Abstract

A detailed theoretical study of the transmission function of birefringent filters using exact 4 × 4 matrix formalisms is presented. The Brewster-angle effect for the filters acting as intracavity wavelength selectors is also analyzed. Finally, the change in phase of the transmitted wave is calculated and the resonance condition of a cavity with a birefringent-type selector is obtained. The results are compared with those obtained from the usual Jones method.

© 1992 Optical Society of America

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References

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  1. F. P. Schäfer, Dye Laser (Springer-Verlag, Berlin, 1977).
  2. E. Bernabeu, F. Moreno, “Shift of tunable laser modes by effect of intracavity wavelength selectors with short-duration pulses,” J. Opt. Soc. Am. 71, 175–179 (1981).
    [CrossRef]
  3. E. Bernabeu, J. C. Amaré, J. M. Alvarez, F. Moreno, “Intensity transmitted by a Fabry–Perot etalon with another internal Fabry–Perot interferometer,” Appl. Opt. 20, 2117–2120 (1981).
    [CrossRef] [PubMed]
  4. A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 64, 447–452 (1974).
    [CrossRef]
  5. D. R. Preuss, J. L. Gole, “Three-stage birefringent filter tuning smoothly over the visible region: theoretical treatment and experimental design,” Appl. Opt. 19, 702–710 (1980).
    [CrossRef] [PubMed]
  6. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  7. P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
    [CrossRef]
  8. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  9. R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.7, pp. 340 ff.
  10. P. J. Lin-Chung, S. Teitler, “4 × 4 Matrix formalisms for optics in stratified anisotropic media,” J. Opt. Soc. Am. A 1, 703–705 (1984).
    [CrossRef]
  11. H. Wöhler, G. Haas, M. Fritsch, D. A. Mlynski, “Faster 4 × 4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [CrossRef]
  12. B. V. Boranev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosk. 60, 814–819 (1986). [Opt. Spectrosc. 60, 501–504 (1986)].
  13. P. J. Valle, F. Moreno, “Study of birefringent-type tuning devices through 4 × 4 matrix algebra,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 43 (1990).
  14. R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristic of anisotropic multilayered structures,” J. Opt. Soc. Am. A 4, 1720–1740 (1987).
    [CrossRef]

1988 (1)

1987 (1)

1986 (1)

B. V. Boranev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosk. 60, 814–819 (1986). [Opt. Spectrosc. 60, 501–504 (1986)].

1984 (1)

1981 (2)

1980 (2)

1979 (1)

1974 (1)

1972 (1)

Alvarez, J. M.

Amaré, J. C.

Azzam, R. M. A.

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.7, pp. 340 ff.

Bashara, N. H.

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.7, pp. 340 ff.

Bernabeu, E.

Berreman, D. W.

Bloom, A. L.

Boranev, B. V.

B. V. Boranev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosk. 60, 814–819 (1986). [Opt. Spectrosc. 60, 501–504 (1986)].

Fritsch, M.

Gaylord, T. K.

Gole, J. L.

Haas, G.

Kobtsev, S. M.

B. V. Boranev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosk. 60, 814–819 (1986). [Opt. Spectrosc. 60, 501–504 (1986)].

Lin-Chung, P. J.

Mlynski, D. A.

Moreno, F.

Preuss, D. R.

Schäfer, F. P.

F. P. Schäfer, Dye Laser (Springer-Verlag, Berlin, 1977).

Teitler, S.

Valle, P. J.

P. J. Valle, F. Moreno, “Study of birefringent-type tuning devices through 4 × 4 matrix algebra,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 43 (1990).

Weis, R. S.

Wöhler, H.

Yeh, P.

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Opt. Spectrosk. (1)

B. V. Boranev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosk. 60, 814–819 (1986). [Opt. Spectrosc. 60, 501–504 (1986)].

Surf. Sci. (1)

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

Other (3)

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.7, pp. 340 ff.

