Abstract

The previously developed ray-optical method for unstable, symmetric, bare resonators with sharp-edged strip and circular mirrors is reviewed here. A deductive stepwise procedure is presented, with emphasis on the physical implications. It is shown how the method can accommodate other edge configurations such as those produced by rounding, and also more complicated nonaxial structures such as the half-symmetric resonator with internal axicon. For the latter, the ray approach categorizes those rays that must be eliminated from the equivalent aligned unfolded symmetric resonator, and it identifies the canonical diffraction problems that must be addressed to account for shadowing and scattering due to the axicon tip. Effects due to shielding or truncation of the axicon tip are also considered. Approximate calculations of the eigenvalues for the lowest-loss modes illustrate the effects due to various tip shielding lengths and spacings of the axicon from the output mirror.

© 1980 Optical Society of America

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References

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  1. P. Horwitz, J. Opt. Soc. of Am. 63, 1528 (1973).
    [CrossRef]
  2. R. R. Butts, P. Avizonis, J. Opt. Soc. of Am. 68, 1072 (1978).
    [CrossRef]
  3. S. H. Cho, S. Y. Shin, L. B. Felsen, J. Opt. Soc. of Am. 69, 563 (1979).
    [CrossRef]
  4. S. H. Cho, L. B. Felsen, J. Opt. Soc. of Am. 69, 1377 (1979).
    [CrossRef]
  5. S. J. Maurer, L. B. Felsen, Proc. IEEE 55, 1718 (1967).
    [CrossRef]
  6. A. E. Siegman, Appl. Opt. 13, 353 (1974).
    [CrossRef] [PubMed]
  7. C. Santana, L. B. Felsen, Appl. Opt. 15, 1470 (1976).
    [CrossRef] [PubMed]
  8. J. B. Keller, J. Opt. Soc. of Am. 52, 116 (1962).
    [CrossRef]
  9. L. Kaminetsky, J. B. Keller, SIAM J. Appl. Math. 22, 109 (1972).
    [CrossRef]
  10. C. Santana, L. B. Felsen, Appl. Opt. 17, 2239 (1978).
    [CrossRef] [PubMed]
  11. C. Santana, S. H. Cho, “Ray-Optical Analysis of Unstable Laser Resonators with Rounded Edges” (in preparation).
  12. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  13. A. Paxton, “Annular Resonator Calculations,” Report AFWL-TR-75-311, Air Force Weapons Laboratory, Albuquerque, New Mexico.
  14. L. B. Felsen, T. Ishihara, Radio Sci. 14, 205 (1979).
    [CrossRef]
  15. L. B. Felsen, T. Ishihara, J. Acoust. Soc. Am. 63, 595 (1979).
    [CrossRef]

1979 (4)

S. H. Cho, S. Y. Shin, L. B. Felsen, J. Opt. Soc. of Am. 69, 563 (1979).
[CrossRef]

S. H. Cho, L. B. Felsen, J. Opt. Soc. of Am. 69, 1377 (1979).
[CrossRef]

L. B. Felsen, T. Ishihara, Radio Sci. 14, 205 (1979).
[CrossRef]

L. B. Felsen, T. Ishihara, J. Acoust. Soc. Am. 63, 595 (1979).
[CrossRef]

1978 (2)

C. Santana, L. B. Felsen, Appl. Opt. 17, 2239 (1978).
[CrossRef] [PubMed]

R. R. Butts, P. Avizonis, J. Opt. Soc. of Am. 68, 1072 (1978).
[CrossRef]

1976 (1)

1974 (1)

1973 (1)

P. Horwitz, J. Opt. Soc. of Am. 63, 1528 (1973).
[CrossRef]

1972 (1)

L. Kaminetsky, J. B. Keller, SIAM J. Appl. Math. 22, 109 (1972).
[CrossRef]

1970 (1)

1967 (1)

S. J. Maurer, L. B. Felsen, Proc. IEEE 55, 1718 (1967).
[CrossRef]

1962 (1)

J. B. Keller, J. Opt. Soc. of Am. 52, 116 (1962).
[CrossRef]

Avizonis, P.

