Abstract

It is shown that, for a tilted birefringent plate (BP) with its optic axis parallel to the plate surface, an explicit Jones transformation matrix can be derived without neglecting the birefringent splitting.

Differences in calculating the desired thickness(es) of a BP by use of different formulas of phase difference between ordinary and extraordinary waves are also presented.

© 1994 Optical Society of America

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References

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  1. A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 64, 447–452 (1974).
    [Crossref]
  2. D. R. Preuss, J. L. Gole, “Three-stage birefringent filter tuning smoothly over the visible region: the theoretical treatment and experimental design,” Appl. Opt. 19, 702–710 (1980).
    [Crossref] [PubMed]
  3. A. Yariv, P. Yen, Optical Waves in Crystals, Wiley Series on Pure and Applied Optics (Wiley, New York, 1984), pp. 121–154.
  4. S. A. Zenchenko, V. I. Ivanov, I. A. Malevich, S. F. Shulekin, “Switching of the emission wavelengths of a copper vapor laser,” Sov. J. Quantum Electron. 15, 124–125 (1985).
    [Crossref]
  5. B. V. Bonarev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosc. (USSR) 60, 501–504 (1986).
  6. X. Zhu, P. N. Kean, W. Sibbett, “Coupled-cavity mode-locking of a KCl:Tl laser using an erbium-doped optical fiber,” Opt. Lett. 14, 1192–1194 (1989).
    [Crossref] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 694–702.
  8. This formula is the same as Eq. (3) in Ref. 2, but be aware that there are typographical errors in Preuss and Gole’s paper. [Equations (A22) and (A13) are the correct forms of Eqs. (3) and (4), respectively.]
  9. Strictly speaking, cos θo here should be cos θ and θ is the average angle between θo and θe (see Ref. 7), but in much of the published literature it is often quoted as cos θo and when the incident angle equals the Brewster angle it is written as sin θi.
  10. S. Zhu, “Birefringent filter with tilted optic axis for tuning dye laser: theory and design,” Appl. Opt. 29, 410–415 (1990).
    [Crossref] [PubMed]
  11. J. Mentel, E. Schmidt, T. Mavrudis, “Birefringent filter with arbitrary orientation of the optic axis: an analysis of improved accuracy,” Appl. Opt. 31, 5022–5029 (1992).
    [Crossref] [PubMed]
  12. K. Naganuma, G. Lenz, E. P. Ippen, “Variable bandwidth birefringent filter for tunable femtosecond lasers,” IEEE J. Quantum Electron. 28, 2142–2150 (1992).
    [Crossref]

1992 (2)

J. Mentel, E. Schmidt, T. Mavrudis, “Birefringent filter with arbitrary orientation of the optic axis: an analysis of improved accuracy,” Appl. Opt. 31, 5022–5029 (1992).
[Crossref] [PubMed]

K. Naganuma, G. Lenz, E. P. Ippen, “Variable bandwidth birefringent filter for tunable femtosecond lasers,” IEEE J. Quantum Electron. 28, 2142–2150 (1992).
[Crossref]

1990 (1)

1989 (1)

1986 (1)

B. V. Bonarev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosc. (USSR) 60, 501–504 (1986).

1985 (1)

S. A. Zenchenko, V. I. Ivanov, I. A. Malevich, S. F. Shulekin, “Switching of the emission wavelengths of a copper vapor laser,” Sov. J. Quantum Electron. 15, 124–125 (1985).
[Crossref]

1980 (1)

1974 (1)

Bloom, A. L.

Bonarev, B. V.

B. V. Bonarev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosc. (USSR) 60, 501–504 (1986).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 694–702.

Gole, J. L.

Ippen, E. P.

K. Naganuma, G. Lenz, E. P. Ippen, “Variable bandwidth birefringent filter for tunable femtosecond lasers,” IEEE J. Quantum Electron. 28, 2142–2150 (1992).
[Crossref]

Ivanov, V. I.

S. A. Zenchenko, V. I. Ivanov, I. A. Malevich, S. F. Shulekin, “Switching of the emission wavelengths of a copper vapor laser,” Sov. J. Quantum Electron. 15, 124–125 (1985).
[Crossref]

Kean, P. N.

Kobtsev, S. M.

B. V. Bonarev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosc. (USSR) 60, 501–504 (1986).

Lenz, G.

K. Naganuma, G. Lenz, E. P. Ippen, “Variable bandwidth birefringent filter for tunable femtosecond lasers,” IEEE J. Quantum Electron. 28, 2142–2150 (1992).
[Crossref]

Malevich, I. A.

S. A. Zenchenko, V. I. Ivanov, I. A. Malevich, S. F. Shulekin, “Switching of the emission wavelengths of a copper vapor laser,” Sov. J. Quantum Electron. 15, 124–125 (1985).
[Crossref]

Mavrudis, T.

Mentel, J.

Naganuma, K.

