Abstract

The transmission characteristics of a birefringent quartz filter inserted under Brewster’s angle into a linear laser resonator is investigated theoretically and experimentally for arbitrary orientations of the optic axis. Exact expressions for filter curves of zero losses as functions of wavelength and orientation of the optic axis are given and confirmed by measurements. The Jones matrix of birefringent filter, which describes the propagation of plane waves through the filter with improved accuracy, is derived. Transmission curves are calculated, showing that the optimum selectivity strongly depends on the orientation of the optic axis. For a special filter the results are verified by measurements.

© 1992 Optical Society of America

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References

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  1. I. F. Mollenauer, J. C. White, eds., Tunable Lasers, Topics in Applied Physics (Springer, Berlin, 1987), pp. 13–15.
  2. A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 64, 447–452 (1974).
    [CrossRef]
  3. S. Zhu, “Birefringent filter with tilted optic axis for tuning dye lasers: theory and design,” Appl. Opt. 29, 410–415 (1990).
    [CrossRef] [PubMed]
  4. W. Luhs, B. Struve, G. Litfin, “Tunable multiline He–Ne laser,” Laser Optoelectron. 18, 319–357 (1986).
  5. F. Ortwein, J. Mentel, E. Schmidt, “A birefringent filter as a tuning element for a multiline He-Se+ laser,” J. Phys. D 22, 488–491 (1989).
    [CrossRef]
  6. E. Schmidt, J. Mentel, K. H. Krahn, “Three-color He-Se+ laser with optimized output power,” Appl. Opt. 25, 1383–1388 (1986).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), pp. 665–703.
  8. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 71–77, 152.
  9. N. Reich, J. Mentel, E. Schmidt, F. Gekat, “Determination of the spectrally resolved gain profile of He–Se+ laser lines from the beat frequency spectrum,” IEEE J. Quantum Electron. 27, 454–458 (1991).
    [CrossRef]
  10. G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
    [CrossRef]
  11. I. J. Hodgkinson, J. I. Vukusic, “Birefringent filters for tuning flash-lamp-pumped dye lasers: simplified theory and design,” Appl. Opt. 17, 1944–1948 (1978).
    [CrossRef] [PubMed]
  12. 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]
  13. A. Wünsche, “Neve Formeln für die Reflexion und Brechung des Lichtes an anisotropen Medien,” Ann. Phys. 25, 201–210 (1970).
    [CrossRef]

1991

N. Reich, J. Mentel, E. Schmidt, F. Gekat, “Determination of the spectrally resolved gain profile of He–Se+ laser lines from the beat frequency spectrum,” IEEE J. Quantum Electron. 27, 454–458 (1991).
[CrossRef]

1990

1989

F. Ortwein, J. Mentel, E. Schmidt, “A birefringent filter as a tuning element for a multiline He-Se+ laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

1986

W. Luhs, B. Struve, G. Litfin, “Tunable multiline He–Ne laser,” Laser Optoelectron. 18, 319–357 (1986).

E. Schmidt, J. Mentel, K. H. Krahn, “Three-color He-Se+ laser with optimized output power,” Appl. Opt. 25, 1383–1388 (1986).
[CrossRef] [PubMed]

1980

1978

1974

A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 64, 447–452 (1974).
[CrossRef]

G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
[CrossRef]

1970

A. Wünsche, “Neve Formeln für die Reflexion und Brechung des Lichtes an anisotropen Medien,” Ann. Phys. 25, 201–210 (1970).
[CrossRef]

Bloom, A. L.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), pp. 665–703.

Gekat, F.

N. Reich, J. Mentel, E. Schmidt, F. Gekat, “Determination of the spectrally resolved gain profile of He–Se+ laser lines from the beat frequency spectrum,” IEEE J. Quantum Electron. 27, 454–458 (1991).
[CrossRef]

Gole, J. L.

Hodgkinson, I. J.

Holtom, G.

G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
[CrossRef]

Krahn, K. H.

Litfin, G.

W. Luhs, B. Struve, G. Litfin, “Tunable multiline He–Ne laser,” Laser Optoelectron. 18, 319–357 (1986).

Luhs, W.

W. Luhs, B. Struve, G. Litfin, “Tunable multiline He–Ne laser,” Laser Optoelectron. 18, 319–357 (1986).

Mentel, J.

N. Reich, J. Mentel, E. Schmidt, F. Gekat, “Determination of the spectrally resolved gain profile of He–Se+ laser lines from the beat frequency spectrum,” IEEE J. Quantum Electron. 27, 454–458 (1991).
[CrossRef]

F. Ortwein, J. Mentel, E. Schmidt, “A birefringent filter as a tuning element for a multiline He-Se+ laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

E. Schmidt, J. Mentel, K. H. Krahn, “Three-color He-Se+ laser with optimized output power,” Appl. Opt. 25, 1383–1388 (1986).
[CrossRef] [PubMed]

Ortwein, F.

