Abstract

All optically biaxial crystals exhibit orientations in which they behave as thermally insensitive (TI) wave plates. Given sufficient electro-optic coupling at these orientations, biaxial, ferroelectric crystals may be advantageously employed as single-crystal, temperature-insensitive electro-optic modulators. I have experimentally confirmed the TI orientation in the principal dielectric plane of the biaxial crystal KTiOPO4 (KTP). KTP behaves as a TI wave plate at 33.7 deg from the Z axis in the X–Z principal plane. The effective electro-optic coupling, Reff, at this orientation is 19 pm/V. The electro-optic coupling and utility of these TI orientations in KTiOAsO4 and KNbO3 are also examined. The calculated Reff of KNbO3 at the TI orientation is expected to be >2000 pm/V.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics (Van Nostrand, Princeton, N.J., 1956).
  2. D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
    [CrossRef]
  3. J. D. Beasley, "Multi-element electro-optic modulators with crystal axes oriented obliquely to direction of electric field," U.S. Patent 5,157,539 (October 20, 1992).
  4. S. P. Velsko, C. Ebbers, W. Cook, and S. Sutton, "Design and performance of average power electro-optic modulators," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994).
  5. P. Günter, "Electro-optical properties of KNbO3," Opt. Commun. 11, 285–290 (1974).
    [CrossRef]
  6. J. D. Bierlein and C. B. Arweiler, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917–919 (1986).
    [CrossRef]
  7. H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
    [CrossRef]
  8. J. D. Bierlein and H. Vanherzeele, "Potassium titanyl phosphate: properties and new applications," J. Opt. Soc. Am. B 6, 622–633 (1989).
    [CrossRef]
  9. D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LilO3, and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
    [CrossRef]
  10. K. Kato, "Temperature insensitive SHG at 0.5321 μm in KTP, IEEE J. Quantum Electron. 28, 1974–1976 (1992).
    [CrossRef]
  11. W. Wiechmann, T. Fuku, H. Masuda, and S. Kubota, "Refractive index temperature derivatives of potassium titanyl phosphate," Opt. Lett. 18, 1208–1210 (1993).
    [CrossRef] [PubMed]
  12. B. Zysset, I. Biaggio, and P. Günter, "Refractive indices of orthorhombic KNbO3. I. Dispersion and temperature dependence," J. Opt. Soc. Am. B 9, 380–386 (1992).
    [CrossRef]
  13. Y. Uematsu, "Nonlinear optical properties of KNbO3 single crystal in the orthorhombic phase," Jpn. J. Appl. Phys. 13, 1362–1368 (1974).
    [CrossRef]
  14. D. K. T. Chu, "Piezoelectric and acoustic properties of KTP and its isomorphs," thesis (University of Delaware, Newark, Del., 1991).
  15. G. Mizell, Virgo Optics, Port Richey, Fla. (personal communication).
  16. C. A. Ebbers, "Linear electro-optic effect in β-BaB2O4," Appl. Phys. Lett. 52, 1948–1949 (1988).
    [CrossRef]
  17. S. P. Velsko, C. A. Ebbers, B. Comaskey, G. Albrecht, and S. Mitchell, "250-watt average power electro-optically Q-switched power oscillator," in Advanced Solid State Lasers, T. Y. Fan and B. Chai, eds., Vol. 20 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 308–313.
  18. D. Eimerl, "Thermal aspects of high average power electrooptic switches," IEEE J. Quantum Electron. QE-23, 2238–2251 (1987).
    [CrossRef]
  19. R. A. Phillips, "Temperature variation of the index of refraction of ADP, KDP, and deuterated KDP," J. Opt. Soc. Am. 56, 629–632 (1966).
    [CrossRef]
  20. S. Z. Kurtev, O. E. Denchev, and S. D. Savov, "Effects of thermally induced birefringence in high-output-power electrooptically Q-switched Nd:YAG lasers and their compensation," Appl. Opt. 32, 278–285 (1993).
    [CrossRef] [PubMed]
  21. Properties of Lithium Niobate, EMIS Data Reviews Series No. 5 (INSPEC, Institution of Electrical Engineers, London, 1989).

