Abstract

In this paper, a detailed description of the optical coupling into a Whispering Gallery Mode (WGM) resonator through a prism via frustrated total internal reflection (FTIR) is presented. The problem is modeled as three media with planar interfaces and closed expressions for FTIR are given. Then, the curvature of the resonator is taken into account and the mode overlap is theoretically studied. A new analytical expression giving the optimal geometry of a disc-shaped or ring-shaped resonator for maximizing the intra-cavity circulating power is presented. Such expression takes into consideration the spatial distribution of the WGM at the surface of the resonator, thus being more accurate than the currently used expressions. It also takes into account the geometry of the prism. It is shown an improvement in the geometry values used with the current expressions of about 30%. The reason why the pump laser signal can be seen in experiments under critical coupling is explained on this basis. Then, the conditions required for exciting the highest possible optical power inside the resonator are obtained. The aim is to achieve a highly-efficient up-conversion of a THz signal into the optical domain via the second-order nonlinearity of the resonator material.

© 2016 Optical Society of America

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References

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  1. JPL, “BICEP2. II. Experiment and three-year data set,” Astrophys. J. 792(1), 62 (2014).
    [Crossref]
  2. G. Carpintero, E. G. Muñoz, H. Hartnagel, S. Preu, and A. Raisanen, Semiconductor TeraHertz Technology: Devices and Systems at Room Temperature Operation (Wiley-IEEE Press, 2015).
    [Crossref]
  3. B. S. Karasik and A. V. Sergeev, “THz Hot-Electron Photon Counter,” IEEE Trans. Appl. Superconductivity 15(2), 618–621 (2005).
    [Crossref]
  4. D.V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Towards Counting microwave photons at room temperature,” Laser Phys. Lett. 6(2), 129–134 (2009).
    [Crossref]
  5. A. B. Shehata, C. Scarcella, and A. Tosi, “Photon counting module based on InGaAs/InP Single-Photon Avalanche Diodes for near-infrared counting up to 1.7 μm,” in Ph.D. Research in Microelectronics and Electronics (PRIME), 2011 7th Conference on, 177–180 (2011).
    [Crossref]
  6. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).
  7. A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, “Efficient microwave to optical photon conversion: an electro-optical realization,” Optica 3(6), 597–604 (2016).
    [Crossref]
  8. D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, “Nonlinear and Quantum Optics with Whispering Gallery Resonators,” arXiv:1605.07972 [physics, Physics:quant-Ph] (2016).
  9. F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).
  10. P.E. Powers., Fundamentals of Nonlinear Optics (CRC Press, 2011).
  11. M. R. Foreman, F. Sedlmeir, H. G. L. Schwefel, and G. Leuchs, “Dielectric tuning and coupling of whispering gallery modes using an anisotropic prism,” J. Opt. Soc. Am. B33, 2177 (2016).
  12. M. L. Gorodetsky and V. S. Ilchenko, “Optical Microsphere Resonators: Optimal Coupling to High-Q Whispering-Gallery Modes,” J. Opt. Soc. Am. B 16(1) 147–154 (1999).
    [Crossref]
  13. A. N. Oraevsky, “Whispering-Gallery waves,” Quantum Electron. 32(5), 377–400 (2002).
    [Crossref]
  14. B.E. Little, J. P. Laine, and H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17(4), 704–715 (1999).
    [Crossref]
  15. A. Yariv., “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
    [Crossref]
  16. I. Breunig, B. Sturman, F. Sedlmeir, H. G. L. Schwefel, and K. Buse, “Whispering gallery modes at the rim of an axisymmetric optical resonator: Analytical versus numerical description and comparison with experiment,” Opt. Express 21(25), 30683–30692 (2013).
    [Crossref]
  17. D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of sub-THz radiation in a high-Q whispering gallery resonator,” Opt. Lett. 34(6), 713–715 (2009).
    [Crossref] [PubMed]

2016 (1)

2014 (1)

JPL, “BICEP2. II. Experiment and three-year data set,” Astrophys. J. 792(1), 62 (2014).
[Crossref]

2013 (1)

2009 (2)

D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of sub-THz radiation in a high-Q whispering gallery resonator,” Opt. Lett. 34(6), 713–715 (2009).
[Crossref] [PubMed]

D.V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Towards Counting microwave photons at room temperature,” Laser Phys. Lett. 6(2), 129–134 (2009).
[Crossref]

2005 (1)

B. S. Karasik and A. V. Sergeev, “THz Hot-Electron Photon Counter,” IEEE Trans. Appl. Superconductivity 15(2), 618–621 (2005).
[Crossref]

2002 (1)

A. N. Oraevsky, “Whispering-Gallery waves,” Quantum Electron. 32(5), 377–400 (2002).
[Crossref]

2000 (1)

A. Yariv., “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

1999 (2)

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

Breunig, I.