F. Ortwein, J. Mentel, E. Schmidt, “A birefringent filter as a tuning element for a multiline He-Se+ laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

Preuss, D. R.

Reich, N.

N. Reich, J. Mentel, E. Schmidt, F. Gekat, “Determination of the spectrally resolved gain profile of He–Se+ laser lines from the beat frequency spectrum,” IEEE J. Quantum Electron. 27, 454–458 (1991).
[CrossRef]

Schmidt, E.

N. Reich, J. Mentel, E. Schmidt, F. Gekat, “Determination of the spectrally resolved gain profile of He–Se+ laser lines from the beat frequency spectrum,” IEEE J. Quantum Electron. 27, 454–458 (1991).
[CrossRef]

F. Ortwein, J. Mentel, E. Schmidt, “A birefringent filter as a tuning element for a multiline He-Se+ laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

E. Schmidt, J. Mentel, K. H. Krahn, “Three-color He-Se+ laser with optimized output power,” Appl. Opt. 25, 1383–1388 (1986).
[CrossRef] [PubMed]

Struve, B.

W. Luhs, B. Struve, G. Litfin, “Tunable multiline He–Ne laser,” Laser Optoelectron. 18, 319–357 (1986).

Teschke, O.

G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
[CrossRef]

Vukusic, J. I.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), pp. 665–703.

Wünsche, A.

A. Wünsche, “Neve Formeln für die Reflexion und Brechung des Lichtes an anisotropen Medien,” Ann. Phys. 25, 201–210 (1970).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 71–77, 152.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 71–77, 152.

Zhu, S.

Ann. Phys.

A. Wünsche, “Neve Formeln für die Reflexion und Brechung des Lichtes an anisotropen Medien,” Ann. Phys. 25, 201–210 (1970).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

N. Reich, J. Mentel, E. Schmidt, F. Gekat, “Determination of the spectrally resolved gain profile of He–Se+ laser lines from the beat frequency spectrum,” IEEE J. Quantum Electron. 27, 454–458 (1991).
[CrossRef]

G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. D

F. Ortwein, J. Mentel, E. Schmidt, “A birefringent filter as a tuning element for a multiline He-Se+ laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

Laser Optoelectron.

W. Luhs, B. Struve, G. Litfin, “Tunable multiline He–Ne laser,” Laser Optoelectron. 18, 319–357 (1986).

Other

I. F. Mollenauer, J. C. White, eds., Tunable Lasers, Topics in Applied Physics (Springer, Berlin, 1987), pp. 13–15.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), pp. 665–703.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 71–77, 152.

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Figures (9)

Fig. 1
Fig. 1

Orientation of the refracted eo beam given by keo and the orientation of the OA given by eζ within the birefringent filter.

Fig. 2
Fig. 2

The wavelength λ as a function of the tuning angle δ for phase differences 2πN, which constitutes filter curves of zero losses. Thickness of the filter, d = 800 μm. The inclination ∊ of the OA is (a) ∊ = 0°, (b) ∊ = 20°, (c) ∊ = 57°, (d) ∊ = 80°. The selectivity of the birefringent filter is zero for δz; it has a maximum for δm+ and δm.

Fig. 3
Fig. 3

Zero-loss curves of a filter with d = 1817 μm, ∊ = 56.81°, and α = 56.68° from theory (curves) and experiment (circles).

Fig. 4
Fig. 4

Transmission as a function of δ of a BF and a subsequent polarizer that is transparent in the plane of incidence for a wave with λ = 522.75 nm linearly polarized in the plane of incidence. d = 1817 μm, ∊ = 56.81°, and α = 56.68°. Curves, theory; circles, experiment.

Fig. 5
Fig. 5

Transmissions T1 and T2 as functions of the tuning angle δ from the eigenvalue of the Jones matrix of a BF in a linear resonator with two Brewster windows. Filter parameters: d = 800 μm, ∊ = 67°, α = 57°, and λ = 522.75 nm.

Fig. 6
Fig. 6

Transmission T1 as a function of δ of a BF with d = 800 μm in a resonator with two Brewster windows. The inclination of the OA is (a) ∊ = 0°, (b) = 20°, (c) ∊ = 57°, (d) ∊ = 80°.

Fig. 7
Fig. 7

Tuning angles δm+ and δm of the optimum selectivity and δz of zero selectivity as a function of the inclination of ∊ of the OA.

Fig. 8
Fig. 8

Transmission as a function of δ of a BF with d = 1817 μm and ∊ = 56.81° in a resonator of a 441-nm Cd laser. Curves, theory; open circles, experiment.

