Abstract

A simple expression is given for the response of degenerate cavities suffering from arbitrary misalignments, and numerical results are presented. The method of resonance excitation is carried out analytically with the help of a complex ray representation of gaussian beams. It is first shown that the modulus and phase of such complex rays can be identified with, respectively, the beam radius and the phase of the on-axis field. This identification simplifies the calculation of the coupling factor between two gaussian beams, which is needed in deriving the expression for the response. For the case of conventional cavities, the results are in exact agreement with results derived from the Laguerre-Gauss or Hermite-Gauss mode theory. The case of degenerate cavities with large and possibly nonorthogonal misalignments, of interest in various nonresonant multipath systems, is also discussed.

© 1969 Optical Society of America

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  1. J. A. Arnaud, Appl. Opt. 8, 189 (1969).
    [CrossRef] [PubMed]
  2. D. R. Herriott, H. J. Schulte, Appl. Opt. 4, 883 (1965).
    [CrossRef]
  3. H. Kogelnik, T. J. Bridges, IEEE J. Quantum Electron. QE-3, 95 (1967).
    [CrossRef]
  4. A. G. Fox, T. Li, IEEE J. Quantum Electron, QE-4, 460 (1968).
    [CrossRef]
  5. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  6. H. Kogelnik, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).
  7. D. Gloge, Arch. Elek. Ueberti 20, 82 (1966).
  8. J. T. Verdeyen, J. B. Gerards, Appl. Opt. 7, 1467 (1968).The expression of the coupling coefficient derived in Appendix A of this paper can also be found, in a more general form, in Ref. 7.
    [CrossRef] [PubMed]
  9. J. A. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
    [CrossRef] [PubMed]
  10. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Los Angeles, 1964).
  11. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. I.E.E.E. 53, 129 (1965);E. A. J. Marcatili, Bell Syst. Tech. J. 46, 1733 (1967).
    [CrossRef]
  12. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  13. V. P. Bykov, L. A. Vainshtein, Zh. Eksperim. Teor. Fiz. 47, 508 (1964);W. K. Kahn, Appl. Opt. 4, 758 (1965).
    [CrossRef]
  14. W. H. Steier, Appl. Opt. 5, 1229 (1966).
    [CrossRef] [PubMed]
  15. S. A. Collins, Appl. Opt. 3, 1263 (1964);T. Li, Appl. Opt. 3, 1315 (1964);J. P. Gordon, Bell Syst. Tech. J. 43, 1826 (1964);T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).
    [CrossRef]
  16. G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964);P. Laures, Appl. Opt. 6, 747 (1967).
    [CrossRef] [PubMed]
  17. R. A. Frazer, W. J. Duncan, A. R. Collar, Elementary Matrices (Cambridge University Press, New York, 1957).
  18. J. A. Arnaud, I.E.E.E. J. Quantum Electron QE-4, 893 (1968).
    [CrossRef]
  19. R. V. Pole, J. Opt. Soc. Amer. 55, 254 (1965);see also D. R. Herriott, J. Opt. Soc. Amer. 56, 719 (1966).
    [CrossRef]
  20. J. A. Arnaud, Bell Telephone Labs.; private communicatiod

1969 (2)

1968 (3)

1967 (1)

H. Kogelnik, T. J. Bridges, IEEE J. Quantum Electron. QE-3, 95 (1967).
[CrossRef]

1966 (3)

1965 (4)

R. V. Pole, J. Opt. Soc. Amer. 55, 254 (1965);see also D. R. Herriott, J. Opt. Soc. Amer. 56, 719 (1966).
[CrossRef]

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. I.E.E.E. 53, 129 (1965);E. A. J. Marcatili, Bell Syst. Tech. J. 46, 1733 (1967).
[CrossRef]

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

D. R. Herriott, H. J. Schulte, Appl. Opt. 4, 883 (1965).
[CrossRef]

1964 (2)

Arnaud, J. A.

J. A. Arnaud, Appl. Opt. 8, 189 (1969).
[CrossRef] [PubMed]

J. A. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
[CrossRef] [PubMed]

J. A. Arnaud, I.E.E.E. J. Quantum Electron QE-4, 893 (1968).
[CrossRef]

J. A. Arnaud, Bell Telephone Labs.; private communicatiod

Bridges, T. J.

H. Kogelnik, T. J. Bridges, IEEE J. Quantum Electron. QE-3, 95 (1967).
[CrossRef]

Bykov, V. P.

V. P. Bykov, L. A. Vainshtein, Zh. Eksperim. Teor. Fiz. 47, 508 (1964);W. K. Kahn, Appl. Opt. 4, 758 (1965).
[CrossRef]

Collar, A. R.