Buse, K.

Carpintero, G.

G. Carpintero, E. G. Muñoz, H. Hartnagel, S. Preu, and A. Raisanen, Semiconductor TeraHertz Technology: Devices and Systems at Room Temperature Operation (Wiley-IEEE Press, 2015).
[Crossref]

Collodo, M. C.

Fink, J. M.

Foreman, M. R.

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

M. R. Foreman, F. Sedlmeir, H. G. L. Schwefel, and G. Leuchs, “Dielectric tuning and coupling of whispering gallery modes using an anisotropic prism,” J. Opt. Soc. Am. B33, 2177 (2016).

Gorodetsky, M. L.

Hartnagel, H.

G. Carpintero, E. G. Muñoz, H. Hartnagel, S. Preu, and A. Raisanen, Semiconductor TeraHertz Technology: Devices and Systems at Room Temperature Operation (Wiley-IEEE Press, 2015).
[Crossref]

Haus, H. A.

Ilchenko, V. S.

Karasik, B. S.

B. S. Karasik and A. V. Sergeev, “THz Hot-Electron Photon Counter,” IEEE Trans. Appl. Superconductivity 15(2), 618–621 (2005).
[Crossref]

Laine, J. P.

Leuchs, G.

A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, “Efficient microwave to optical photon conversion: an electro-optical realization,” Optica 3(6), 597–604 (2016).
[Crossref]

D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, “Nonlinear and Quantum Optics with Whispering Gallery Resonators,” arXiv:1605.07972 [physics, Physics:quant-Ph] (2016).

M. R. Foreman, F. Sedlmeir, H. G. L. Schwefel, and G. Leuchs, “Dielectric tuning and coupling of whispering gallery modes using an anisotropic prism,” J. Opt. Soc. Am. B33, 2177 (2016).

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

Little, B.E.

Marquardt, C.

A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, “Efficient microwave to optical photon conversion: an electro-optical realization,” Optica 3(6), 597–604 (2016).
[Crossref]

D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, “Nonlinear and Quantum Optics with Whispering Gallery Resonators,” arXiv:1605.07972 [physics, Physics:quant-Ph] (2016).

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

Matsko, A. B.

D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of sub-THz radiation in a high-Q whispering gallery resonator,” Opt. Lett. 34(6), 713–715 (2009).
[Crossref] [PubMed]

D.V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Towards Counting microwave photons at room temperature,” Laser Phys. Lett. 6(2), 129–134 (2009).
[Crossref]

D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, “Nonlinear and Quantum Optics with Whispering Gallery Resonators,” arXiv:1605.07972 [physics, Physics:quant-Ph] (2016).

Muñoz, E. G.

G. Carpintero, E. G. Muñoz, H. Hartnagel, S. Preu, and A. Raisanen, Semiconductor TeraHertz Technology: Devices and Systems at Room Temperature Operation (Wiley-IEEE Press, 2015).
[Crossref]

Oraevsky, A. N.

A. N. Oraevsky, “Whispering-Gallery waves,” Quantum Electron. 32(5), 377–400 (2002).
[Crossref]

Painter, O.

Powers., P.E.

P.E. Powers., Fundamentals of Nonlinear Optics (CRC Press, 2011).

Preu, S.

G. Carpintero, E. G. Muñoz, H. Hartnagel, S. Preu, and A. Raisanen, Semiconductor TeraHertz Technology: Devices and Systems at Room Temperature Operation (Wiley-IEEE Press, 2015).
[Crossref]

Raisanen, A.

G. Carpintero, E. G. Muñoz, H. Hartnagel, S. Preu, and A. Raisanen, Semiconductor TeraHertz Technology: Devices and Systems at Room Temperature Operation (Wiley-IEEE Press, 2015).
[Crossref]

Rueda, A.

Savchenkov, A. A.

