Abstract

An intuitive technique has been developed in which the lowest- and the higher-order modes for unstable resonators having rectangular or circular mirrors are derived. Instead of two mirrors, it considers the resonator to consist of a train of diffractive edges that are virtual images of the edge of the feedback mirror. Circular-mirror unstable resonators are shown not to exhibit the exact geometrical mode even at high Fresnel numbers.

© 1986 Optical Society of America

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References

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  1. A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
    [CrossRef]
  2. A. E. Siegman, R. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-4, 156–163 (1967).
    [CrossRef]
  3. W. Streifer, “Unstable optical resonators and waveguides,” IEEE J. Quantum Electron. QE-4, 229–230 (1968).
    [CrossRef]
  4. E. A. Sziklas, A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef] [PubMed]
  5. A. E. Siegman, H. Y. Miller, “Unstable optical resonator loss calculations using Prony method,” Appl. Opt. 9, 2729–2736 (1970).
    [CrossRef] [PubMed]
  6. P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [CrossRef]
  7. W. H. Southwell, “Virtual-source theory of unstable resonator modes,” Opt. Lett. 6, 487–489 (1981).
    [CrossRef] [PubMed]
  8. P. Horwitz, “Modes in misaligned unstable resonators,” Appl. Opt. 15, 167–178 (1976).
    [CrossRef] [PubMed]
  9. W. H. Southwell, “Asymptotic solution of the Huygens-Fresnel integral in circular coordinates,” Opt. Lett. 3, 100–102 (1978). The solution derived in this reference was rewritten in terms of Bessel functions instead of Hankel functions by using the asymptotic identities:Jl(x)=[2/(πx)]1/2cos(x−lπ/2−π/4),Jl+1(x)=[2/(πx)]1/2sin(x−lπ/2−π/4).
    [CrossRef]
  10. R. R. Butts, P. B. Avizonis, “Asymptotic analysis of unstable laser resonators with circular mirrors,” J. Opt. Soc. Am. 68, 1072–1078 (1978).
    [CrossRef]

1981 (1)

1978 (2)

1976 (1)

1975 (1)

1973 (1)

1970 (1)

1968 (1)

W. Streifer, “Unstable optical resonators and waveguides,” IEEE J. Quantum Electron. QE-4, 229–230 (1968).
[CrossRef]

1967 (1)

A. E. Siegman, R. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-4, 156–163 (1967).
[CrossRef]

1965 (1)

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[CrossRef]

Arrathoon, R.

A. E. Siegman, R. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-4, 156–163 (1967).
[CrossRef]

Avizonis, P. B.

Butts, R. R.

Horwitz, P.

Miller, H. Y.

Siegman, A. E.

E. A. Sziklas, A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[CrossRef] [PubMed]

A. E. Siegman, H. Y. Miller, “Unstable optical resonator loss calculations using Prony method,” Appl. Opt. 9, 2729–2736 (1970).
[CrossRef] [PubMed]

A. E. Siegman, R. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-4, 156–163 (1967).
[CrossRef]

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[CrossRef]

Southwell, W. H.

Streifer, W.

W. Streifer, “Unstable optical resonators and waveguides,” IEEE J. Quantum Electron. QE-4, 229–230 (1968).
[CrossRef]

Sziklas, E. A.

Appl. Opt. (3)

IEEE J. Quantum Electron. (2)

A. E. Siegman, R. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-4, 156–163 (1967).
[CrossRef]

W. Streifer, “Unstable optical resonators and waveguides,” IEEE J. Quantum Electron. QE-4, 229–230 (1968).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (2)

Proc. IEEE (1)

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Confocal unstable resonator with magnification M = (L + f)/f, where L is the mirror separation and f is the focal length of the feedback mirror, which has semidiameter a.

Fig. 2
Fig. 2

Schematic diagram of an unfolded confocal unstable resonator in terms of the virtual image of opposite edges of the feedback mirror. In the virtual-source theory, the resonator mode is obtained by summing edge-diffraction fields from these virtual images as they are illuminated by an initial plane wave.

Fig. 3
Fig. 3

Lowest-loss odd mode across the feedback mirror for an unstable strip resonator with M = 1.9 and Neq = 49.4.

Fig. 4
Fig. 4

The second lowest-loss even mode across the feedback mirror for an unstable strip resonator with M = 1.9 and Neq = 49.4.

Fig. 5
Fig. 5

Magnitude of the normalized eigenvalues for the even modes of a strip resonator with M = 2.9.

Fig. 6
Fig. 6

Magnitude of the normalized eigenvalues for the odd modes of a strip resonator with M = 2.9.

Fig. 7
Fig. 7

Magnitude of the normalized eigenvalues for the l = 0 modes for a circular-mirror resonator with M = 2.