R. A. Frazer, W. J. Duncan, A. R. Collar, Elementary Matrices (Cambridge University Press, New York, 1957).

Collins, S. A.

Deschamps, G. A.

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964);P. Laures, Appl. Opt. 6, 747 (1967).
[CrossRef] [PubMed]

Duncan, W. J.

R. A. Frazer, W. J. Duncan, A. R. Collar, Elementary Matrices (Cambridge University Press, New York, 1957).

Fox, A. G.

A. G. Fox, T. Li, IEEE J. Quantum Electron, QE-4, 460 (1968).
[CrossRef]

Frazer, R. A.

R. A. Frazer, W. J. Duncan, A. R. Collar, Elementary Matrices (Cambridge University Press, New York, 1957).

Gerards, J. B.

Gloge, D.

D. Gloge, Arch. Elek. Ueberti 20, 82 (1966).

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. I.E.E.E. 53, 129 (1965);E. A. J. Marcatili, Bell Syst. Tech. J. 46, 1733 (1967).
[CrossRef]

Herriott, D. R.

Kogelnik, H.

J. A. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
[CrossRef] [PubMed]

H. Kogelnik, T. J. Bridges, IEEE J. Quantum Electron. QE-3, 95 (1967).
[CrossRef]

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

H. Kogelnik, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).

Li, T.

A. G. Fox, T. Li, IEEE J. Quantum Electron, QE-4, 460 (1968).
[CrossRef]

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

Luneburg, R. K.

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

Mast, P. E.

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964);P. Laures, Appl. Opt. 6, 747 (1967).
[CrossRef] [PubMed]

Pole, R. V.

R. V. Pole, J. Opt. Soc. Amer. 55, 254 (1965);see also D. R. Herriott, J. Opt. Soc. Amer. 56, 719 (1966).
[CrossRef]

Schulte, H. J.

Steier, W. H.

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. I.E.E.E. 53, 129 (1965);E. A. J. Marcatili, Bell Syst. Tech. J. 46, 1733 (1967).
[CrossRef]

Vainshtein, L. A.

V. P. Bykov, L. A. Vainshtein, Zh. Eksperim. Teor. Fiz. 47, 508 (1964);W. K. Kahn, Appl. Opt. 4, 758 (1965).
[CrossRef]

Verdeyen, J. T.

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. I.E.E.E. 53, 129 (1965);E. A. J. Marcatili, Bell Syst. Tech. J. 46, 1733 (1967).
[CrossRef]

Appl. Opt. (8)

Arch. Elek. Ueberti (1)

D. Gloge, Arch. Elek. Ueberti 20, 82 (1966).

I.E.E.E. J. Quantum Electron (1)

J. A. Arnaud, I.E.E.E. J. Quantum Electron QE-4, 893 (1968).
[CrossRef]

IEEE J. Quantum Electron (1)

A. G. Fox, T. Li, IEEE J. Quantum Electron, QE-4, 460 (1968).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. Kogelnik, T. J. Bridges, IEEE J. Quantum Electron. QE-3, 95 (1967).
[CrossRef]

J. Opt. Soc. Amer. (1)

R. V. Pole, J. Opt. Soc. Amer. 55, 254 (1965);see also D. R. Herriott, J. Opt. Soc. Amer. 56, 719 (1966).
[CrossRef]

Proc. I.E.E.E. (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. I.E.E.E. 53, 129 (1965);E. A. J. Marcatili, Bell Syst. Tech. J. 46, 1733 (1967).
[CrossRef]

Zh. Eksperim. Teor. Fiz. (1)

V. P. Bykov, L. A. Vainshtein, Zh. Eksperim. Teor. Fiz. 47, 508 (1964);W. K. Kahn, Appl. Opt. 4, 758 (1965).
[CrossRef]

Other (5)

J. A. Arnaud, Bell Telephone Labs.; private communicatiod

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964);P. Laures, Appl. Opt. 6, 747 (1967).
[CrossRef] [PubMed]

R. A. Frazer, W. J. Duncan, A. R. Collar, Elementary Matrices (Cambridge University Press, New York, 1957).

H. Kogelnik, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964).

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

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Figures (6)

Fig. 1
Fig. 1

This figure represents a fundamental gaussian beam as it propagates in free space. Its surface (hyperboloid of revolution) can be generated by a skew ray, represented by a plain line (or by its symmetric, represented by a dotted line), as it rotates about the axis. The bissectrix of the two skew rays intersects the axis at the wavefront center C.

Fig. 2
Fig. 2

x ¯ a and x ¯ b represent the axes of two ray pencils whose centers are Ca and Cb, respectively. χ is the ray which belongs to the two ray pencils. The five circles drawn in this figure for the construction of Δ1 and 1 2 ( x ¯ a , x ¯ b ) are normally intersecting the rays and the z axis; they can be considered as wave surfaces. The relation between the phase of the coupling factor and these quantities is outlined in the text.

