Abstract

By the use of geometric optics and matrix methods an analysis of the spherical mirror Herriott cell is presented. The hyperboloidal beam envelope inside the cell is derived, and equations are given for the position and magnitude of its waist with respect to design parameters. The analysis is then repeated for the cell within a Fabry-Perot resonator. Here the stability condition is derived, and its region of validity is investigated for a range of parameters. Finally a theoretical study of the beam radius distribution at the optics is presented.

© 1989 Optical Society of America

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References

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  1. J. G. Xin, D. R. Hall, “Multipass Coaxial Radiofrequency Discharge CO2 Laser,” Opt. Commun. 58420–422 (1986).
    [CrossRef]
  2. G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
    [CrossRef]
  3. G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).
  4. L. W. Casperson, P. M. Schienert, “Multipass Resonators for Annular Gain Lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
    [CrossRef]
  5. T. R. Ferguson, M. E. Smithers, “Toric Unstable Resonators,” Appl. Opt. 23, 2122–2126 (1984).
    [CrossRef] [PubMed]
  6. D. Herriott, H. Kogelnik, R. Kompfner, “Off-Axis Paths in Spherical Mirror Interferometers,” Appl. Opt. 3, 523–526 (1964).
    [CrossRef]
  7. J. G. Xin, D. R. Hall, “Compact, Multipass, Single Transverse Mode CO2 Laser,” Appl. Phys. Lett. 51, 469–471 (1987).
    [CrossRef]
  8. H. Schulke, G. Herziger, R. Webster, “Multipass Resonators for Laser Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 801, 45–50 (1987).
  9. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  10. A. Yariv, Optoelectronics (Holt Saunders, 1985).
  11. H. Kogelnik, T. Bridges, “A Nonresonant Multipass CO2 Laser Amplifier,” IEEE J. Quantum Electron. QE-3, 95–96 (1967).
    [CrossRef]
  12. J. G. Xin, A. Duncan, D. R. Hall, “Optical Field Distribution in a Hyperboloidal Ray Envelope Multipass Laser Resonator,” in preparation.

1987

J. G. Xin, D. R. Hall, “Compact, Multipass, Single Transverse Mode CO2 Laser,” Appl. Phys. Lett. 51, 469–471 (1987).
[CrossRef]

H. Schulke, G. Herziger, R. Webster, “Multipass Resonators for Laser Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 801, 45–50 (1987).

1986

J. G. Xin, D. R. Hall, “Multipass Coaxial Radiofrequency Discharge CO2 Laser,” Opt. Commun. 58420–422 (1986).
[CrossRef]

1984

1981

L. W. Casperson, P. M. Schienert, “Multipass Resonators for Annular Gain Lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
[CrossRef]

1967

H. Kogelnik, T. Bridges, “A Nonresonant Multipass CO2 Laser Amplifier,” IEEE J. Quantum Electron. QE-3, 95–96 (1967).
[CrossRef]

1966

1964

1961

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[CrossRef]

G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).

Bridges, T.

H. Kogelnik, T. Bridges, “A Nonresonant Multipass CO2 Laser Amplifier,” IEEE J. Quantum Electron. QE-3, 95–96 (1967).
[CrossRef]

Casperson, L. W.

L. W. Casperson, P. M. Schienert, “Multipass Resonators for Annular Gain Lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
[CrossRef]

Duncan, A.

J. G. Xin, A. Duncan, D. R. Hall, “Optical Field Distribution in a Hyperboloidal Ray Envelope Multipass Laser Resonator,” in preparation.

Ferguson, T. R.

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).

Goubau, G.

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[CrossRef]

Hall, D. R.

J. G. Xin, D. R. Hall, “Compact, Multipass, Single Transverse Mode CO2 Laser,” Appl. Phys. Lett. 51, 469–471 (1987).
[CrossRef]

J. G. Xin, D. R. Hall, “Multipass Coaxial Radiofrequency Discharge CO2 Laser,” Opt. Commun. 58420–422 (1986).
[CrossRef]

J. G. Xin, A. Duncan, D. R. Hall, “Optical Field Distribution in a Hyperboloidal Ray Envelope Multipass Laser Resonator,” in preparation.

Herriott, D.

Herziger, G.

H. Schulke, G. Herziger, R. Webster, “Multipass Resonators for Laser Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 801, 45–50 (1987).

Kogelnik, H.

Kompfner, R.

Li, T.

Schienert, P. M.

L. W. Casperson, P. M. Schienert, “Multipass Resonators for Annular Gain Lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
[CrossRef]

Schulke, H.

H. Schulke, G. Herziger, R. Webster, “Multipass Resonators for Laser Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 801, 45–50 (1987).

Schwering, F.

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[CrossRef]

Smithers, M. E.

