Abstract

Second harmonic generation (SHG) can be used as a technique for controlling the spatial mode structure of optical beams. We demonstrate experimentally the generation of higher-order spatial modes, and the possibility to use nonlinear phase matching as a predictable and robust technique for the conversion of transverse electric modes of the second harmonic output. The details of this effect are well described by our wave propagation models, which include mode dependent phase shifts. This is, to our knowledge, the first detailed study of spatial mode conversion in SHG. We discuss potential applications of this effect.

©2007 Optical Society of America

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References

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  1. V. KhokhlovRadiotek. Electron 6, 1116–1130 (1961).
  2. P.A. Franken, A.E. Hill, C.W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [Crossref]
  3. R.W. Boyd, Nonlinear Optics, (Academic Press, 1992).
  4. G. Gurzadian, D.N. Nikogosian, and V.G. Dmitriev, Handbook of Nonlinear Optical Crystals, (Springer, 2006).
  5. R. Paschotta, P. Kurz, R. Henking, S. Schiller, and J. Mlynek, “82% efficient continuous-wave frequency doubling of 1.06 mm with a monolithic MgO:LiNbO3 resonator,” Opt. Lett. 19, 1325–1327 (1994).
    [Crossref] [PubMed]
  6. G. Breitenbach, S. Schiller, and J. Mlynek, “81% Conversion efficiency in frequency-stable continuous-wave parametric oscillation,” J. Opt. Soc. Am. B 12, 2095–2101 (1995).
    [Crossref]
  7. A.E. Siegman, Lasers, (University Science, Mill Valley California, 1986).
  8. G.D. Boyd and D.A. Kleinman, “Parametric Interaction of Focused Gaussian Light Beams,” J. Appl. Phys.,  39, 3597–3639 (1968).
    [Crossref]
  9. J.T. Lue and C.J. Sun, “Limiting factors for parametric generation with focused high-order transverse- and multilongitudinal-mode lasers,” J. Opt. Soc. Am. B 4, 1958–1963 (1987).
    [Crossref]
  10. L.G. Gouy, “Sur la propagation anormale des ondes,” Compt. Rendue Acad. Sci. 111, 33–35 (1890).
  11. R.W. Boyd, “Intuitive Explanation of the Phase Anomaly of Focused Light Beams,” J. Opt. Soc. Am 70, 877–880 (1980).
    [Crossref]
  12. M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
    [Crossref]
  13. V. Delaubert, M. Lassen, D.R.N. Pulford, H-A. Bachor, and C.C. Harb, Generalized Boyd-Kleinman Calculation of High Order Transverse Modes Second Harmonic Generation, To be submitted (2007).
  14. T. Kasamatsu, H. Kubomura, and H Kan, “Numerical Simulation of Conversion Efficiency and Beam Quality Factor in Second Harmonic Generation with Diffraction and Pump Depletion,” Jpn. J. Appl. Phys.,  44, 8495–8497 (2005).
    [Crossref]
  15. S. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A,  21, 599–606 (1980).
    [Crossref]
  16. N. Lastzkaet. al., arXiv.org:physics/0611257, (2006).
  17. M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).
  18. J.R. Kurz, J. Huang, X. Xie, T. Saida, and M.M. Fejer, “Mode multiplexing in optical f requency mixers,” Opt. Lett. 29, 551–553 (2004).
    [Crossref] [PubMed]
  19. H. He, M.E.J. Friese, N.R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with phase singularity,” Phys. Rev. Lett. 75, 826829–826833 (1995).
    [Crossref]
  20. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and Steven Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–291 (1986).
    [Crossref] [PubMed]
  21. W.M. Lee, X. Yuan, and D. Tang, “Optical tweezers with multiple optical forces using double-hologram interference,” Opt. Express,  11, 199–207 (2002).
    [Crossref]

2007 (1)

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

2006 (1)

M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).