D.V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Towards Counting microwave photons at room temperature,” Laser Phys. Lett. 6(2), 129–134 (2009).
[Crossref]

D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of sub-THz radiation in a high-Q whispering gallery resonator,” Opt. Lett. 34(6), 713–715 (2009).
[Crossref] [PubMed]

Scarcella, C.

A. B. Shehata, C. Scarcella, and A. Tosi, “Photon counting module based on InGaAs/InP Single-Photon Avalanche Diodes for near-infrared counting up to 1.7 μm,” in Ph.D. Research in Microelectronics and Electronics (PRIME), 2011 7th Conference on, 177–180 (2011).
[Crossref]

Schunk, G.

A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, “Efficient microwave to optical photon conversion: an electro-optical realization,” Optica 3(6), 597–604 (2016).
[Crossref]

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

Schwefel, H. G. L.

A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, “Efficient microwave to optical photon conversion: an electro-optical realization,” Optica 3(6), 597–604 (2016).
[Crossref]

I. Breunig, B. Sturman, F. Sedlmeir, H. G. L. Schwefel, and K. Buse, “Whispering gallery modes at the rim of an axisymmetric optical resonator: Analytical versus numerical description and comparison with experiment,” Opt. Express 21(25), 30683–30692 (2013).
[Crossref]

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

M. R. Foreman, F. Sedlmeir, H. G. L. Schwefel, and G. Leuchs, “Dielectric tuning and coupling of whispering gallery modes using an anisotropic prism,” J. Opt. Soc. Am. B33, 2177 (2016).

D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, “Nonlinear and Quantum Optics with Whispering Gallery Resonators,” arXiv:1605.07972 [physics, Physics:quant-Ph] (2016).

Sedlmeir, F.

A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, “Efficient microwave to optical photon conversion: an electro-optical realization,” Optica 3(6), 597–604 (2016).
[Crossref]

I. Breunig, B. Sturman, F. Sedlmeir, H. G. L. Schwefel, and K. Buse, “Whispering gallery modes at the rim of an axisymmetric optical resonator: Analytical versus numerical description and comparison with experiment,” Opt. Express 21(25), 30683–30692 (2013).
[Crossref]

M. R. Foreman, F. Sedlmeir, H. G. L. Schwefel, and G. Leuchs, “Dielectric tuning and coupling of whispering gallery modes using an anisotropic prism,” J. Opt. Soc. Am. B33, 2177 (2016).

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

Sergeev, A. V.

B. S. Karasik and A. V. Sergeev, “THz Hot-Electron Photon Counter,” IEEE Trans. Appl. Superconductivity 15(2), 618–621 (2005).
[Crossref]

Shehata, A. B.

A. B. Shehata, C. Scarcella, and A. Tosi, “Photon counting module based on InGaAs/InP Single-Photon Avalanche Diodes for near-infrared counting up to 1.7 μm,” in Ph.D. Research in Microelectronics and Electronics (PRIME), 2011 7th Conference on, 177–180 (2011).
[Crossref]

Stiller, B.

Strekalov, D. V.

A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, “Efficient microwave to optical photon conversion: an electro-optical realization,” Optica 3(6), 597–604 (2016).
[Crossref]

D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of sub-THz radiation in a high-Q whispering gallery resonator,” Opt. Lett. 34(6), 713–715 (2009).
[Crossref] [PubMed]

D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, “Nonlinear and Quantum Optics with Whispering Gallery Resonators,” arXiv:1605.07972 [physics, Physics:quant-Ph] (2016).

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

Strekalov, D.V.

D.V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Towards Counting microwave photons at room temperature,” Laser Phys. Lett. 6(2), 129–134 (2009).
[Crossref]

Sturman, B.

Tosi, A.

A. B. Shehata, C. Scarcella, and A. Tosi, “Photon counting module based on InGaAs/InP Single-Photon Avalanche Diodes for near-infrared counting up to 1.7 μm,” in Ph.D. Research in Microelectronics and Electronics (PRIME), 2011 7th Conference on, 177–180 (2011).
[Crossref]

Vogl, U.

A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, “Efficient microwave to optical photon conversion: an electro-optical realization,” Optica 3(6), 597–604 (2016).
[Crossref]

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

Yariv., A.

A. Yariv., “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

Yu, N.