R. R. Butts, P. Avizonis, J. Opt. Soc. of Am. 68, 1072 (1978).
[CrossRef]

Butts, R. R.

R. R. Butts, P. Avizonis, J. Opt. Soc. of Am. 68, 1072 (1978).
[CrossRef]

Cho, S. H.

S. H. Cho, S. Y. Shin, L. B. Felsen, J. Opt. Soc. of Am. 69, 563 (1979).
[CrossRef]

S. H. Cho, L. B. Felsen, J. Opt. Soc. of Am. 69, 1377 (1979).
[CrossRef]

C. Santana, S. H. Cho, “Ray-Optical Analysis of Unstable Laser Resonators with Rounded Edges” (in preparation).

Felsen, L. B.

L. B. Felsen, T. Ishihara, Radio Sci. 14, 205 (1979).
[CrossRef]

L. B. Felsen, T. Ishihara, J. Acoust. Soc. Am. 63, 595 (1979).
[CrossRef]

S. H. Cho, L. B. Felsen, J. Opt. Soc. of Am. 69, 1377 (1979).
[CrossRef]

S. H. Cho, S. Y. Shin, L. B. Felsen, J. Opt. Soc. of Am. 69, 563 (1979).
[CrossRef]

C. Santana, L. B. Felsen, Appl. Opt. 17, 2239 (1978).
[CrossRef] [PubMed]

C. Santana, L. B. Felsen, Appl. Opt. 15, 1470 (1976).
[CrossRef] [PubMed]

S. J. Maurer, L. B. Felsen, Proc. IEEE 55, 1718 (1967).
[CrossRef]

Horwitz, P.

P. Horwitz, J. Opt. Soc. of Am. 63, 1528 (1973).
[CrossRef]

Ishihara, T.

L. B. Felsen, T. Ishihara, J. Acoust. Soc. Am. 63, 595 (1979).
[CrossRef]

L. B. Felsen, T. Ishihara, Radio Sci. 14, 205 (1979).
[CrossRef]

Kaminetsky, L.

L. Kaminetsky, J. B. Keller, SIAM J. Appl. Math. 22, 109 (1972).
[CrossRef]

Keller, J. B.

L. Kaminetsky, J. B. Keller, SIAM J. Appl. Math. 22, 109 (1972).
[CrossRef]

J. B. Keller, J. Opt. Soc. of Am. 52, 116 (1962).
[CrossRef]

Maurer, S. J.

S. J. Maurer, L. B. Felsen, Proc. IEEE 55, 1718 (1967).
[CrossRef]

Miller, H. Y.

Paxton, A.

A. Paxton, “Annular Resonator Calculations,” Report AFWL-TR-75-311, Air Force Weapons Laboratory, Albuquerque, New Mexico.

Santana, C.

C. Santana, L. B. Felsen, Appl. Opt. 17, 2239 (1978).
[CrossRef] [PubMed]

C. Santana, L. B. Felsen, Appl. Opt. 15, 1470 (1976).
[CrossRef] [PubMed]

C. Santana, S. H. Cho, “Ray-Optical Analysis of Unstable Laser Resonators with Rounded Edges” (in preparation).

Shin, S. Y.

S. H. Cho, S. Y. Shin, L. B. Felsen, J. Opt. Soc. of Am. 69, 563 (1979).
[CrossRef]

Siegman, A. E.

