Abstract

Polarisation eigenmode theory is well established for laser cavities in which the principal axes for gain and polarisation elements are parallel. Here we generalise the theory to include the case for gain axes at arbitrary angle to the birefringence, which is the case for Raman lasers based on cubic-class gain crystals that contain stress-induced birefringence. The theory describes regimes dominated by gain, linear or circular birefringence, and the intermediate regime in which elliptically polarised output modes are obtained. Previously reported behaviour for diamond Raman lasers are found to be in accord with the findings. Design criteria are obtained to enable prediction of polarisation behaviour as functions of birefringence and resonator design.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2018 (1)

2016 (1)

2015 (2)

S. Zhang, Y. Tan, and S. Zhang, “Effect of gain and loss anisotropy on polarization dynamics in Nd:YAG microchip lasers,” J. Opt. 17, 045703 (2015).
[Crossref]

R. J. Williams, J. Nold, M. Strecker, O. Kitzler, A. McKay, T. Schreiber, and R. P. Mildren, “Efficient Raman frequency conversion of high-power fiber lasers in diamond,” Laser & Photonics Rev. 9, 405–411 (2015).
[Crossref]

2012 (1)

2010 (1)

2000 (1)

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, “Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal,” IEEE J. Quantum Electron. 36, 228–235 (2000).
[Crossref]

1999 (2)

T. Basiev, A. Sobol, P. Zverev, L. Ivleva, V. Osiko, and R. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11, 307 – 314 (1999).
[Crossref]

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped vcsels,” IEEE J. Quantum Electron. 35, 342–351 (1999).
[Crossref]

1997 (1)

J. Martin-Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[Crossref]

1996 (1)

1995 (1)

M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[Crossref] [PubMed]

1994 (1)

1972 (1)

V. Y. Molchanov and G. V. Skrotskii, “Matrix method for the calculation of the polarization eigenstates of anisotrpic optical resonators,” Soviet J. Quantum Electron. 1, 3–26 (1972).
[Crossref]

1970 (1)

W. Koechner and D. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557–566 (1970).
[Crossref]

1966 (1)

1965 (1)

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, 1787–1805 (1965).
[Crossref]

1964 (1)

R. Loudon, “The Raman effect in crystals,” Adv. Phys. 13, 423–482 (1964).
[Crossref]

1948 (1)

1941 (1)

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[Crossref]

Alouini, M.

Baili, G.

Balle, S.

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped vcsels,” IEEE J. Quantum Electron. 35, 342–351 (1999).
[Crossref]

Basiev, T.

T. Basiev, A. Sobol, P. Zverev, L. Ivleva, V. Osiko, and R. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11, 307 – 314 (1999).
[Crossref]

Bloembergen, N.

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, 1787–1805 (1965).
[Crossref]

Conroy, R. S.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, “Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal,” IEEE J. Quantum Electron. 36, 228–235 (2000).
[Crossref]

Dolfi, D.

Feng, Q.

M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[Crossref] [PubMed]

Friel, G. J.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, “Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal,” IEEE J. Quantum Electron. 36, 228–235 (2000).
[Crossref]

Frougier, J.

Gahl, A.

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped vcsels,” IEEE J. Quantum Electron. 35, 342–351 (1999).
[Crossref]

George, J.-M.

Hecht, E.

E. Hecht, Optics(Addison-Wesley Longman, Inc., 2002).

Helmfrid, S.

Hurwitz, H.

Ivleva, L.

T. Basiev, A. Sobol, P. Zverev, L. Ivleva, V. Osiko, and R. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11, 307 – 314 (1999).
[Crossref]

Jackson, S. D.

Jasbeer, H.

Joly, A.

Jones, C. R.

Kemp, A. J.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, “Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal,” IEEE J. Quantum Electron. 36, 228–235 (2000).
[Crossref]

Kitzler, O.

Koechner, W.

W. Koechner and D. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557–566 (1970).
[Crossref]

Kogelnik, H.