Webster, R.

H. Schulke, G. Herziger, R. Webster, “Multipass Resonators for Laser Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 801, 45–50 (1987).

Xin, J. G.

J. G. Xin, D. R. Hall, “Compact, Multipass, Single Transverse Mode CO2 Laser,” Appl. Phys. Lett. 51, 469–471 (1987).
[CrossRef]

J. G. Xin, D. R. Hall, “Multipass Coaxial Radiofrequency Discharge CO2 Laser,” Opt. Commun. 58420–422 (1986).
[CrossRef]

J. G. Xin, A. Duncan, D. R. Hall, “Optical Field Distribution in a Hyperboloidal Ray Envelope Multipass Laser Resonator,” in preparation.

Yariv, A.

A. Yariv, Optoelectronics (Holt Saunders, 1985).

Appl. Opt.

Appl. Phys. Lett.

J. G. Xin, D. R. Hall, “Compact, Multipass, Single Transverse Mode CO2 Laser,” Appl. Phys. Lett. 51, 469–471 (1987).
[CrossRef]

Bell Syst. Tech. J.

G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).

IEEE J. Quantum Electron.

H. Kogelnik, T. Bridges, “A Nonresonant Multipass CO2 Laser Amplifier,” IEEE J. Quantum Electron. QE-3, 95–96 (1967).
[CrossRef]

IRE Trans. Antennas Propag.

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[CrossRef]

Opt. Commun.

J. G. Xin, D. R. Hall, “Multipass Coaxial Radiofrequency Discharge CO2 Laser,” Opt. Commun. 58420–422 (1986).
[CrossRef]

Opt. Quantum Electron.

L. W. Casperson, P. M. Schienert, “Multipass Resonators for Annular Gain Lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

H. Schulke, G. Herziger, R. Webster, “Multipass Resonators for Laser Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 801, 45–50 (1987).

Other

A. Yariv, Optoelectronics (Holt Saunders, 1985).

J. G. Xin, A. Duncan, D. R. Hall, “Optical Field Distribution in a Hyperboloidal Ray Envelope Multipass Laser Resonator,” in preparation.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the multipass optical resonator.

Fig. 2
Fig. 2

Schematic diagram of the beam patterns on the Herriott cell mirror (circle, ellipse, and straight line).

Fig. 3
Fig. 3

Multipass resonator: (a) symmetric case; (b) unsymmetric case.

Fig. 4
Fig. 4

Calibrated distance (z/l) from the envelope waist to a Herriott cell mirror vs g-parameter for different Φ values.

Fig. 5
Fig. 5

Ratio of the beam envelope waist radius at the Herriott cell mirror vs the g-parameter.

Fig. 6
Fig. 6

Schematic diagram of the equivalent multipass optical resonator.

Fig. 7
Fig. 7

Maximum distance of l1 (=l2) for stability against the number of round trips: l = 30 cm,(a) R1 = R2 = R = 394cm,R3 = ∞,(b) R1 = R2 = R = 613cm,R4 = 50, 100, 500 cm,(c) R1 = R2 = R = 880cm.

Fig. 8
Fig. 8

Beam spot radius on one of the Herriott cell mirrors.

Equations (76)