F. P. Schäfer, Dye Laser (Springer-Verlag, Berlin, 1977).

P. J. Valle, F. Moreno, “Study of birefringent-type tuning devices through 4 × 4 matrix algebra,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 43 (1990).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the BF. The faces of the plate are parallel to the XY plane. The incidence plane is the XZ plane. The tuning angle φ and the incidence angle α are defined as shown.

Fig. 2
Fig. 2

Transmission factors of a BF calculated by the JF (dashed curves) and the EM (continuous curves) as a function of the wavelength. The filter is composed of a single quartz plate whose thickness is d = 0.8 mm, incidence angle is α = 28.6°, and tuning angles are (a) φ = 30°, (b) φ = 45°, and (c) φ = 60°.

Fig. 3
Fig. 3

Visibility factor of the Jones transmission function of a filter as a function of the tuning angle φ, and for incidence angles α = 28.6° (dashed curve) and α = 57.2° (continuous curve).

Fig. 4
Fig. 4

Details of the fine structure (continuous curves) in transmission maxima of a single quartz plate filter (d = 0.8 mm, φ = 45°) for incidence angles (a) α = 28.6°, (b) α = 50°, (c) α = 57.2°, and (d) α = 65°. The dashed curves are the Jones transmission factor.

Fig. 5
Fig. 5

Visibility factor of the fine structure in Jones maxima as a function of the incidence angle. The Y axis must be scaled by 10−3.

Fig. 6
Fig. 6

(a) Triple fine structure of a filter consisting of three quartz plates whose thickness are 0.3, 0.9, and 3.0 mm (φ = 43° and α = 50°). (b) Transmission factor of the same filter calculated by the JF; the marked zone corresponds to (a).

Fig. 7
Fig. 7

(a) Detail of the maximum of the transmission factor of a single quartz plate filter (d = 0.8 mm, α = 50°, and φ = 45°). (b) Change in phase because of the filter in the same wavelength interval. JF and EM give the same phase.

Fig. 8
Fig. 8

(a) Change in phase of a single LiNbO3 plate filter whose refractive indices are no = 2.300 and nex = 2.708. The EM (continuous curve) gives a different phase from the JF (dashed curve). (b) Change in phase produced by an isotropic plate whose refractive index is a mean value between no and nex.

Fig. 9
Fig. 9

Axial modes of a 1-m length laser cavity with (continuous lines) and without (dashed lines) a BF inside it. (a) The filter is composed of a quartz plate whose thickness is 0.8 mm. (b) The filter consists of three quartz plates whose thicknesses are 0.8, 3.2, and 12 mm.

Tables (1)

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Table I FSR of a F–P and a Single Quartz Plate Filter for Several Incidence Angles αa

Equations (21)