K. Naganuma, G. Lenz, E. P. Ippen, “Variable bandwidth birefringent filter for tunable femtosecond lasers,” IEEE J. Quantum Electron. 28, 2142–2150 (1992).
[Crossref]

Preuss, D. R.

Schmidt, E.

Shulekin, S. F.

S. A. Zenchenko, V. I. Ivanov, I. A. Malevich, S. F. Shulekin, “Switching of the emission wavelengths of a copper vapor laser,” Sov. J. Quantum Electron. 15, 124–125 (1985).
[Crossref]

Sibbett, W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 694–702.

Yariv, A.

A. Yariv, P. Yen, Optical Waves in Crystals, Wiley Series on Pure and Applied Optics (Wiley, New York, 1984), pp. 121–154.

Yen, P.

A. Yariv, P. Yen, Optical Waves in Crystals, Wiley Series on Pure and Applied Optics (Wiley, New York, 1984), pp. 121–154.

Zenchenko, S. A.

S. A. Zenchenko, V. I. Ivanov, I. A. Malevich, S. F. Shulekin, “Switching of the emission wavelengths of a copper vapor laser,” Sov. J. Quantum Electron. 15, 124–125 (1985).
[Crossref]

Zhu, S.

Zhu, X.

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

K. Naganuma, G. Lenz, E. P. Ippen, “Variable bandwidth birefringent filter for tunable femtosecond lasers,” IEEE J. Quantum Electron. 28, 2142–2150 (1992).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

B. V. Bonarev, S. M. Kobtsev, “Calculation and optimization of a birefringent filter for a cw dye laser,” Opt. Spectrosc. (USSR) 60, 501–504 (1986).

Sov. J. Quantum Electron. (1)

S. A. Zenchenko, V. I. Ivanov, I. A. Malevich, S. F. Shulekin, “Switching of the emission wavelengths of a copper vapor laser,” Sov. J. Quantum Electron. 15, 124–125 (1985).
[Crossref]

Other (4)

A. Yariv, P. Yen, Optical Waves in Crystals, Wiley Series on Pure and Applied Optics (Wiley, New York, 1984), pp. 121–154.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 694–702.

This formula is the same as Eq. (3) in Ref. 2, but be aware that there are typographical errors in Preuss and Gole’s paper. [Equations (A22) and (A13) are the correct forms of Eqs. (3) and (4), respectively.]

Strictly speaking, cos θo here should be cos θ and θ is the average angle between θo and θe (see Ref. 7), but in much of the published literature it is often quoted as cos θo and when the incident angle equals the Brewster angle it is written as sin θi.

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Figures (3)

Fig. 1
Fig. 1

Geometrical structure of a BP: OO′, normal of the plate surface; OGBCO′ and OPH, incidence plane of the light ray; OF, projection of the optic axis on the plane surface; OFAEO, plain that contains the optic axis, vertical to the plate surface.

Fig. 2
Fig. 2

Beam splitting inside a BP: OO′, normal of the plate surface; OC, extraordinary refractive ray; OB, ordinary refractive ray; BD, distance between the exiting ordinary and extraordinary rays.

Fig. 3
Fig. 3

Variations in the three effective refractive-index differences Δn, Δn′, Δn″ as functions of the rotation angle ϕ. [The definitions of Δn, Δn′, Δn″ are in Eq. (21) and approximations (23) and (29), respectively. Our further calculations show that (Δn − Δn″)/Δn < 0.5% for any rotation angle.] The dashed line indicates the value of (n e n o ) for quartz material.

Equations (29)