Fig. 3
Fig. 3

This figure schematically represents a ring type cavity with an input–output mirror (M) of transmissivity t. (P) is an arbitrary reference plane in the ring path.

Fig. 4
Fig. 4

Response of a degenerate cavity as a function of frequency for various degrees of misalignments, expressed by the parameter m (m = 0 to 4), for a round trip loss equal to 0.9 dB. The circles correspond to the small losses–small misalignment approximation.

Fig. 5
Fig. 5

This figure is a continuation of Fig. 4 for the case of large misalignments (m = 20 to ∞), with different scales. The circles correspond to the small losses–small misalignment approximation.

Fig. 6
Fig. 6

A ring type cavity, degenerate in the ring plane, is represented in (a). Its theoretical response and bandwidth are shown in (b) as functions of the misalignment parameter 2δ/w, where δ is the lens offset and w the input beam waist radius, for a cavity round trip loss equal to 4.5 dB, and a round trip path length equal to 960 mm.

Equations (64)

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N ( x , y , z ) = n + n x n x + n y n y + n x x n x 2 + n y y n y 2 + n x y n x y ,
2 E x 2 + 2 E y 2 + 2 E z 2 + k 2 ( n 2 + 2 n x x + 2 n y y + 2 n x x x 2 + 2 n y y y 2 + 2 n x y x y ) E = 0 ,
Ψ = n 1 2 Ψ ,
Ψ = E exp ( j k 0 z n d z ) ,
z = 0 z d z / n ,
2 Ψ x 2 + 2 Ψ y 2 2 j k Ψ z + 2 k 2 ( n x x + n y y + n x x x 2 + n y y y 2 + n x y x y ˙ ) Ψ = 0 .
x ¯ ¨ = n x + 2 n x x x ¯ + n x y y ¯ ,
y ¯ ¨ = n y + 2 n y y y ¯ + n x y x ¯ ,
Ψ ( x , y , z ) = ψ ( x x ¯ , y y ¯ , z ) × exp [ j k ( x x ¯ ˙ + y y ¯ ˙ x ¯ x ¯ ˙ 2 y ¯ y ¯ ˙ 2 + 1 2 0 x z n x x ¯ d z + 1 2 0 y z n y y ¯ d z ) ] ,
ζ ˙ = N ( x ¯ , y ¯ , z ) d S / d z N ( 0 , 0 , z ) ( 1 + n x x ¯ + n x x x ¯ 2 + n y y ¯ + n y y y ¯ 2 + n x y x ¯ y ¯ ) ( 1 + x ¯ ˙ 2 2 + y ¯ ˙ 2 2 ) 1 1 2 d d z ( x ¯ x ¯ ˙ + y ¯ y ¯ ˙ ) + 1 2 ( n x x ¯ + n y y ¯ ) .
ψ ( x , y , z ) = [ X 1 2 exp ( j k 2 X x 2 ) ] [ Y 1 2 exp ( j k 2 Y ˙ Y y 2 ) ] ,
Θ phase of ψ ( 0 , 0 , z ) = 1 2 [ phase of X + phase of Y ] .
θ = cos 1 ( A + D ) / 2 ;
j k 2 X x 2 j k 2 x 2 R ( x W ) 2 ,
1 W 2 = imaginary part of k 2 X = k 2 ξ η ˙ η ξ ˙ X X * ,
1 R = real part of X = 1 2 d d z log ( X X * ) .
X X * = W 2 ,
R = W / .
X = W e j Θ ,
= W e j Θ [ ( 1 / R ) j ( 2 / k W 2 ) ] ,
( X * X * ) = 4 j / k .
X ( z ) = w j ( 2 / k w ) z .
C a b = + Ψ a Ψ b * d x d y .
C a b x = ( 4 j k ) 1 2 ( X a , X b * ) 1 2 exp [ j k 2 ( x ¯ a x ¯ b , X a ) ( x ¯ a x ¯ b , X b * ) ( X a , X b * ) ] × exp { j k 2 [ ( x ¯ a , x ¯ b ) 0 a z n x x ¯ a d z + 0 b z n x x ¯ b d z ] }