2005 (1)

T. Kasamatsu, H. Kubomura, and H Kan, “Numerical Simulation of Conversion Efficiency and Beam Quality Factor in Second Harmonic Generation with Diffraction and Pump Depletion,” Jpn. J. Appl. Phys.,  44, 8495–8497 (2005).
[Crossref]

2004 (1)

2002 (1)

1995 (2)

H. He, M.E.J. Friese, N.R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with phase singularity,” Phys. Rev. Lett. 75, 826829–826833 (1995).
[Crossref]

G. Breitenbach, S. Schiller, and J. Mlynek, “81% Conversion efficiency in frequency-stable continuous-wave parametric oscillation,” J. Opt. Soc. Am. B 12, 2095–2101 (1995).
[Crossref]

1994 (1)

1987 (1)

1986 (1)

1980 (2)

S. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A,  21, 599–606 (1980).
[Crossref]

R.W. Boyd, “Intuitive Explanation of the Phase Anomaly of Focused Light Beams,” J. Opt. Soc. Am 70, 877–880 (1980).
[Crossref]

1968 (1)

G.D. Boyd and D.A. Kleinman, “Parametric Interaction of Focused Gaussian Light Beams,” J. Appl. Phys.,  39, 3597–3639 (1968).
[Crossref]

1961 (2)

V. KhokhlovRadiotek. Electron 6, 1116–1130 (1961).

P.A. Franken, A.E. Hill, C.W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

1890 (1)

L.G. Gouy, “Sur la propagation anormale des ondes,” Compt. Rendue Acad. Sci. 111, 33–35 (1890).

Ashkin, A.

Bachor, H-A.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).

V. Delaubert, M. Lassen, D.R.N. Pulford, H-A. Bachor, and C.C. Harb, Generalized Boyd-Kleinman Calculation of High Order Transverse Modes Second Harmonic Generation, To be submitted (2007).

Bjorkholm, J. E.

Boyd, G.D.

G.D. Boyd and D.A. Kleinman, “Parametric Interaction of Focused Gaussian Light Beams,” J. Appl. Phys.,  39, 3597–3639 (1968).
[Crossref]

Boyd, R.W.

R.W. Boyd, “Intuitive Explanation of the Phase Anomaly of Focused Light Beams,” J. Opt. Soc. Am 70, 877–880 (1980).
[Crossref]

R.W. Boyd, Nonlinear Optics, (Academic Press, 1992).

Breitenbach, G.

Buchhave, P.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

Chu, Steven

Delaubert, V.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).

V. Delaubert, M. Lassen, D.R.N. Pulford, H-A. Bachor, and C.C. Harb, Generalized Boyd-Kleinman Calculation of High Order Transverse Modes Second Harmonic Generation, To be submitted (2007).

Dmitriev, V.G.

G. Gurzadian, D.N. Nikogosian, and V.G. Dmitriev, Handbook of Nonlinear Optical Crystals, (Springer, 2006).

Dziedzic, J. M.

Fabre, C.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

Fejer, M.M.

Franken, P.A.

P.A. Franken, A.E. Hill, C.W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Friese, M.E.J.

H. He, M.E.J. Friese, N.R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with phase singularity,” Phys. Rev. Lett. 75, 826829–826833 (1995).
[Crossref]

Gouy, L.G.

L.G. Gouy, “Sur la propagation anormale des ondes,” Compt. Rendue Acad. Sci. 111, 33–35 (1890).

Gurzadian, G.

G. Gurzadian, D.N. Nikogosian, and V.G. Dmitriev, Handbook of Nonlinear Optical Crystals, (Springer, 2006).

Harb, C.C.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).

V. Delaubert, M. Lassen, D.R.N. Pulford, H-A. Bachor, and C.C. Harb, Generalized Boyd-Kleinman Calculation of High Order Transverse Modes Second Harmonic Generation, To be submitted (2007).

He, H.

H. He, M.E.J. Friese, N.R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with phase singularity,” Phys. Rev. Lett. 75, 826829–826833 (1995).
[Crossref]

Heckenberg, N.R.

H. He, M.E.J. Friese, N.R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with phase singularity,” Phys. Rev. Lett. 75, 826829–826833 (1995).
[Crossref]

Henking, R.

Hill, A.E.

P.A. Franken, A.E. Hill, C.W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Huang, J.

Janousek, J.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

Kan, H

T. Kasamatsu, H. Kubomura, and H Kan, “Numerical Simulation of Conversion Efficiency and Beam Quality Factor in Second Harmonic Generation with Diffraction and Pump Depletion,” Jpn. J. Appl. Phys.,  44, 8495–8497 (2005).
[Crossref]

Kasamatsu, T.

T. Kasamatsu, H. Kubomura, and H Kan, “Numerical Simulation of Conversion Efficiency and Beam Quality Factor in Second Harmonic Generation with Diffraction and Pump Depletion,” Jpn. J. Appl. Phys.,  44, 8495–8497 (2005).
[Crossref]

Khokhlov, V.

V. KhokhlovRadiotek. Electron 6, 1116–1130 (1961).

Kleinman, D.A.