Lake, T. K.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, “Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal,” IEEE J. Quantum Electron. 36, 228–235 (2000).
[Crossref]

Li, T.

Lin, J.

Loudon, R.

R. Loudon, “The Raman effect in crystals,” Adv. Phys. 13, 423–482 (1964).
[Crossref]

Martin-Regalado, J.

J. Martin-Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[Crossref]

McKay, A.

Miguel, M. S.

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped vcsels,” IEEE J. Quantum Electron. 35, 342–351 (1999).
[Crossref]

J. Martin-Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[Crossref]

Mildren, R. P.

Molchanov, V. Y.

V. Y. Molchanov and G. V. Skrotskii, “Matrix method for the calculation of the polarization eigenstates of anisotrpic optical resonators,” Soviet J. Quantum Electron. 1, 3–26 (1972).
[Crossref]

Moloney, J. V.

M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[Crossref] [PubMed]

Nold, J.

R. J. Williams, J. Nold, M. Strecker, O. Kitzler, A. McKay, T. Schreiber, and R. P. Mildren, “Efficient Raman frequency conversion of high-power fiber lasers in diamond,” Laser & Photonics Rev. 9, 405–411 (2015).
[Crossref]

Osiko, V.

T. Basiev, A. Sobol, P. Zverev, L. Ivleva, V. Osiko, and R. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11, 307 – 314 (1999).
[Crossref]

Piper, J. A.

Powell, R.

T. Basiev, A. Sobol, P. Zverev, L. Ivleva, V. Osiko, and R. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11, 307 – 314 (1999).
[Crossref]

Prati, F.

J. Martin-Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[Crossref]

Rice, D.

W. Koechner and D. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557–566 (1970).
[Crossref]

Sabella, A.

San Miguel, M.

M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[Crossref] [PubMed]

Sarang, S.

Schreiber, T.

R. J. Williams, J. Nold, M. Strecker, O. Kitzler, A. McKay, T. Schreiber, and R. P. Mildren, “Efficient Raman frequency conversion of high-power fiber lasers in diamond,” Laser & Photonics Rev. 9, 405–411 (2015).
[Crossref]

Shen, Y. R.

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, 1787–1805 (1965).
[Crossref]

Sinclair, B. D.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, “Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal,” IEEE J. Quantum Electron. 36, 228–235 (2000).
[Crossref]

Skrotskii, G. V.

V. Y. Molchanov and G. V. Skrotskii, “Matrix method for the calculation of the polarization eigenstates of anisotrpic optical resonators,” Soviet J. Quantum Electron. 1, 3–26 (1972).
[Crossref]

Sobol, A.

T. Basiev, A. Sobol, P. Zverev, L. Ivleva, V. Osiko, and R. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11, 307 – 314 (1999).
[Crossref]

Strecker, M.

R. J. Williams, J. Nold, M. Strecker, O. Kitzler, A. McKay, T. Schreiber, and R. P. Mildren, “Efficient Raman frequency conversion of high-power fiber lasers in diamond,” Laser & Photonics Rev. 9, 405–411 (2015).
[Crossref]

Tan, Y.

S. Zhang, Y. Tan, and S. Zhang, “Effect of gain and loss anisotropy on polarization dynamics in Nd:YAG microchip lasers,” J. Opt. 17, 045703 (2015).
[Crossref]

Tatsuno, K.

Williams, R. J.

H. Jasbeer, R. J. Williams, O. Kitzler, A. McKay, S. Sarang, J. Lin, and R. P. Mildren, “Birefringence and piezo-Raman analysis of single crystal CVD diamond and effects on Raman laser performance,” J. Opt. Soc. Am. B 33, B56–B64 (2016).
[Crossref]

R. J. Williams, J. Nold, M. Strecker, O. Kitzler, A. McKay, T. Schreiber, and R. P. Mildren, “Efficient Raman frequency conversion of high-power fiber lasers in diamond,” Laser & Photonics Rev. 9, 405–411 (2015).
[Crossref]

Zhang, S.

