! m 2 ! ) 1 2 × He m 1 m 2 m 1 m 2 ( 2 N 1 V ; N ) ,
V [ ( β ; α ) 1 ( β ; β ¯ α ¯ ) ( α ; β ) 1 ( α ; β ¯ α ¯ ) ] .
H e ( n ) ( x ; ν ) = ξ n ν ξ n 2 + ν 2 ξ n 4 + { ν n / 2 , n even ( ν ) ( n 1 ) / 2 ξ , n odd .
H e ( n ) ( x ; ν ) H e m ( x ; ν ) = m ! s = 0 [ m / 2 ] ( ν / 2 ) s ξ m 2 s ( m 2 s ) ! s ! = ν m / 2 H e m ( ν 1 2 x ) ,
H e m ( x ) = m ! s = 0 [ m / 2 ] ( 1 2 ) s x m 2 s ( m 2 s ) ! s ! .
H m ( x ) 2 m / 2 H e m ( 2 1 2 x ) .
H e m 1 m 2 ( x ; ν ) = ξ 1 m 1 ξ 2 m 2 ( 1 2 m 1 ( m 1 1 ) 1 ν 11 ξ 1 m 1 2 ξ 2 m 2 + m 1 m 2 1 ν 12 ξ 1 m 1 1 ξ 2 m 2 1 + 1 2 m 2 ( m 2 1 ) 1 ν 22 ξ 1 m 1 ξ 2 m 2 2 ) + + α , β = 0 [ exp ] ( ) γ m 1 ! m 2 ! ν 11 α ν 22 β ν 12 γ α β ξ 1 m 1 γ α + β ξ 2 m 2 γ β + α 2 ( α + β ) α ! β ! ( γ α β ) ! ( m 1 γ α + β ) ! ( m 2 γ + α β ) ! + .
H e m 1 m 2 ( 0 ; ν ) = α = 0 θ ( ) n / 2 m 1 ! m 2 ! ν 11 α ν 22 α θ ν 12 m 1 2 α 2 2 α θ α ! ( α θ ) ! ( m 1 2 α ) ! ,
H e m 0 ( x ; ν ) = ν 11 m / 2 H e m ( ν 11 1 2 ξ 1 ) ,
H e m 0 ( 0 ; ν ) = m ! ( ν 11 / 2 ) m / 2 ( m / 2 ) ! .