Fig. 9
Fig. 9

Three-dimensional representation of the losses of a BF in a linear resonator as a function of δ and λ. Filter parameters: d = 400 μm and in (a), ∊ = 0°; (b), ∊ = 20°; in (c), ∊ = 57°; in (d), ∊ = 67°.

Equations (41)

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1 n eo 2 = s ξ 2 + s η 2 n eo 2 - n o 2 + s ζ 2 n eo 2 - n e 2 ,
n eo 2 s ζ 2 ( n e 2 - n o 2 ) = ( n e 2 - n eo 2 ) n o 2 .
s = e y cos β eo + e z sin β eo ,
e ζ = e x cos sin δ + e y sin + e z cos cos δ ,
s ζ = e ζ s = sin cos β eo + cos cos δ sin β eo .
( sin m + cos cos δ sin α ) 2 ( n e 2 - n o 2 ) = ( n e 2 - m 2 - sin 2 α ) n o 2 .
n eo = [ m 2 ( , δ , α ) + sin 2 α ] 1 / 2 ,
ϕ = 2 π λ d ( n eo cos β eo - n o cos β o ) ,
cos δ sin α = ± n o cos { 1 - n o 2 - sin 2 α n e 2 - n o 2 × [ ( N λ d ( n o 2 - sin 2 α ) 1 / 2 + 1 ) 2 - 1 ] } 1 / 2 - tan ( n o 2 - sin 2 α ) 1 / 2 × ( N λ d ( n o 2 - sin 2 α ) 1 / 2 + 1 ) .
tan e = sin α ( n e 2 - sin 2 α ) 1 / 2 .
D i = D i o + D i e
D i = D i o + D i e ,
D t o D i o = n o 2 2 sin β o cos α sin ( α + β o ) = 1 a 3 ,
D t e D i e = n eo 2 2 sin β eo cos α sin ( α + β eo ) = 1 a 4 ,
D t o D i o = 1 a 3 cos ( α - β o ) = 1 a 5 ,
D t e D i e = n eo 2 sin β eo cos α sin ( α + β eo ) cos ( α - β eo ) + tan τ sin 2 β eo = 1 a 6 .
cos τ = E t e D t e E t e D t e .
E t e s ξ e ξ + s η e η n eo 2 - n o 2 + s ζ e ζ n eo 2 - n e 2 ,
D t e o n o 2 ( s ξ e ξ + s η e η ) n eo 2 - n o 2 + o n e 2 s ζ e ζ n eo 2 - n e 2 .
e ζ D t o = 0 D t o = D t o cos sin δ cos cos δ cos β o - sin sin β o = D t o a 7 .
D t e ( e ζ × s ) = 0 D t e = D t e sin β eo sin - cos β eo cos cos δ cos sin δ = D t e a 8
( D t o D t o ) = 1 N ( a 6 a 8 - a 4 a 6 a 7 a 8 - a 4 a 7 ) ( D i D i ) ,
( D t e D t e ) = 1 N ( - a 5 a 7 a 3 - a 5 a 7 a 8 a 3 a 8 ) ( D i D i ) ,
N = a 3 a 6 a 8 - a 4 a 5 a 7 .
D e o D t o = 2 sin α cos β o n o 2 sin ( α + β o ) = a 9 ,
D e e D t e = a 10 ,
D e o D t o = a 11
D e e D t e = 2 sin α cos β eo n eo 2 [ sin ( α + β eo ) cos ( α - β eo ) ] = a 12 .
( D e o D e o ) = ( a 9 0 0 a 11 ) ( D t o D t o ) ,
( D e e D e e ) = ( a 10 0 0 a 12 ) ( D t 2 D t e ) .
( D e D e ) = ( D e o D e o ) + exp ( j ϕ ) ( D e e D e e ) .
D c = A D i , A = 1 N { a 6 a 8 a 9 - a 5 a 7 a 10 exp ( j ϕ ) - a 4 a 9 + a 3 a 10 exp ( j ϕ ) a 7 a 8 [ a 6 a 11 - a 5 a 12 exp ( j ϕ ) ] - a 4 a 7 a 11 + a 3 a 8 a 12 exp ( j ϕ ) } .
D e 2 = A 22 2 D i 2 .
B 11 = sin 2 α w sin 2 β w sin 2 ( α w + β w ) , B 22 = B 11 cos 2 ( α w - β w ) .
( B 4 A 2 - t ) D = 0 ,
T 1 , 2 = t 1 , 2 t 1 , 2 * .
a 7 = 0 , a 8 = 0 , respectively .
cos δ z = tan n eo .
a 7 = - a 8
cos δ m = sin α B cos α B 1 + sin 2 α B tan ± [ ( sin α B cos α B 1 + sin 2 α B tan ) 2 + 1 - cos 2 α B tan 2 1 + sin 2 α B ] 1 / 2 .
tan s = ( 1 + sin 2 α B ) 1 / 2 cos α B ,

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