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r = ( x , S x , y , S y ) t .
r = T r ,
T ( l ) = [ T x ( l ) 0 0 T y ( l ) ] ,
T x ( l ) = T y ( l ) = [ 0 l 1 0 ] ,
T ( R ) = [ T x ( R ) 0 0 T y ( R ) ] ,
T x ( R ) = T y ( R ) = [ 1 0 2 / R 1 ] .
T = T ( R 1 ) T ( l ) ( R 2 ) T ( l ) ,
T = [ T x T y ] ,
T x = T y = [ 2 g 2 1 2 l g 2 2 ( 2 g 1 g 2 g 1 g 2 ) / l 4 g 1 g 2 2 g 2 1 ] ,
r N = T N r 0 .
T x N = T y N = 1 sin Φ × [ 2 [ g 1 g 2 ( 1 g 1 g 2 ) ] cos N Φ + ( g 2 2 g 1 g 2 ) sin N Φ 2 l g 2 sin N Φ 2 [ ( 2 g 1 g 2 g 1 g 2 ) / l ] sin N Φ ( 2 g 1 g 2 2 g 2 ) sin N Φ + 2 / [ g 1 g 2 ( 1 g 1 g 2 ) ] cos N Φ ] .
T x , y N = { cos N Φ + ( P W ) sin N Φ l P sin N Φ [ ( 2 W P Q ) sin N Φ ] / l cos N Φ ( P W ) sin N Φ } ,
P = g 2 / [ g 1 g 2 ( 1 g 1 g 2 ) ] ,
Q = g 1 / [ g 1 g 2 ( 1 g 1 g 2 ) ] ,
W = g 1 g 2 / [ g 1 g 2 ( 1 g 1 g 2 ) ] ,
cos Φ = 2 g 1 g 2 1 .
| cos Φ | = | 2 g 1 g 2 1 | 1 0 g 1 g 2 1 .
x N = U sin ( N Φ + α ) ,
y N = V sin ( N Φ + β ) ,
U = { x 0 2 + [ x 0 ( P W ) + S x 0 l P ] 2 } 1 / 2 , V = { y 0 2 + [ y 0 ( P W ) + S y 0 l P ] 2 } 1 / 2 , tan α = x 0 / [ x 0 ( P W ) + S x 0 l P ] , tan β = y 0 / [ y 0 ( P W ) + S y 0 l P ] .
α = β ± π .
x N = U sin ( N Φ + α ) , y N = ± V sin ( N Φ + α ) ,
y N = ± ( V U ) x N .
α = β ± π / 2 .
x N = U sin ( N Φ + α ) , y N = ± V cos ( N Φ + α ) ,
( x N U ) 2 + ( y N V ) 2 = 1 .
Φ N = N Φ + α , Φ N = N Φ + β .
N Φ = 2 π ,
r 0 2 = [ sin ( N Φ ) / U ] 2 + [ cos ( N Φ ) / V ] 2 ,
x N = r 0 sin N Φ ,
y N = r 0 cos N Φ .
x = x N + z d x / d z = x N + z S xN ,
y = y N + z d y / d z = y N + z S yN .
S xN = x 0 ( 2 W P Q ) sin N Φ / l + S x 0 [ cos N Φ ( P W ) sin N Φ ] ,
S yN = y 0 ( 2 W P Q ) sin N Φ / l + S y 0 [ cos N Φ ( P W ) sin N Φ ] ,
y 0 = V , x 0 = 0 , α = β ± π / 2 .
S x 0 = U / l P ,
S y 0 = V ( P W ) / l P .
S xN = U [ ( P W ) sin N Φ + cos N Φ ] / ( l P ) ,
S yN = V [ ( W 2 P Q ) sin N Φ ( P W ) cos N Φ ] / ( l P ) .
r 2 = x N 2 + y N 2 + z 2 ( S xN 2 + S yN 2 ) + 2 z ( x N S xN + y N S yN ) .
r 2 / [ ( x N S yN y N S xN ) 2 / ( S xN 2 + S yN 2 ) ] [ z + ( x N S xN + y N S yN ) / ( S xN 2 + S yN 2 ) ] 2 / [ x N S yN y N S xN ) / ( S xN 2 + S yN 2 ) ] 2 = 1 .
x N = U sin ( N Φ ) ,
y N = V cos ( N Φ ) .
r 2 / { U 2 / [ 1 + ( P W ) 2 ] } { z [ l P ( P W ) ] / [ 1 + ( P W ) 2 ] } 2 / { l 2 P 2 / [ 1 + ( P W ) 2 ] 2 } = 1 .
z [ l P ( P W ) ] / [ 1 + ( P W ) 2 ] = 0 ,
z w = [ l P ( P W ) ] / [ 1 + ( P W ) 2 ] .
z w = [ l ( 1 g 1 ) cos 2 ( Φ / 2 ) ] / { g 1 cos 2 ( Φ / 2 ) ] 2 + [ sin 2 ( Φ / 2 ) cos 2 ( Φ / 2 ) ] } .
r min = U / [ 1 + ( P W ) 2 ] 1 / 2 .
r min / U = 1 [ 1 + ( P W ) 2 ] 1 / 2 .
r min / U = [ g 1 ( sin ( Φ / 2 ) ] / [ g 1 2 sin 2 ( Φ / 2 ) + ( 1 g 1 ) 2 cos 2 ( Φ / 2 ) ] 1 / 2 .