1993

1992

1989

1988

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LilO3, and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

C. A. Ebbers, "Linear electro-optic effect in β-BaB2O4," Appl. Phys. Lett. 52, 1948–1949 (1988).
[CrossRef]

1987

D. Eimerl, "Thermal aspects of high average power electrooptic switches," IEEE J. Quantum Electron. QE-23, 2238–2251 (1987).
[CrossRef]

1986

J. D. Bierlein and C. B. Arweiler, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917–919 (1986).
[CrossRef]

1974

P. Günter, "Electro-optical properties of KNbO3," Opt. Commun. 11, 285–290 (1974).
[CrossRef]

Y. Uematsu, "Nonlinear optical properties of KNbO3 single crystal in the orthorhombic phase," Jpn. J. Appl. Phys. 13, 1362–1368 (1974).
[CrossRef]

1966

Albrecht, G.

S. P. Velsko, C. A. Ebbers, B. Comaskey, G. Albrecht, and S. Mitchell, "250-watt average power electro-optically Q-switched power oscillator," in Advanced Solid State Lasers, T. Y. Fan and B. Chai, eds., Vol. 20 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 308–313.

Arweiler, C. B.

J. D. Bierlein and C. B. Arweiler, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917–919 (1986).
[CrossRef]

Barnes, N. P.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LilO3, and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

Beasley, J. D.

J. D. Beasley, "Multi-element electro-optic modulators with crystal axes oriented obliquely to direction of electric field," U.S. Patent 5,157,539 (October 20, 1992).

Biaggio, I.

Bierlein, J. D.

J. D. Bierlein and H. Vanherzeele, "Potassium titanyl phosphate: properties and new applications," J. Opt. Soc. Am. B 6, 622–633 (1989).
[CrossRef]

J. D. Bierlein and C. B. Arweiler, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917–919 (1986).
[CrossRef]

Bonner, C.

D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
[CrossRef]

Chu, D. K. T.

D. K. T. Chu, "Piezoelectric and acoustic properties of KTP and its isomorphs," thesis (University of Delaware, Newark, Del., 1991).

Comaskey, B.

S. P. Velsko, C. A. Ebbers, B. Comaskey, G. Albrecht, and S. Mitchell, "250-watt average power electro-optically Q-switched power oscillator," in Advanced Solid State Lasers, T. Y. Fan and B. Chai, eds., Vol. 20 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 308–313.

Cook, W.

S. P. Velsko, C. Ebbers, W. Cook, and S. Sutton, "Design and performance of average power electro-optic modulators," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994).

Denchev, O. E.

Ebbers, C.

S. P. Velsko, C. Ebbers, W. Cook, and S. Sutton, "Design and performance of average power electro-optic modulators," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994).

Ebbers, C. A.

C. A. Ebbers, "Linear electro-optic effect in β-BaB2O4," Appl. Phys. Lett. 52, 1948–1949 (1988).
[CrossRef]

S. P. Velsko, C. A. Ebbers, B. Comaskey, G. Albrecht, and S. Mitchell, "250-watt average power electro-optically Q-switched power oscillator," in Advanced Solid State Lasers, T. Y. Fan and B. Chai, eds., Vol. 20 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 308–313.

Eimerl, D.

D. Eimerl, "Thermal aspects of high average power electrooptic switches," IEEE J. Quantum Electron. QE-23, 2238–2251 (1987).
[CrossRef]

Fuku, T.

Gettemy, D. J.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LilO3, and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

Günter, P.

Harker, W. C.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LilO3, and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

Huang, C.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Kato, K.

K. Kato, "Temperature insensitive SHG at 0.5321 μm in KTP, IEEE J. Quantum Electron. 28, 1974–1976 (1992).
[CrossRef]

Kubota, S.

Kurtev, S. Z.

Lian, T.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Liao, H.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Lindholm, G.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LilO3, and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

Mason, W. P.

W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics (Van Nostrand, Princeton, N.J., 1956).

Masuda, H.

Miller, D.

D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
[CrossRef]

Mitchell, S.

S. P. Velsko, C. A. Ebbers, B. Comaskey, G. Albrecht, and S. Mitchell, "250-watt average power electro-optically Q-switched power oscillator," in Advanced Solid State Lasers, T. Y. Fan and B. Chai, eds., Vol. 20 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 308–313.

Mizell, G.