S. Zhang, Y. Tan, and S. Zhang, “Effect of gain and loss anisotropy on polarization dynamics in Nd:YAG microchip lasers,” J. Opt. 17, 045703 (2015).
[Crossref]

S. Zhang, Y. Tan, and S. Zhang, “Effect of gain and loss anisotropy on polarization dynamics in Nd:YAG microchip lasers,” J. Opt. 17, 045703 (2015).
[Crossref]

Zverev, P.

T. Basiev, A. Sobol, P. Zverev, L. Ivleva, V. Osiko, and R. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11, 307 – 314 (1999).
[Crossref]

Adv. Phys. (1)

R. Loudon, “The Raman effect in crystals,” Adv. Phys. 13, 423–482 (1964).
[Crossref]

Appl. Opt. (2)

IEEE J. Quantum Electron. (4)

J. Martin-Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[Crossref]

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped vcsels,” IEEE J. Quantum Electron. 35, 342–351 (1999).
[Crossref]

W. Koechner and D. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557–566 (1970).
[Crossref]

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, “Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal,” IEEE J. Quantum Electron. 36, 228–235 (2000).
[Crossref]

J. Opt. (1)

S. Zhang, Y. Tan, and S. Zhang, “Effect of gain and loss anisotropy on polarization dynamics in Nd:YAG microchip lasers,” J. Opt. 17, 045703 (2015).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (2)

Laser & Photonics Rev. (1)

R. J. Williams, J. Nold, M. Strecker, O. Kitzler, A. McKay, T. Schreiber, and R. P. Mildren, “Efficient Raman frequency conversion of high-power fiber lasers in diamond,” Laser & Photonics Rev. 9, 405–411 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Opt. Mater. (1)

T. Basiev, A. Sobol, P. Zverev, L. Ivleva, V. Osiko, and R. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11, 307 – 314 (1999).
[Crossref]

Phys. Rev. (1)

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, 1787–1805 (1965).
[Crossref]

Phys. Rev. A (1)

M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[Crossref] [PubMed]

Soviet J. Quantum Electron. (1)

V. Y. Molchanov and G. V. Skrotskii, “Matrix method for the calculation of the polarization eigenstates of anisotrpic optical resonators,” Soviet J. Quantum Electron. 1, 3–26 (1972).
[Crossref]

Other (1)

E. Hecht, Optics(Addison-Wesley Longman, Inc., 2002).

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Figures (8)

Fig. 1
Fig. 1 Crystallographic directions in diamond with respect to usual propagation direction used in Raman lasers and an example of birefringence fast f and slow s axis orientation.
Fig. 2
Fig. 2 (a) χ as a function of pump and Stokes polarisation angles. Full black and white lines show maxima γ1 and minima γ2, respectively. (b) χcross as a function of pump and Stokes polarisation angles.
Fig. 3
Fig. 3 Output polarisations shown as ellipses as a function of linear pump polarisations angles and birefringence phase shift. Top axis indicates corresponding Γ. The colours red/blue indicate left/right handedness. R = 99% and τ = 0°, 20°, 45°, and 70° in (a), (b), (c), and (d), respectively.
Fig. 4
Fig. 4 Output polarisations shown as ellipses as a function of induced rotation angles and birefringence phase shift. (a) and (b) show the polarisation behaviour for R = 99% Γ = 3.8 × 10−3, τ = 20°, and α = 35° for a linear and ring cavity, respectively.
Fig. 5
Fig. 5 (a) Normalised gain as a function of pump and birefringence fast axis angle τ for strong birefringence ξ = 15 mrad and low gain R = 99%. Black dashed lines indicate gain maxima. White stripes indicate the region where the output polarisation is parallel to the slow birefringence axis. (b) Gain as a function of birefringence fast axis angle for pumping polarisations parallel to the [110], [111], [001], and [ 1 ¯ 1 ¯ 0 ]. The lineschange from full to dashed for the Stokes polarisation parallel to fast and slow axis, respectively.
Fig. 6
Fig. 6 Ellipticity of the Stokes mode as a function of R and birefringence phase shift ξ for pump polarisation angles of (a) 5° and (b) 90°. The dashed black line indicates the magnitude of the gain difference Γ (also shown asthe top axis) as a function of R on the ξ axis. The insets show magnified regions for typical values observed in diamond and used in cw DRLs.
Fig. 7
Fig. 7 Measured Stokes polarisation angles (full thick lines) compared to modelled Stokes polarisations (dashed). (i) this work, R = 0.3, ξ = 0.3 rad, τ unknown; (ii), (iii), (iv) from [3], R = 0.99 (R used was 99.5% plus 0.5% additional roundtrip loss), ξ = 0.06 rad, 0.2 rad, 0.27 rad, τ = 45°, 71°, 1.7°, and θ = 1.25°, 9.15°, 6.6°, respectively, measured by Mueller polarimetry.
Fig. 8
Fig. 8 Stokes output polarisation angle as a function of pump polarisation angle. Dashed lines show fast f and slow s axis of the linear birefringence determined by Mueller polarimetry. Blue circles and full line show measuredand modelled Stokes polarisation angles for given pump polarisation angles for Γ ≫ ξ. The triangle, square and diamond symbols show measurements (ii), (iii), (iv) corresponding to spots A, C, D in [3] respectively and show the opposite case of Γ ≪ ξ for varying values of birefringence and its orientation. Full lines of the same colour are corresponding modelled results and dashed lines indicate the orientation of the linear birefringence.