| C a b x | X a 1 2 X b 1 2 ( a X a b X b ) 1 2 = ( X a , X b ) 1 2 ,
Δ T = C a C b ¯ + C b 0 b ¯ + 0 b 0 a ¯ + 0 a C a ¯ ,
Δ T = ( C a C b ¯ + C b α b ¯ + + α a C a ¯ ) + ( 0 a α a ¯ + α b 0 b ¯ + 0 b 0 a ¯ ) Δ 1 + Δ 2 ,
( x ¯ a χ , X a ) = ( x ¯ b χ , X b ) = 0 .
Δ 2 = ζ α a ζ α b = 1 2 [ x ¯ a x ¯ ˙ a + 0 x a z n x x ¯ a d z x ¯ b x ¯ ˙ b 0 x b z n x x ¯ b d z ( x ¯ a x ¯ b ) ( x ¯ ˙ a + x ¯ ˙ b ) ] = 1 2 [ ( x ¯ a , x ¯ b ) 0 x a z n x x ¯ a d z + 0 x b z n x x ¯ b d z ] .
P = + r = 0 E r ( x , y ) r s = 0 E s * ( x , y ) s d x d y ,
P = r = 0 s = 0 r + s C r s ,
C r s = C 0 s r if r s , C r s = C s r * = C 0 s r * if r s ,
T = 1 + 2 real part of r = 1 r C o r .
[ x ¯ r x ¯ ˙ r 1 ] = [ A B δ C D δ ˙ 0 0 1 ] r [ x ¯ 0 x ¯ ˙ 0 1 ] .
[ X r r ] = [ A B C D ] r [ X 0 0 ] ,
[ X r r ] = { cos r θ [ 1 0 0 1 ] + sin r θ sin θ [ ( A D ) / 2 B C ( A D ) / 2 ] } [ X 0 0 ] ,
l r = r l 1 + 1 2 ρ = 1 r ( x ¯ ρ x ¯ 0 , x ¯ 1 x ¯ 0 ) .
l 1 l 1 1 2 ( x ¯ 0 x ¯ 1 , x ¯ 0 x ¯ 1 ) .
T = 1 + 2 real part of r = 1 r × ( 4 j k ) ( X 0 , X r * ) 1 exp [ j k 2 ( x ¯ 0 x ¯ r , X 0 ) ( x ¯ 0 x ¯ r , X r * ) ( X 0 , X r * ) ] × exp [ j k r l 1 j k 2 ρ = 1 r ( x ¯ ρ x ¯ 0 , x ¯ 1 x ¯ 0 ) ] ,
x ¯ r = [ 1 + r ( A 1 ) ] x ¯ 0 + r B x ¯ ˙ 0 + r ( r 1 ) 2 ( A δ + B δ ˙ ) + r ( 3 r ) 2 δ ,
x ¯ ˙ r = r C x ¯ 0 + [ 1 r ( D 1 ) ] x ¯ ˙ 0 + r ( r 1 ) 2 ( C δ + D δ ˙ ) + r ( 3 r ) 2 δ ˙ ;
X r = [ 1 + r ( A 1 ) ] X 0 + r B 0 ,
r = r C X 0 + [ 1 + r ( D 1 ) ] 0 .
x ¯ r = x ¯ 0 + r δ ,
x ¯ ˙ r = x ¯ ˙ 0 + r δ ˙ ,
X r = X 0 ,
r = 0 ,
l r = r l 1 ,
l 1 = l 1 + 1 2 ( x ¯ 0 x ¯ 0 , δ ) .
T = 1 + 2 r = 1 r cos r φ exp [ 1 8 r 2 ( k Δ ) 2 ] ,
φ k l 1 ,
Δ 2 | ( δ x , X ) | 2 + | ( δ y , Y ) | 2 .
T 1 = 1 + 1 π 1 2 ( 1 m ) exp ( 1 m 2 ) erfc ( 1 m ) ,
m k Δ 0 / ( 2 1 2 π ) ,
l r = r l 0 + ( x 0 , x r ) / 2 ,
C o r = ( cos r θ + j Q sin r θ ) 1 e j r φ ,
Q = 1 2 [ ( W W m ) 2 + ( W m W ) 2 ] + k 2 8 W 2 W m 2 ( 1 R 1 R m ) 2 .
1 R m j 2 k W m 2 = D A 2 B j sin θ B .
T = 1 + 2 real part of r = 1 r e j r φ cos r θ + j Q sin r θ .
T 1 = 1 + 1 2 Q + 1 ( Q 1 Q + 1 ) p ,
C o r = exp ( j r θ ) exp [ ( 1 cos r θ ) | a | 2 ] exp ( j r φ j sin r θ | a | 2 ) = exp ( | a | 2 ) exp [ | a | 2 exp ( j r θ ) ] exp [ j r ( φ θ ) ] ,
| a | 2 ( x ¯ 0 / w ) 2 + ( k w x ¯ ˙ 0 / 2 ) 2 .
T 1 = 1 + 1 | a | 2 q q ! exp ( | a | 2 ) ,
Θ out Θ in = cot 1 k W 2 2 ( 1 R + A B ) .

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