G.D. Boyd and D.A. Kleinman, “Parametric Interaction of Focused Gaussian Light Beams,” J. Appl. Phys.,  39, 3597–3639 (1968).
[Crossref]

Kubomura, H.

T. Kasamatsu, H. Kubomura, and H Kan, “Numerical Simulation of Conversion Efficiency and Beam Quality Factor in Second Harmonic Generation with Diffraction and Pump Depletion,” Jpn. J. Appl. Phys.,  44, 8495–8497 (2005).
[Crossref]

Kurz, J.R.

Kurz, P.

Lam, P.K.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).

Lassen, M.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).

V. Delaubert, M. Lassen, D.R.N. Pulford, H-A. Bachor, and C.C. Harb, Generalized Boyd-Kleinman Calculation of High Order Transverse Modes Second Harmonic Generation, To be submitted (2007).

Lastzka, N.

N. Lastzkaet. al., arXiv.org:physics/0611257, (2006).

Lee, W.M.

Lue, J.T.

Mlynek, J.

Nikogosian, D.N.

G. Gurzadian, D.N. Nikogosian, and V.G. Dmitriev, Handbook of Nonlinear Optical Crystals, (Springer, 2006).

Paschotta, R.

Peters, C.W.

P.A. Franken, A.E. Hill, C.W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Pulford, D.R.N.

V. Delaubert, M. Lassen, D.R.N. Pulford, H-A. Bachor, and C.C. Harb, Generalized Boyd-Kleinman Calculation of High Order Transverse Modes Second Harmonic Generation, To be submitted (2007).

Rubinsztein-Dunlop, H.

H. He, M.E.J. Friese, N.R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with phase singularity,” Phys. Rev. Lett. 75, 826829–826833 (1995).
[Crossref]

Saida, T.

Schiller, S.

Sheng, S.

S. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A,  21, 599–606 (1980).
[Crossref]

Siegman, A. E.

S. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A,  21, 599–606 (1980).
[Crossref]

Siegman, A.E.

A.E. Siegman, Lasers, (University Science, Mill Valley California, 1986).

Sun, C.J.

Tang, D.

Treps, N.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).

Wagner, K.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

Weinreich, G.

P.A. Franken, A.E. Hill, C.W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Xie, X.

Yuan, X.

Compt. Rendue Acad. Sci. (1)

L.G. Gouy, “Sur la propagation anormale des ondes,” Compt. Rendue Acad. Sci. 111, 33–35 (1890).

J. Appl. Phys. (1)

G.D. Boyd and D.A. Kleinman, “Parametric Interaction of Focused Gaussian Light Beams,” J. Appl. Phys.,  39, 3597–3639 (1968).
[Crossref]

J. of the Euro Opt. Soc.-RP. (1)

M. Lassen, V. Delaubert, C.C. Harb, P.K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. of the Euro Opt. Soc.-RP. 1, 06003–1–06003–7 (2006).

J. Opt. Soc. Am (1)

R.W. Boyd, “Intuitive Explanation of the Phase Anomaly of Focused Light Beams,” J. Opt. Soc. Am 70, 877–880 (1980).
[Crossref]

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

Jpn. J. Appl. Phys. (1)

T. Kasamatsu, H. Kubomura, and H Kan, “Numerical Simulation of Conversion Efficiency and Beam Quality Factor in Second Harmonic Generation with Diffraction and Pump Depletion,” Jpn. J. Appl. Phys.,  44, 8495–8497 (2005).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

S. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction,” Phys. Rev. A,  21, 599–606 (1980).
[Crossref]

Phys. Rev. Lett. (3)

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H-A. Bachor, P.K. Lam, N. Treps, P. Buchhave, C. Fabre, and C.C. Harb, “Tools for spatial multi-mode quantum information: modulation, detection and quantum correlations,” Phys. Rev. Lett. 98, 083602–1–083602–4 (2007).
[Crossref]

H. He, M.E.J. Friese, N.R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with phase singularity,” Phys. Rev. Lett. 75, 826829–826833 (1995).
[Crossref]

P.A. Franken, A.E. Hill, C.W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Other (6)

R.W. Boyd, Nonlinear Optics, (Academic Press, 1992).

G. Gurzadian, D.N. Nikogosian, and V.G. Dmitriev, Handbook of Nonlinear Optical Crystals, (Springer, 2006).

A.E. Siegman, Lasers, (University Science, Mill Valley California, 1986).

V. Delaubert, M. Lassen, D.R.N. Pulford, H-A. Bachor, and C.C. Harb, Generalized Boyd-Kleinman Calculation of High Order Transverse Modes Second Harmonic Generation, To be submitted (2007).