Equations (13)

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B = ( e i ξ 2 0 0 e i ξ 2 ) ,
G = ( e G 0 γ 1 0 0 e G 0 γ 2 ) ,
R e 2 G 0 γ 1 = 1
G 0 = 1 γ 1 ln  ( R 1 2 ) ,
G = ( e Γ 2 0 0 e Γ 2 ) ,
M = G R ( τ ) B R ( τ ) R ( τ ) B R ( τ ) G ,
d S k d z = g 2 i j l χ i j k l ( 3 ) S i P j P l * d P l d z = g 2 η i j k χ i j k l ( 3 ) S i P j S k * ,
d S 1 d z = g 2 ( χ 1 1 1 1 ( 3 ) S 1 P 1 P 1 * + χ 2 1 1 1 ( 3 ) S 2 P 1 P 1 * ) d S 2 d z = g 2 ( χ 2 1 2 1 ( 3 ) S 2 P 1 P 1 * + χ 1 1 2 1 ( 3 ) S 1 P 1 P 1 * ) d P 1 d z = g 2 η i , k = 1 , 2 χ i 1 k 1 ( 3 ) S i P 1 S k * .
χ ( α , β ) = i = 1 , 2 , 3 ( e s ( β ) R i e p ( α ) ) ( e s ( β ) R i e p ( α ) ) *
χ cross ( α , β ) = i = 1 , 2 , 3 ( e s ( β ) R i e p ( α ) ) ( e s ( β + π / 2 ) R i e p ( α ) ) * ,
d S 1 d z = g γ 1 2 S 1 P 1 P 1 * d P 1 d z = g 2 η ( γ 1 S 1 P 1 S 1 * + γ 2 S 2 P 1 S 2 * ) .
γ 1 = cos 2 { 1 2 arctan   [ 2 cot   ( α ) ] } cos 2 ( α ) + sin 2 { α + 1 2 arctan   [ 2 cot   ( α ) ] } γ 2 = cos 2 { π 2 + 1 2 arctan   [ 2 cot   ( α ) ] } cos 2 ( α ) + sin 2 { π 2 + α + 1 2 arctan   [ 2 cot   ( α ) ] } ,
J = ( cos  T 2 i ξ c o s   ( 2 τ ) Γ T sin  T 2 Θ i ξ s i n   ( 2 τ ) T sin  T 2 Θ + i ξ s i n   ( 2 τ ) T sin  T 2 cos  T 2 + i ξ c o s   ( 2 τ ) Γ T sin  T 2 ) ,

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