D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of sub-THz radiation in a high-Q whispering gallery resonator,” Opt. Lett. 34(6), 713–715 (2009).
[Crossref] [PubMed]

D.V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Towards Counting microwave photons at room temperature,” Laser Phys. Lett. 6(2), 129–134 (2009).
[Crossref]

Zeltner, R.

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

Astrophys. J. (1)

JPL, “BICEP2. II. Experiment and three-year data set,” Astrophys. J. 792(1), 62 (2014).
[Crossref]

Electron. Lett. (1)

A. Yariv., “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

IEEE Trans. Appl. Superconductivity (1)

B. S. Karasik and A. V. Sergeev, “THz Hot-Electron Photon Counter,” IEEE Trans. Appl. Superconductivity 15(2), 618–621 (2005).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Laser Phys. Lett. (1)

D.V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Towards Counting microwave photons at room temperature,” Laser Phys. Lett. 6(2), 129–134 (2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Quantum Electron. (1)

A. N. Oraevsky, “Whispering-Gallery waves,” Quantum Electron. 32(5), 377–400 (2002).
[Crossref]

Other (7)

G. Carpintero, E. G. Muñoz, H. Hartnagel, S. Preu, and A. Raisanen, Semiconductor TeraHertz Technology: Devices and Systems at Room Temperature Operation (Wiley-IEEE Press, 2015).
[Crossref]

D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, “Nonlinear and Quantum Optics with Whispering Gallery Resonators,” arXiv:1605.07972 [physics, Physics:quant-Ph] (2016).

F. Sedlmeir, M. R. Foreman, U. Vogl, R. Zeltner, G. Schunk, D. V. Strekalov, C. Marquardt, G. Leuchs, and H. G. L. Schwefel, “Polarization-selective out-coupling of whispering gallery modes,” arXiv:1608.07660 [physics] (2016).

P.E. Powers., Fundamentals of Nonlinear Optics (CRC Press, 2011).

M. R. Foreman, F. Sedlmeir, H. G. L. Schwefel, and G. Leuchs, “Dielectric tuning and coupling of whispering gallery modes using an anisotropic prism,” J. Opt. Soc. Am. B33, 2177 (2016).

A. B. Shehata, C. Scarcella, and A. Tosi, “Photon counting module based on InGaAs/InP Single-Photon Avalanche Diodes for near-infrared counting up to 1.7 μm,” in Ph.D. Research in Microelectronics and Electronics (PRIME), 2011 7th Conference on, 177–180 (2011).
[Crossref]

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

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Figures (10)
Fig. 1
Fig. 1 Nonlinear up-conversion THz receiver. The laser beam is incident into the coupling prism and an optic WGM is excited in the resonator. By exciting another WGM for the THz signal the nonlinear interaction takes place and the up-converted signal is coupled out to the prism and detected by an optical spectrum analyzer.

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Fig. 2
Fig. 2 Three media propagation model. n1 and n2 are the indices of refraction of the prism and resonator respectively. (a) Propagation from the prism. E 1 a + and E 1 a − are the incident and reflected electric fields evaluated in the prism-air interface respectively, whereas E 2 a + is the transmitted electric field evaluated in the air-resonator interface. (b) Propagation from the resonator. E 2 b + and E 2 b − are the incident and reflected electric fields evaluated in the resonator-air interface respectively, whereas E 1 b + is the transmitted electric field evaluated in the air-prism interface.

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Fig. 3
Fig. 3 Two-port network scheme.

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Fig. 4
Fig. 4 WGM power relative to incident laser power versus |ρ1| for three different values of α. Note that the maximum intracavity power (critical coupling) is achieved at the marked positions corresponding to α = |ρ1|.

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Fig. 5
Fig. 5 Power enhancement as a function of h for three different incidence angles greater than the critical angle between prism and air θc1 = arcsin(1/n1) (evanescent field is generated), and less than the critical angle between prism and resonator θc2 = arcsin(n2/n1) (to avoid total internal reflection in the whole system). For this particular example, the prism is made of diamond (n1 = 2.39) and the resonator is made of LiNbO3 (n2 = 2.21). The absortion coefficient of LiNbO3 is about κ″ = 5 × 10−2, so for a resonator with radius R0 = 2.5mm, α ≈ e−2πκ″ R0 = 0.999.