q 2 = ( A q 1 + B ) / ( C q 1 + D ) ,
1 / q i = 1 / R i j λ / ( π ω 1 2 ) ( i = 1 , 2 ) ,
1 / q 1 = ( C + D / q 1 ) / A + B / q 1 ) .
1 / q 1 = ( D A ) / ( 2 B ) ± j { 1 [ ( A + D ) / 2 ] 2 } 1 / 2 ( 1 / B ) .
R = 2 B / ( D A ) ,
ω R 2 = λ | B | { 1 [ ( A + D ) / 2 ] 2 } 1 / 2 / π .
0 1 [ ( A + D ) / 2 ] 2 , 1 ( A + D ) / 2 1 ,
T = T ( R 3 ) T ( l + l 1 ) T N 1 T ( l 2 ) T ( R 4 ) T ( l + l 2 ) T N 1 T ( l 1 ) ,
T N 1 = [ T ( R 1 ) T ( l ) T ( R 2 ) T ( l ) ] N 1 = { cos ( N 1 ) Φ + ( P W ) sin ( N 1 ) Φ l P sin ( N 1 ) Φ [ ( 2 W P Q ) sin ( N 1 ) Φ ] / l cos ( N 1 ) Φ ( P W ) sin ( N 1 ) Φ } ,
T N 1 = [ T ( R 2 ) T ( l ) T ( R 1 ) T ( l ) ] N 1 = { cos ( N 1 ) Φ + ( Q W ) sin ( N 1 ) Φ l Q sin ( N 1 ) Φ [ ( 2 W P Q ) sin ( N 1 ) Φ ] / l cos ( N 1 ) Φ ( Q W ) sin ( N 1 ) Φ } ,
[ A B C D ]
A = a 1 a 2 + b 1 c 2 , B = a 1 b 2 + b 1 d 2 , C = a 2 c 1 + c 2 d 1 , D = c 1 b 2 + d 1 d 2 ,
a 1 = cos ( N 1 ) Φ + [ W ( 1 + 2 l 1 / l ) P ( 1 + l 1 / l ) Q ( l 1 / l ) ] sin ( N 1 ) Φ , b 1 = L cos ( N 1 ) Φ + [ W ( L + 2 l 1 l 2 / l ) P ( l 2 + l 1 l 2 / l ) Q ( l 1 + l 1 l 2 / l ) ] sin ( N 1 ) Φ , c 1 = [ 2 / R 3 ] cos ( N 1 ) Φ + [ 2 W ( 1 / l 1 / R 3 2 l 1 / l R 3 ) + P ( 2 / R 3 + 2 l 1 / l R 3 1 / l ) + Q ( 2 l 1 / l R 3 1 / l ) ] sin ( N 1 ) Φ , d 1 = [ 1 2 L / R 3 ] cos ( N 1 ) Φ + [ 2 W ( l 2 / l L / R 3 2 l 1 l 2 / l R 3 + 1 / 2 ) + P ( 2 l 1 l 2 / l R 3 + 2 l 2 / R 3 l 2 / l ) + Q ( 2 l 1 l 2 / l R 3 + 2 l 1 / R 3 l 2 / l 1 ) sin ( N 1 ) Φ , }
a 2 = cos ( N 1 ) Φ + [ W ( 1 + 2 l 2 / l ) Q ( 1 + l 2 / l ) P ( l 2 / l ) ] sin ( N 1 ) Φ , b 2 = L cos ( N 1 ) Φ + [ W ( L + 2 l 1 l 2 / l ) Q ( l 1 + l 1 l 2 / l ) P ( l 2 + l 1 l 2 / l ) ] sin ( N 1 ) Φ , c 2 = [ 2 / R 4 ] cos ( N 1 ) Φ + [ 2 W ( 1 / l 1 / R 4 2 l 2 / l R 4 ) + Q ( 2 / R 4 + 2 l 2 / l R 4 1 / l ) + P ( 2 l 2 / l R 4 1 / l ) ] sin ( N 1 ) Φ , d 2 = [ 1 2 L / R 4 ] cos ( N 1 ) Φ + [ 2 W ( l 1 / l L / R 4 2 l 1 l 2 / l R 4 + 1 / 2 ) , + Q ( 2 l 1 l 2 / l R 4 + 2 l 1 / R 4 l 1 / l ) + P ( 2 l 1 l 2 / l R 4 + 2 l 2 / R 4 l 1 / l 1 ) sin ( N 1 ) Φ , }
R 1 = R 2 , R 3 = , l 1 = l 2 .
Φ ( N p 1 ) = π , R 4 = ,
T N p 1 , T N p 1 = [ 1 0 0 1 ] .
A = 1 ; B = 2 ( l + l 1 + l 2 ) ; C = 0 ; D = 1 .
A + D 2 = 1 ,
Φ ( N p 1 ) = π ,
R ( n ) = { [ a + b / R α ] 2 + [ λ b / ( π ω α 2 ) ] } { [ c + d / R α ] [ a + b / R α ] + [ λ / π ω α 2 ] b d } ,
ω ( n ) 2 = ω α 2 { [ a + b / R α ] 2 + [ λ b / ( π ω α 2 ) ] 2 } .
a = cos ( n Φ ) + ( P W ) sin ( n Φ ) , b = l P sin ( n Φ ) , c = [ ( 2 W P Q ) sin ( n Φ ) ] / l , d = cos ( n Φ ) ( P W ) sin ( n Φ ) .
R α = R 1 , ω α = ( λ l P / π ) 1 / 2 ,
R ( n ) = R 1 ω ( n ) = ω R ,

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