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δ Z Φ = i K 0 M ( z ) Φ ,
M ( z ) = [ 0 m 12 0 0 m 21 0 m 23 0 0 0 0 1 m 23 0 m 43 0 ] ,
m 12 = 1 γ 2 , m 21 = Δ cos 2 φ, m 23 = 1 2 Δ sin 2 φ, m 43 = ( Δ sin 2 φ ) γ 2 ,
Φ ( d ) = P ( d , 0 ) Φ ( 0 ) .
P ( d , 0 ) = exp ( i k 0 M d ) = n = 0 ( 1 ) n [ i k 0 d n ! M ] n ,
P 11 = P 22 = 1 Y ( λ 1 2 sin 2 φ cos α 3 + cos 2 φ cos α 1 ) , P 12 = i λ 1 2 Y ( λ 1 2 λ 3 1 sin 2 φ sin α 3 + 1 λ 1 cos 2 φ sin α 1 ) , P 13 = P 42 = λ 1 2 sin φ cos φ Y ( cos α 3 cos α 1 ) , P 14 = P 32 = i sin φ cos φ Y ( λ 1 2 λ 3 sin α 3 λ 1 sin α 1 ) , P 21 = i Y ( λ 3 sin 2 φ sin α 3 + λ 1 cos 2 φ sin α 1 ) , P 23 = P 41 = i sin φ cos φ Y ( λ 3 sin α 3 λ 1 sin α 1 ) , P 24 = P 31 = sin φ cos φ Y ( cos α 3 cos α 1 ) , P 33 = P 44 = 1 Y ( cos 2 φ cos α 3 + λ 1 2 sin 2 φ cos α 1 ) , P 34 = i 1 Y ( λ 3 cos 2 φ sin α 3 + λ 1 sin 2 φ sin α 1 ) , P 43 = i 1 Y ( λ 3 cos 2 φ sin α 3 + λ 1 3 sin 2 φ sin α 1 ) ,
Y = sin 2 φ sin 2 α, α i = ω c λ i d , λ 1 , 2 = ± ( sin 2 α ) 1 / 2 , λ 3 , 4 = ± ( ) 1 / 2 [ ( Δ cos 2 φ ) sin 2 α ] 1 / 2 .
[ E r p E r s ] = [ R p p R p s P s p R s s ] [ E i p E i s ] [ E t p E t s ] = [ T p p T p s T s s T s s ] [ E i p E i s ] ,
R p p = ( a i p b r s a r s b i p ) / ( a r s b r p a r p b r s ) , R p s = ( a i s b r s a r s b i s ) / ( a r s b r p a r p b r s ) , R s p = ( a r s b i p a i p b r p ) / ( a r s b r p a r p b r s ) , R s s = ( a r p b i s a i s b r p ) / ( a r s b r p a r p b r s ) ,
a i p = P 12 P 21 cos 2 α, a r p = P 12 2 P 11 cos α + P 21 cos 2 α , a i s = P 13 + ( P 14 P 23 ) cos α P 24 cos 2 α , a r s = P 13 ( P 14 + P 23 ) cos α + P 24 cos 2 α ,
b i p = P 13 + ( P 14 P 23 ) cos α + P 24 cos 2 α , b r p = P 13 + ( P 14 + P 23 ) cos α P 24 cos 2 α , b i s = P 43 + P 34 cos 2 α , b r s = P 43 + 2 P 33 cos α P 34 cos 2 α ,
T p p = ( P 21 cos α + P 11 ) + R p p ( P 21 cos α + P 11 ) + R s p ( P 23 P 24 cos α ) , T p s = ( P 24 cos α + P 23 ) + R p s ( P 21 cos α + P 11 ) + R s p ( P 23 P 24 cos α ) , T s p = ( P 24 cos α + P 14 ) + R p p ( P 24 cos α + P 14 ) + R s p ( P 33 P 34 cos α ) , T s s = ( P 34 cos α + P 33 ) + R p s ( P 24 cos α + P 14 ) + R s s ( P 33 P 34 cos α ) .
T p p = T p p ( λ, α, φ, d , n o , n e x ) ,
V = | T pp | EM 2 | T p p | JF 2 λ ,
( Δ ν ) FSR = c 2 d ( n 2 sin 2 α ) 1 / 2 ,
Ψ = 2 π λ k L = k π,
Ψ = 2 π λ k L + ξ = k π ,
L = L d cos α ,
tan ( ξ J ) = n o 2 sin 2 φ sin δ o + ( n o 2 sin 2 α ) cos 2 φ sin δ ex n o 2 sin 2 φ cos δ o + ( n o 2 sin 2 α ) cos 2 φ cos δ ex ,
δ o = 2 π d λ n o ( 1 sin 2 α n o 2 ) 1 / 2 , δ e = 2 π d λ n e x [ 1 sin 2 α ( cos 2 φ n o 2 + sin 2 φ n e x 2 ) ] 1 / 2 ,
tan ( ξ ) = R + 1 R 1 tan ( δ 2 ) .

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