Abstract

A technique is presented in which the full vector solution for resonator modes is synthesized from the known scalar solution. Second-order perturbation theory is utilized to predict the behavior of a resonator when an imperfect coating on the rear cone results in a phase shift between the s and p polarizations. The theoretical results agree well with a numerical example.

© 1980 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 32.
  2. D. Fink, “Polarization effects in axicons”, Appl. Opt. 18, 581–582 (1979).
    [CrossRef] [PubMed]
  3. G. C. Dente, “Polarization effects in resonators”, Appl. Opt. 18, 2911–2912 (1979).
    [CrossRef] [PubMed]
  4. W. H. Southwell, “Multilayer coatings producing 90° phase change”, Appl. Opt. 18, 1875 (1979).
    [CrossRef] [PubMed]

1979

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Figures (3)

Fig. 1
Fig. 1

Half-symmetric unstable resonator with a rear cone. The parameters used in the numerical example of Figs. 2 and 3 are a = 1.5916, Z = 10000, λWV = 0.00038, and the focal length of the feedback mirror is 13333.3. This results in an equivalent Fresnel number Neq = 0.4787 and a magnification M = 3.186 (all values are in centimeters).

Fig. 2
Fig. 2

Modulus of the eigenvalue versus phase error from a perfect 90° phase-shift coating on the rear cone: λ0 = 0.52285 + 0.04106i, λ2 = −0.04706 + 0.11669i, L1 = −0.08682 − 0.18120i, L2 = −0.09533 − 0.21024i.

Fig. 3
Fig. 3

Percentage of the total energy in the l = 0 part of the vector resonator solution.

Equations (30)

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K ^ P ^ E = λ E ,
r p = ρ p e i δ p ,             r s = ρ s e i δ s ,
E p = r 2 p E p ,             E s = r 2 s E s ,
P ^ = ( cos ϕ - sin ϕ sin ϕ cos ϕ ) ( ρ 2 p e i ( 2 δ p - π ) 0 0 ρ 2 s e i 2 δ s ) × ( cos ϕ sin ϕ - sin ϕ cos ϕ ) = ( a + b cos 2 ϕ b sin 2 ϕ b sin 2 ϕ a - b cos 2 ϕ ) ,
a = ½ [ ρ 2 p e i ( 2 δ p - π ) + ρ 2 s e i 2 δ s ] , b = ½ [ ρ 2 p e i ( 2 δ p - π ) - ρ 2 s e i 2 δ s ] .
i ^ R = c o s ϕ i ^ + s i n ϕ j ^ , i ^ A = - s i n ϕ i ^ + c o s ϕ j ^ .
i ^ R = 1 2 [ e - i ϕ i ^ + + e + i ϕ i ^ - ] ,
i ^ ± = 1 2 ( i ^ ± i j ^ ) ,
P ^ = ( a b e - i 2 ϕ b e i 2 ϕ a )
= a ( 1 0 0 1 ) + b ( 0 e - i 2 ϕ e i 2 ϕ 0 ) = P ^ a + P ^ b .
P ^ a E ± = a E ± ,
P ^ b E ± = b e ± 2 ϕ E .
L ^ E = λ E ,
L ^ = K ^ P ^ = K ^ + V ^ ,
V ^ = ( a - 1 ) K ^ + K ^ P ^ b .
E = E l + + m l [ a m + ( 1 ) + a m + ( 2 ) ] E m + + m [ a m - ( 1 ) + a m - ( 2 ) ] E m - ,
λ = λ l + λ ( 1 ) + λ ( 2 ) .
K ^ E l + = λ l E l + m l a m + ( 1 ) K ^ E m + + m a m - ( 1 ) K ^ E m - + V ^ E l +
= λ l m l a m + ( 1 ) E m + + λ l m a m - ( 1 ) E m - + λ ( 1 ) E l + m l a m + ( 2 ) K ^ E m + + m a m - ( 2 ) K ^ E m - + m l a m + ( 1 ) V ^ E m + + m a m - ( 1 ) V ^ E m -
= λ l m l a m + ( 2 ) E m + + λ l m a m - ( 2 ) E m - + λ ( 1 ) m l a m + ( 1 ) E m + + λ ( 1 ) m a m - ( 1 ) E m - + λ ( 2 ) E l + .
E n + * E m + d A = δ m n ,             E n - * E m + d A = 0 ,
E l + = f l ( r ) e i l ϕ .
a n + ( 1 ) = 0 ,             n l .
λ ( 1 ) = ( a - 1 ) λ l .
a l + 2 - ( 1 ) = b E l + 2 - * K ^ e i 2 ϕ E l - d A λ l - λ l + 2 .
a n + ( 2 ) = 0 ,
λ ( 2 ) = a l + 2 - ( 1 ) b E l + * K ^ e - i 2 ϕ E l + 2 + d A ,
a l + 2 - ( 2 ) = ( 1 - a ) a l + 2 - ( 1 ) .
λ = a λ l + b 2 L 1 L 2 λ l - λ l + 2 ,
E = E l + + ( 2 - a ) b L 1 λ l - λ l + 2 E l + 2 - ,

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