G. Mizell, Virgo Optics, Port Richey, Fla. (personal communication).

Palese, S.

D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
[CrossRef]

Pereira, M.

D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
[CrossRef]

Phillips, R. A.

Savov, S. D.

Schilling, L.

D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
[CrossRef]

Shen, H.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Sutton, S.

S. P. Velsko, C. Ebbers, W. Cook, and S. Sutton, "Design and performance of average power electro-optic modulators," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994).

Uematsu, Y.

Y. Uematsu, "Nonlinear optical properties of KNbO3 single crystal in the orthorhombic phase," Jpn. J. Appl. Phys. 13, 1362–1368 (1974).
[CrossRef]

Vanherzeele, H.

Velsko, S. P.

S. P. Velsko, C. Ebbers, W. Cook, and S. Sutton, "Design and performance of average power electro-optic modulators," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994).

S. P. Velsko, C. A. Ebbers, B. Comaskey, G. Albrecht, and S. Mitchell, "250-watt average power electro-optically Q-switched power oscillator," in Advanced Solid State Lasers, T. Y. Fan and B. Chai, eds., Vol. 20 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 308–313.

Walmsley, I.

D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
[CrossRef]

Wang, O.

D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
[CrossRef]

Wiechmann, W.

Yu, G.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Zeng, R.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Zheng, Z.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Zhou, Y.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Zysset, B.

Appl. Opt.

Appl. Phys. Lett.

J. D. Bierlein and C. B. Arweiler, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917–919 (1986).
[CrossRef]

C. A. Ebbers, "Linear electro-optic effect in β-BaB2O4," Appl. Phys. Lett. 52, 1948–1949 (1988).
[CrossRef]

IEEE J. Quantum Electron.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LilO3, and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

K. Kato, "Temperature insensitive SHG at 0.5321 μm in KTP, IEEE J. Quantum Electron. 28, 1974–1976 (1992).
[CrossRef]

D. Eimerl, "Thermal aspects of high average power electrooptic switches," IEEE J. Quantum Electron. QE-23, 2238–2251 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

Y. Uematsu, "Nonlinear optical properties of KNbO3 single crystal in the orthorhombic phase," Jpn. J. Appl. Phys. 13, 1362–1368 (1974).
[CrossRef]

Opt. Commun.

P. Günter, "Electro-optical properties of KNbO3," Opt. Commun. 11, 285–290 (1974).
[CrossRef]

Opt. Laser Technol.

H. Liao, H. Shen, Z. Zheng, T. Lian, Y. Zhou, C. Huang, R. Zeng, and G. Yu, "Accurate values for the index of refraction and the optimum phase match parameters in a flux grown KTiOPO4 crystal," Opt. Laser Technol. 20, 103–104 (1988).
[CrossRef]

Opt. Lett.

Other

S. P. Velsko, C. A. Ebbers, B. Comaskey, G. Albrecht, and S. Mitchell, "250-watt average power electro-optically Q-switched power oscillator," in Advanced Solid State Lasers, T. Y. Fan and B. Chai, eds., Vol. 20 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 308–313.

D. K. T. Chu, "Piezoelectric and acoustic properties of KTP and its isomorphs," thesis (University of Delaware, Newark, Del., 1991).

G. Mizell, Virgo Optics, Port Richey, Fla. (personal communication).

W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics (Van Nostrand, Princeton, N.J., 1956).

D. Miller, C. Bonner, S. Palese, M. Pereira, L. Schilling, I. Walmsley, and O. Wang, "Development and applications of electro-optics for high power systems," in Diode Pumping of Average-Power Solid State Lasers, G. F. Albrecht, R. J. Beach, and S. P. Velsko, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1865, 100–110 (1993).
[CrossRef]

J. D. Beasley, "Multi-element electro-optic modulators with crystal axes oriented obliquely to direction of electric field," U.S. Patent 5,157,539 (October 20, 1992).

S. P. Velsko, C. Ebbers, W. Cook, and S. Sutton, "Design and performance of average power electro-optic modulators," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994).