Appl. Opt. (4)

J. Acoust. Soc. Am. (1)

L. B. Felsen, T. Ishihara, J. Acoust. Soc. Am. 63, 595 (1979).
[CrossRef]

J. Opt. Soc. of Am. (5)

J. B. Keller, J. Opt. Soc. of Am. 52, 116 (1962).
[CrossRef]

P. Horwitz, J. Opt. Soc. of Am. 63, 1528 (1973).
[CrossRef]

R. R. Butts, P. Avizonis, J. Opt. Soc. of Am. 68, 1072 (1978).
[CrossRef]

S. H. Cho, S. Y. Shin, L. B. Felsen, J. Opt. Soc. of Am. 69, 563 (1979).
[CrossRef]

S. H. Cho, L. B. Felsen, J. Opt. Soc. of Am. 69, 1377 (1979).
[CrossRef]

Proc. IEEE (1)

S. J. Maurer, L. B. Felsen, Proc. IEEE 55, 1718 (1967).
[CrossRef]

Radio Sci. (1)

L. B. Felsen, T. Ishihara, Radio Sci. 14, 205 (1979).
[CrossRef]

SIAM J. Appl. Math. (1)

L. Kaminetsky, J. B. Keller, SIAM J. Appl. Math. 22, 109 (1972).
[CrossRef]

Other (2)

A. Paxton, “Annular Resonator Calculations,” Report AFWL-TR-75-311, Air Force Weapons Laboratory, Albuquerque, New Mexico.

C. Santana, S. H. Cho, “Ray-Optical Analysis of Unstable Laser Resonators with Rounded Edges” (in preparation).

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Figures (14)

Fig. 1
Fig. 1

Modal ray congruences for self-consistent field between infinite hyperbolic strip mirrors: (a) 2kL: k = (2π/wavelength). Modal caustic is an ellipse or hyperbola. (b) 2kL = . Caustic degenerates into the straight line between the foci. Resulting ray fields from the foci describe the self-replicating cylindrical waves of Siegman.6

Fig. 2
Fig. 2

Modal wave fronts and ray congruences. For self-replicating (modal) fields, the optical phase accumulation along the ray path from 2 to 1 is an integral multiple of 2π.

Fig. 3
Fig. 3

Actual and canonical problems for edge diffraction: (a) upgoing modal ray congruence incident on edge A on upper mirror; when edge is rounded (dashed line), the scattering occurs from the curvature discontinuity; (b) equivalent canonical problem for determination of edge-diffracted fields.

Fig. 4
Fig. 4

Multiply reflected, edge-diffracted ray species: (a) rays crossing resonator axis; (b) rays not crossing resonator axis.

Fig. 5
Fig. 5

Ray-optical determination of edge-diffracted fields at an observation point x on the lower mirror. Gi is the incident modal ray field arriving at the edge with angle θi. Gn represents an (|n| − 1) times reflected ray field originating at the right-hand (n > 0) and left-hand (n < 0) edges, respectively. Corresponding ray departure angles at the edges are denoted by θn and the arrival points x by xn. (a) Direct rays originating at upper edges. (b) Multiply reflected rays originating at upper edges. (c) Multiply reflected rays originating at lower edges. (d) Divergence of ray tube due to reflection from the curved mirror surfaces. Δxn are the ray tube cross sections.

Fig. 6
Fig. 6

Collective ray field G0 accounting for all multiply reflected, edge-diffracted rays with |n| > N.

Fig. 7
Fig. 7

Total field due to 2N + 1 rays at lower edge x = 1 due to primary diffraction of incident field.

Fig. 8
Fig. 8

Doubly diffracted ray field Gl,n reaching an observation point x on lower mirror after (|l| − 1) reflections, the excitation being a singly diffracted incident ray field (shown dashed) with angle θn at the edge. The index n characterizes one of the singly diffracted fields in Fig. 7; the index l characterizes the number of reflections of the corresponding doubly diffracted fields. G0,n represents the collective ray field for |l| > N.

Fig. 9
Fig. 9

Resonance condition: Cl = Dl, −NlN. (a) After a certain number of diffractions. (b) After an additional diffraction.

Fig. 10
Fig. 10

HSURIA. Structure is rotationally symmetric about the (dotted) axis. I—spherical output mirror (same as in symmetrical resonator in Fig. 3, etc.); II—internal conical reflector (axicon); III—external conical reflector; IV—flat mirror terminating annular region. The length parameters L1, L2, and L3 specify the geometrical arrangement.