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Fig. 6
Fig. 6 Optimal h as a function of θ1/θc2 for four different values of α. The media considered are the ones of Fig. 5. θc2 = arcsin(n2/n1) is the critical angle between prism and resonator. In real resonators, however, angles close to θc2 cannot be used due to the finite coupling length of the curved surface and the plane prism interface which will be shown in Section 5.

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Fig. 7
Fig. 7 Gaussian beam refraction.

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Fig. 8
Fig. 8 Left: Resonator-prism configuration. Right: Disc-shaped WGM resonator geometry.

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Fig. 9
Fig. 9 Optimal ratio r0/R0 of a LiNbO3 resonator as a function of the incidence angle θ1 calculated by neglecting the dependence of the WGM with γ and by Eq. (26) for the cases of β = θ1 and β = 68° which is the required incidence angle for a diamond prism and a LiNbO3 resonator.

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Fig. 10
Fig. 10 Reflected beam’s normalized power as a function of the ratio r0/R0 for three different prism angles and θ1 = 68°.

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Equations (27)

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

κ wgm = ω c n eff ≈ ω c n 2
sin θ 1 = n eff n 1 ≈ n 2 n 1
τ 21 = n 2 n 1 cos θ 2 cos θ 1 τ 12
| E 1 a + | 2 − | E 1 a − + E 1 b + | 2 | E 2 a + + E 2 b − | 2 − | E 2 b + | 2 = n 2 cos θ 2 n 1 cos θ 1 ,
ρ 2 = − τ 12 τ 12 * ρ 1 *
[ E 1 − E 2 − ] = [ ρ 1 τ 21 τ 12 ρ 2 ]     [ E 1 + E 2 + ]
[ b 1 b 2 ] = [ ρ 1 k k ρ 2 ]     [ a 1 a 2 ] ,
a 2 = α e j ξ b 2
| b 1 | 2 = ( α − | ρ 1 | ) 2 ( 1 − α | ρ 1 | ) 2 | a 1 | 2
| b 2 | 2 = 1 − | ρ 1 | 2 ( 1 − α | ρ 1 | ) 2 | a 1 | 2
ρ 1 = n 1 cos θ 1 ( 1 + r ) − C 0 ( 1 − r ) n 1 cos θ 1 ( 1 + r ) + C 0 ( 1 − r ) ,
r = C 0 − n 2 C 2 C 0 + n 2 C 2 e − j 4 π λ 0 h C 0 ,
C 0 = j n 1 2 sin 2 θ 1 − 1
C 2 = 1 − ( n 1 n 2 ) 2 sin 2 θ 1 ,
Ψ 1 ( x , y ) = e − 1 W 2 ( x 2 + ( y cos θ 1 b ( Δ β ) ) 2 ) ,
b ( Δ β ) = 1 − n 1 2 sin ( Δ β ) 2 cos Δ β
E | r = r 0 ∝ e − γ 2 / 2 γ m 2 e − j m φ
E e ∝ E | r = r 0 e − κ ″ ( r − r 0 )
h = d + R 0 − r 0 ( R 0 r 0 − 1 + 1 − ( x r 0 ) 2 ) 2 − ( y r 0 ) 2
h ≈ d + R 0 − R 0 1 − 1 R 0 [ ( x r 0 ) 2 + ( y R 0 ) 2 ]
h ( x , y ) ≈ r | z = 0 − r 0 ≈ d + 1 2 [ ( x r 0 ) 2 + ( y R 0 ) 2 ]
E e | z = 0 ∝ e − 1 2 [ ( x / Δ x ) 2 + ( y / Δ y ) 2 ] e − j m φ
Δ x 2 = r 0 2 γ m 2 1 + κ ″ r 0 γ m 2
Δ y 2 = R 0 κ ″
[ b ( Δ β ) cos θ 1 ] 2 = Δ x 2 Δ y 2 = r 0 R 0 ( κ ″ r 0 γ m 2 1 + κ ″ r 0 γ m 2 )
[ b ( Δ β ) cos θ 1 ] 2 ≈ r 0 R 0 ( 1 + n 2 n 2 2 − 1 r 0 R 0 ) − 1
P r P i = 1 + 2 b cos ( θ 1 ) Δ x Δ y W 2 − 8 b Δ x Δ y ( 2 Δ x 2 + W 2 ) ( 2 b 2 Δ y 2 + W 2 )

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