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sin θ e = sin θ i n ,
δ = 2 π λ l [ ( n 2 - sin 2 θ i ) 1 / 2 - ( n o 2 - sin 2 θ i ) ] 1 / 2 ,
1 n 2 = [ 1 n o 2 - 1 n e 2 ] cos 2 γ + 1 n e 2 ,
cos γ = sin θ e cos ɛ cos ϕ + sin ɛ cos θ e ,
cos α = cos θ e cos ɛ cos ϕ - sin ɛ sin θ e ( 1 - cos 2 γ ) 1 / 2 .
M b = [ cos 2 α exp ( i δ e ) + sin 2 α exp ( i δ o ) sin α cos α [ exp ( i δ e ) - exp ( i δ o ) ] sin α cos α [ exp ( i δ e ) - exp ( i δ o ) ] sin 2 α exp ( i δ e ) + cos 2 α exp ( i δ o ) ] .
cos γ = sin θ e cos ϕ ,
cos α = cos θ e cos ϕ ( 1 - sin 2 θ e cos 2 ϕ ) 1 / 2 ,
1 n 2 = [ 1 n o 2 - 1 n e 2 ] sin 2 θ e cos 2 ϕ + 1 n e 2 ,
M b = ( 1 - cos 2 ϕ sin 2 θ e ) - 1 × [ cos 2 θ e cos 2 ϕ exp ( i δ e ) + sin 2 ϕ exp ( i δ o ) cos θ e sin ϕ cos ϕ [ exp ( i δ e ) - exp ( i δ o ) ] cos θ e sin ϕ cos ϕ [ exp ( i δ e ) - exp ( i δ o ) ] sin 2 ϕ exp ( i δ e ) + cos 2 θ e cos 2 ϕ exp ( i δ o ) ] .
n = n e [ 1 - ( 1 n o 2 - 1 n e 2 ) sin 2 θ i cos 2 ϕ ] 1 / 2 ,
sin θ e = sin θ i n e [ 1 - ( 1 n o 2 - 1 n e 2 ) sin 2 θ i cos 2 ϕ ] 1 / 2 .
M b = ( 1 - cos 2 ϕ sin 2 θ i q n e 2 ) - 1 × [ ( 1 - sin 2 θ i q n e 2 ) cos 2 ϕ exp ( i δ e ) + sin 2 ϕ exp ( i δ o ) ( 1 - sin 2 θ i q n e 2 ) 1 / 2 sin ϕ cos ϕ [ exp ( i δ e ) - exp ( i δ o ) ] ( 1 - sin 2 θ i q n e 2 ) 1 / 2 sin ϕ cos ϕ [ exp ( i δ e ) - exp ( i δ o ) ] sin 2 ϕ exp ( i δ e ) + ( 1 - sin 2 θ i q n e 2 ) cos 2 ϕ exp ( i δ o ) ] ,
q 1 - ( 1 n o 2 - 1 n e 2 ) sin 2 θ i cos 2 ϕ .
M b ( 1 - cos 2 ϕ sin 2 θ i n o 2 ) - 1 × [ ( 1 - sin 2 θ i n o 2 ) cos 2 ϕ exp ( i δ e ) + sin 2 ϕ exp ( i δ o ) ( 1 - sin 2 θ i n o 2 ) 1 / 2 sin ϕ cos ϕ [ exp ( i δ e ) - exp ( i δ o ) ] ( 1 - sin 2 θ i n o 2 ) 1 / 2 sin ϕ cos ϕ [ exp ( i δ e ) - exp ( i δ o ) ] sin 2 ϕ exp ( i δ e ) + ( 1 - sin 2 θ i n o 2 ) cos 2 ϕ exp ( i δ o ) ] .
δ e = 2 π λ ( n O C ¯ + D C ¯ ) = 2 π λ l ( n cos θ e - n o cos θ o + n o cos θ o ) ,
δ o = 2 π λ n o O B ¯ = 2 π λ l n o cos θ o ,
δ = δ e - δ o = 2 π λ l ( n cos θ e - n o cos θ o ) .
δ e = 2 π λ l [ n e ( 1 - sin 2 θ i sin 2 ϕ n e 2 - sin 2 θ i cos 2 ϕ n o 2 ) 1 / 2 - n o ( 1 - sin 2 θ i n o 2 ) 1 / 2 - n o ( 1 - sin 2 θ i n o 2 ) 1 / 2 ] ,
δ o = 2 π λ l n o ( 1 - sin 2 θ i n o 2 ) 1 / 2 ,
δ = 2 π λ l [ n e ( 1 - sin 2 θ i sin 2 ϕ n e 2 - sin 2 θ i cos 2 ϕ n o 2 ) 1 / 2 - n o ( 1 - sin 2 θ i n o 2 ) 1 / 2 ] = 2 π λ l Δ n .
δ e 2 π λ l [ n cos θ e ] = 2 π λ n e l [ 1 + sin 2 θ i cos 2 ϕ n e 2 - sin 2 θ i cos 2 ϕ n o 2 ( 1 - sin 2 θ i sin 2 ϕ n e 2 - sin 2 θ i cos 2 ϕ n o 2 ) 1 / 2 ] ,
δ 2 π λ l [ n e 1 + sin 2 θ i cos 2 ϕ n e 2 - sin 2     θ i cos 2 ϕ n o 2 ( 1 - sin 2 θ i sin 2 ϕ n e 2 - sin 2 θ i cos 2 ϕ n o 2 ) 1 / 2 - n o 1 ( 1 - sin 2 θ i n o 2 ) 1 / 2 ] = 2 π λ l Δ n .
l m = m λ Δ n ,
l m = m λ Δ n .
l m = m λ 8.859 × 10 - 3 = 0.1693 m ( mm ) ,
l m = m λ 5.081 × 10 - 3 = 0.2952 m ( mm ) .
δ 2 π λ l ( n e - n o ) sin 2 γ cos θ o
δ 2 π λ l ( n e - n o ) ( 1 - sin 2 θ i n o 2 ) 1 / 2 × { 1 - sin 2 θ i cos 2 ϕ n e 2 [ 1 - ( 1 n o 2 - 1 n e 2 ) sin 2 θ i cos 2 ϕ ] } = 2 π λ l Δ n ,

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