Fig. 11
Fig. 11

Ray trajectories in the HSURIA (solid) and the equivalent symmetric unfolded resonator (dashed). Apex of the axicon is at O. Overbarred quantities in the unfolded system correspond to unbarred quantities in the actual system. The mirror spacing in the unfolded structure is 2L = 2(L1 + L2 + L3). (a) n = ±1. n = +1 ray and all others (with positive n) originating at A get back to A without crossing the axis, n = −1 ray escapes from the HSURIA; it does not get to A because its equivalent trajectory from Â′ passes to the right of the axicon tip O. This is true also of all other rays with (negative) odd n originating at A′ or Â′. (b) n = 2. (c) n = −2. This ray and all others with (negative) even n originating at A′ or Â′ get to A.

Fig. 12
Fig. 12

Transition phenomena (in shaded regions) due to rays incident near the tip of the axicon. Transition region near the tip extends over an interval of several wavelengths and also surrounds the reflected ray boundaries. (a) Conical wave field incident from outer cone. Ray 1 is an ordinary reflected ray. Ray 3 passes the axicon and escapes from the resonator. Ray 2 establishes the tip diffracted field and also the boundary of the reflected rays; these coincide along the axis (dashed). (b) Conical wave field incident from output mirror. Ray 1 is an ordinary reflected ray. Ray 2 establishes the tip diffracted field and also the boundary of reflected rays (dashed).

Fig. 13
Fig. 13

Eigenvalues for lowest-loss modes in resonators with magnification M = 2. Symmetric: — from Eq. (16) or from Ref. 2; --- Ref. 12. Half-symmetric: … from Eq. (32). HSURIA: — — — from Eq. (31).

Fig. 14
Fig. 14

Lowest-loss eigenvalues for HSURIA with M = 2, Neq = 2.5 vs L1/L, for various axicon shielding lengths d. Computed points are indicated by dots. With reference to Table I, N+ remains constant whereas N decreases by unity when L1/L decreases by one-half. Thus there are (N − 1) rays from edge A′ when L1/L = ½, (N − 2) rays when L1/L = ¼, etc.

Tables (1)

Tables Icon

Table I Maximum number of rays N+ and N for various axicon tip shielding lengths d in a HSURIA with M = 2, Neq = 2.5, at L1/L = 1

Equations (37)