N. Lastzkaet. al., arXiv.org:physics/0611257, (2006).

V. KhokhlovRadiotek. Electron 6, 1116–1130 (1961).

Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1. Scheme for single pass SHG measurement illustrated for the case of a TEM10 fundamental pump mode.

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Fig. 2.
Fig. 2. SHG efficiency as a function of the focusing parameter ξ, in optimal phase matching conditions, for i) TEM00, ii) TEM10 and iii) TEM20 pump modes. All curves are normalized to the best conversion efficiency obtained for the TEM00 case. We have represented the three most relevant regimes to our analysis : ’thin crystal approximation regime’ corresponding to collimated beams, ’optimal focusing regime’, and ’tight focusing regime’. Two experimental points are represented in the optimal focusing region.

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Fig. 3.
Fig. 3. Modal decomposition of the SH field as a function of the focusing parameter ξ, in optimal phase matching conditions, for fundamental pump modes a) TEM10 and b) TEM20. Traces i), ii) and iii) correspond to the TEM00, TEM20 and TEM40 SH components, respectively. These results have been obtained with the analytical model.

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Fig. 4.
Fig. 4. Mode components of the SH fields as a function of the phase matching temperature, in optimum focusing conditions, for a) TEM103 and b) TEM20 pump modes. Traces i), ii) and iii) correspond to the TEM00, TEM20 and TEM40 SH components, respectively. These results have been obtained with the analytical model.

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Fig. 5.
Fig. 5. Normalized temperature dependence of second harmonic conversion efficiency for pump modes a) TEM00, b) TEM10 and c) TEM20. The experimental data are represented by the black dots with indicative error bars. The solid lines are made with both theoretical models. The decomposition of the SH mode is also shown for each temperature with dotted/dashed lines. Traces i), ii) and iii) correspond to the TEM00, TEM20 and TEM40 SH components, respectively. The maximum normalized theoretical conversion efficiency is 1 for a TEM00 pump, 0.50 for a TEM10 and 0.40 for a TEM20 pump mode. The pump beams were focused in all cases to a waist of w 0 = 35μm, inside a 20 mm long lithium niobate crystal.

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Fig. 6.
Fig. 6. SH profiles generated in the crystal, observed in the far field (FF), for three different phase matching temperatures, with optimal focusing conditions, a) TEM10 and b) TEM20 pump modes. The cross-section traces contain both the experimental data and the theoretical fits. The length of the crystal was l = 5 mm and a focusing parameter of ξ = l/6 was chosen. c) SH profiles for TEM10 and TEM20 collimated pump modes, i.e. the thin crystal approximation. Again the cross-section traces contain both the experimental data and the theoretical fits. The cross-section traces are made with the generalized Boyd-Kleinman analysis. The experimental parameters used are a focusing of ξ ≪ 1, and with optimal phase matching, ∆k = 0.

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Fig. 7.
Fig. 7. This movie has been recorded with a CCD camera and shows the SH profiles generated by the non-linear crystal. The phase matching temperatures is swept from below optimum phase matching to above optimum. The field is a TEM10 mode at 1064 nm and the SH filed moves from a predominantly TEM20 mode to a predominantly TEM00 mode. [Media 1]

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Fig. 8.
Fig. 8. a)Illustration of a scheme for temperature sensing and control of a laser frequency over a wide spectrum. The beam is first converted into a desired higher order mode, in this case a TEM10 mode, with a mode-converter. Pumping the nonlinear material with this beam then generates a SH multi-mode beam. The detection of these modes requires spatial detectors like array or CCD detectors arrays or mode separator associated with individual detectors. b) Normalized SH mode amplitude difference dependence, ∆A, as a function of temperature and wavelength for a TEM10 pump mode using the numerical model.

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

ℰ n ( x ) = ∑ i = 1 n Γ ni ν 2 i ( x )
H n , 2 p ( ξ , Δ k ) = 1 2 π ∫ − ξ ξ ( 1 − iτ ) n − p e − i lΔ kτ 2 ξ ( 1 + iτ ) n − p + 1 dτ
∇ t 2 E 1 − 2 jk 1 ∂ E 1 ∂ z = − 2 d ω 1 2 μ 0 E 1 * E 2 exp ( − j Δ kz )
∇ t 2 E 2 − 2 jk 2 ∂ E 2 ∂ z = − 2 d ω 2 2 μ 0 E 1 2 exp ( j Δ kz )

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