Fig. 8
Fig. 8

Magnitude of the normalized eigenvalues for the l = 1 modes for a circular-mirror resonator with M = 2.

Fig. 9
Fig. 9

Magnitude of the normalized eigenvalues for the l = 2 modes for a circular-mirror resonator with M = 2.

Fig. 10
Fig. 10

Magnitude of the normalized eigenvalues for even modes of a strip resonator with M = 2. There is good mode separation, and the mode at unity corresponds to the geometrical-optics result.

Fig. 11
Fig. 11

Magnitude of the normalized eigenvalues tor l = 0 modes for a circular-mirror resonator with M = 2. Modes continue to interweave even for these high-Fresnel-number resonators.

Fig. 12
Fig. 12

Lowest-loss even mode, irradiance, and phase for a strip resonator with M = 2 and Neq = 2000.5. The normalized eigenvalue had a magnitude of 1.00335 and an argument of −0.00302 rad. The mode at Neq = 2000 was similar but had a slight irradiance dip on axis instead of the slight swell shown here. Also, the eigenvalue was slightly less than unity, as seen in Fig. 10.

Fig. 13
Fig. 13

Lowest-loss l = 0 mode, irradiance, and phase for a circular-mirror resonator with M = 2 and Neq = 2000.5. The normalized eigenvalue had a magnitude of 1.1445 and an argument of 0.00066 rad. Note the sharp spike at the center of the feedback mirror.

Fig. 14
Fig. 14

Lowest-loss l = 0 mode, irradiance, and phase for a circular-mirror resonator with M = 2 and Neq = 2000. The normalized eigenvalue had a magnitude of 1.1189 and an argument of −0.15483 rad. Note that there is some 60° phase change across the feedback mirror.

Equations (37)

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x j = a M j ,
z j = ( M 2 j 1 ) f ,
β = M 1 / 2 exp ( i 2 π λ 2 L ) ,
υ N ( x ) = β N + m = 1 N β N m + 1 ( x ) u m ,
N = ln ( 250 N eq ) ln ( M ) ,
N eq = 1 2 ( M 1 ) a 2 / ( λ L ) .
D j ( x ) = D s ( x j , z j , x ) ± D s ( x j , z j , x ) ,
D s ( X , Z , x ) = i ( i / 2 ) 1 / 2 ϕ ( h ) exp ( i π h 2 / 2 ) ,
h = ( 2 / λ Z ) 1 / 2 | X x | ,
ϕ ( h ) = f ( h ) + i g ( h ) ,
f ( h ) = 1 + 0.926 h 2 + 1.792 h + 3.10 h 2 ,
g ( h ) = 1 2 + 4.142 h + 3.492 h 2 + 6.67 h 3 .
D j ( x ) = D r ( x j , z j , x ) + D r ( x j , z j x ) ,
D r ( A , Z , r ) = i l [ π t A 2 / 2 ] 1 / 2 exp [ i t ( A 2 + r 2 ) ] ϕ ( h ) [ J l ( p ) + i sgn ( A ) J l + 1 ( p ) ] ,
h = ( 2 / λ z ) 1 / 2 | A r | ,
p = | 2 t A r | ,
t = π / ( λ z ) .
υ N + 1 ( x ) = γ υ N ( x ) ,
β N + 1 + m = 1 N + 1 β N m + 2 D N m + 2 ( x ) u m = γ β N + m = 1 N γ β N m + 1 D N m + 1 ( x ) u m .
u m + 1 = γ u m .
D N + 1 ( x ) D N + 1 ( a ) = D N + 1 ,
u 1 = ( α 1 ) / D N + 1 ,
α = γ / β .
u m = γ m 1 u 1 .
υ N ( a ) = u N + 1 .
υ N ( a ) = γ N u 1 .
β N + m = 1 N β N m + 1 D D m + 1 γ N 1 u 1 = γ N u 1 .
α N + 1 ( 1 + D 1 ) α N + m = 0 N 1 ( D N m D N m + 1 ) α m = 0.
υ N ( x ) = 1 + m = 1 N α m 1 D N m + 1 ( x ) .
α β = γ = γ exp ( i 2 π n ) , n = integer .
arg ( α ) 2 π λ 2 L = 2 π n .
λ = 2 L [ n + arg ( α ) ] .
D n + 1 odd ( x ) 0.
α N m = 1 N D N m + 1 α m 1 = 0 ( odd modes ) .
υ N ( x ) = m = 1 N α m 1 D N m + 1 ( x ) ( odd modes ) .
x > z j x 1 z 1 x j z j z 1 ,
Jl(x)=[2/(πx)]1/2cos(xlπ/2π/4),Jl+1(x)=[2/(πx)]1/2sin(xlπ/2π/4).

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