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p j = ( j π 2 k L ) / ln M ,
M = 1 + 2 γ + ( γ 2 + 2 γ ) 1 / 2 , γ = L / r ,
G i = [ sgn ( x ) ] q exp ( i π N eq x 2 ) · { | x | i p 1 / 2 on lower mirror , | x | i p 1 / 2 ( 1 ) j + 1 on upper mirror ,
N eq = N f ( M M 1 ) / 2 = ( k a 2 / 4 π L ) ( M M 1 ) / 2 ,
G ¯ i ( x ) = exp ( i π N eq x 2 ) exp ( i m ψ ) · { | x | i p 1 on lower mirror , | x | i p 1 ( 1 ) j + 1 on upper mirror ,
f ( θ , θ i ) = 2 sin θ i sin θ cos θ + cos θ i ,
f ( θ , θ i ) = 2 i ( r r 2 ) 4 k r r 2 2 sin θ i sin θ ( cos θ + cos θ i ) 3 ,
G ¯ n ( x ) = u n exp ( i S n ) = G n ( x ) exp ( i π N eq x 2 ) ,
G 0 = n = N + 1 ( G n + G n ) .
G T ( x ) = n = 1 N [ G n ( x ) = G n ( x ) ] + G 0 ( x ) .
θ n = π / 2 ( a 4 L ) [ M M 1 + sgn ( n ) ( 2 / M ) M | n | × ( 1 M 2 ) ( 1 ± M | n | ) 1 ] ,
G ¯ T ( x ) = G T ( x ) exp ( i π N eq x 2 ) ,
G T ( x ) = l = N + N D l ( x ) = l = N + N n = N + N C n G l , n ( x ) .
D 0 ( x ) = l = N + 1 [ D l ( x ) + D l ( x ) ] = n = N + N C n l = N + 1 [ G l , n ( x ) + G l , n ( x ) ] = n = N + N C n G 0 , n ( x ) ,
det ( [ G ] [ I ] ) = 0 ,
l = N + N G l , 0 ( 1 ) = 1.
λ = M i p + 1 / 2 for hyperbolic mirrors , M i p + 1 for hyperboloidal mirrors .
G T ( x ) = C l = N + N G l , 0 ( x ) ,
C = l = N N C l
δ = f r ( θ ± 1 , θ n ) / f s ( θ ± 1 , θ n )
= δ ¯ ( 1 M 2 | l | ) 2 [ ( d n 1 ) ( 1 M 2 | l | ) ( 1 M 2 ) 1 + 1 sgn ( l ) x M | l | ] 2 ,
δ ¯ = exp ( i π / 2 ) ( r / r 2 1 ) 4 π N eq · M 1 M + 1 ,
d n = [ 1 + sgn ( n ) M 2 M | n | ] / [ 1 + sgn ( n ) M | n | ] .
G l , n r ( x ) = δ ¯ ( 1 M 2 | l | ) 2 G l , n s ( x ) [ ( d n 1 ) ( 1 M 2 | l | ) ( 1 M 2 ) 1 + 1 sgn ( l ) x M | l | ] 2 .
G l , n r ( x ) = 2 λ | l | E l ( x ) Q ( 1 M 2 | l | ) 5 / 2 [ ( d n 1 ) ( 1 M 2 ) 1 ( 1 M 2 | l | ) + 1 sgn ( l ) x M | l | ] 3 ,
G 0 , n r ( x ) = 2 2 exp ( i 2 π N eq ) , × { λ N λ 1 1 [ ( d n 1 ) ( 1 M 2 ) + 1 ] 3 for even x symmetry ( λ M ) N λ M 1 3 [ ( d n 1 ) ( 1 M 2 ) + 1 ] 4 for odd x symmetry ,
2 = ( r / r 2 1 ) 4 π 2 M 1 M + 1 [ exp ( i π / 4 ) 2 N eq ] 3 / 2 ,
E l ( x ) = exp { i 2 π N eq [ 1 sgn ( l ) x M | l | ] 2 1 M 2 | l | } ,
G l , n r ( x ) = ( i ) m 2 δ ¯ λ | l ) A l ( x ) E l ( x ) ( 1 M 2 | l | ) 2 [ ( d n 1 ) ( 1 M 2 | l | ) ( 1 M 2 ) 1 + ( 1 sgn ( l ) x M | l | ) ] 3 ,
G 0 , n r ( x ) = ( i ) m δ ¯ exp ( i 2 π N eq ) m ! ( 2 π N eq | x | ) M N ) m × λ N λ M m 1 · 1 [ ( d n 1 ) ( 1 M 2 ) 1 + 1 ] 3 ,
A l ( x ) = [ J m ( Y l ) i sgn ( l ) J m + 1 ( Y l ) ] exp [ i sgn ( l ) Y l ] ,
Y l = 4 π N eq 1 M 2 | l | x M | l | ,
l = 1 G 2 l , 0 ( 1 ) + l = 1 G l , 0 ( 1 ) = 1.
l = 1 G l , 0 ( 1 ) = 1.
d l M l / 2 ( M + M 1 ) ( 1 + M l ) 1 · a , l > 0 ,
d l 1 2 M | l | ( M M 1 ) ( 1 M | l | ) 1 ( L 1 / L ) · a , l < 0 ,
l = 1 N G 2 l , 0 ( 1 ) + l = 1 N + G l , 0 ( 1 ) = 1.

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