Properties of Lithium Niobate, EMIS Data Reviews Series No. 5 (INSPEC, Institution of Electrical Engineers, London, 1989).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(A) The location of the principal-plane TI propagation direction occurs for propagation in the plane normal to the axis with the intermediate thermal index (dn2/dT). (B) Light polarized parallel to the rotation axis and propagating in the n1n3 principal plane sees dn2/dT, independent of the propagation direction. Light polarized perpendicular to this axis sees an effective dn/dT, which varies between dn1/dT and dn3/dT. At the TI propagation direction, dn13/dT = dn2/dT. The observed birefringence at this orientation is thus independent of temperature.

Fig. 2
Fig. 2

(A) Wulff net projection of the calculated TI propagation directions in KTP. O.A., location of the optic axis. (B) Wulff net projection of the calculated TI propagation directions in KNbO3. O.A., location of the optic axis.

Fig. 3
Fig. 3

Schematic of apparatus used for thermal birefringence measurements.

Fig. 4
Fig. 4

(A) For propagation near the Z axis, the transmitted power maxima shift away from the Z axis at elevated temperatures. (B) For propagation near the X axis, the transmitted power maxima shift toward the Z axis. (Data are normalized for comparison.)

Fig. 5
Fig. 5

Displacement of the birefringence fringes as a function of propagation direction. At 33.7 deg from the Z axis in the X–Z plane, the change in birefringence as a function of temperature vanishes.

Fig. 6
Fig. 6

Thermal birefringence in Y-cut and TI-cut KTP: change in transmitted power versus temperature. The thermal birefringence of TI-cut KTP is significantly reduced.

Fig. 7
Fig. 7

(A) Calculated Reff in KTP for propagation in the X–Z plane for ± relative values of rxzx. The arrow indicates the two potential values at the TI propagation direction. The lower value (corresponding to −rxzx) has been determined to be the correct value. (B) Calculated Reff in KNbO3 for propagation in the Y–Z plane for ± values of ryzy. The arrow indicates the approximate position of the TI propagation direction.

Tables (5)

Tables Icon

Table 1 Thermal Birefringence

Tables Icon

Table 2 Estimated Location of the Principal-Plane TI Orientation in Degrees Measured from Z axis

Tables Icon

Table 3 Propagation Direction, Field Direction, and Effective Electro-optic Coefficients

Tables Icon

Table 4 Relevant Thermo-optic Parameters at λ = 1.064 μm

Tables Icon

Table 5 Theoretical Comparison of Temperature Rise and Thermal Lensing in Switches (λ = 1.064 μm, 2.5 kV λ/4 Switching Voltage, 1 kW Average Power, 5 × 20 mm2 Input Aperture)

Equations (16)

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

Δ ϕ = 2 π λ ( n 1 n 2 ) L .
Δ ϕ T = 2 π L λ [ n 1 T n 2 T + α L ( n 1 n 2 ) ] .
Δ ϕ T = 2 π L λ ( n 1 n 2 ) .
Δ ϕ T = 2 π L λ [ n β T n α γ T + α L ( n β n α γ ) ] ,
n α γ = n α n γ [ n γ 2 cos 2 ( θ ) + n α 2 sin 2 ( θ ) ] 1 / 2 ,
α L = [ α γ cos 2 ( θ ) + α α sin 2 ( θ ) ] ,
n α γ T = n α γ 3 n α 3 n α T cos 2 ( θ ) + n α γ 3 n γ 3 n γ T sin 2 ( θ ) .
θ TI = arccos ( n γ n β n γ n α ) 1 / 2
I = I 0 sin 2 ( Γ 2 + Φ 0 ) + V 0 ,
Γ = 2 π L λ | n 1 T n 2 T + α 3 ( n 1 n 2 ) | .
I = A + B sin 2 { π L λ C [ n y n x z ( θ ) ] + D } .
V π = λ ( d / L ) R eff ,
R eff = n y 3 r yyz sin ( θ ) n x z 3 [ r xxz cos 2 ( θ ) sin ( θ ) + r zzz sin 3 ( θ ) ± 2 r xzx cos 2 ( θ ) sin ( θ ) ] ,
R eff = n x 3 r xxz sin ( θ ) n y z 3 [ r zzz sin 3 ( θ ) + r yyz sin ( θ ) cos 2 ( θ ) ± 2 r yzy sin ( θ ) cos 2 ( θ ) ] .
n z 3 r zzz s n x 3 r xxz s
n α 3 r α α α T n γ 3 r γ γ α T

Metrics