Abstract

We present a new cost-effective terahertz linear polarizer made from a stack of silicon wafers at Brewster’s angle, andevaluate its performances. We show that this polarizer is wide-band, has a high extinction ratio (> 6 × 103) and very small insertion losses (< 1%). We provide measurements of the temporal waveforms after linearly polarizing the THz beam and show that there is no distortion of the pulse. We compare its performances with a commercial wire-grid polarizer, and show that the Brewster’s angle polarizer can conveniently be used to control the power of a terahertz beam.

© 2011 OSA

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References

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  1. D. Grischkowsky, S. R. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [CrossRef]
  2. Q. Wu and X. C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68(12), 1604–1606 (1996).
    [CrossRef]
  3. D. Mittleman, Sensing with Terahertz Radiation (Springer, 2003).
  4. C.-F. Hsieh, Y.-C. Lai, R.-P. Pan, and C.-L. Pan, “Polarizing terahertz waves with nematic liquid crystals,” Opt. Lett. 33(11), 1174–1176 (2008).
    [CrossRef] [PubMed]
  5. I. Yamada, K. Takano, M. Hangyo, M. Saito, and W. Watanabe, “Terahertz wire-grid polarizers with micrometer-pitch Al gratings,” Opt. Lett. 34(3), 274–276 (2009).
    [CrossRef] [PubMed]
  6. H. Dong, Y. Gong, V. Paulose, and M. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282, 3671–3675 (2009).
    [CrossRef]
  7. J.-B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31(2), 265–267 (2006).
    [CrossRef] [PubMed]
  8. A. E. Costley, K. H. Hursey, G. F. Neill, and J. M. Ward, “Free-standing fine-wire grids; their manufacture, performance, and use at millimeter and submillimeter wavelengths,” J. Opt. Soc. Am. 67(7), 979–981 (1977).
    [CrossRef]
  9. T. Kondo, T. Nagashima, and M. Hangyo, “Fabrication of wire-grid-type polarizers for THz region using a general-purpose color printer,” Jpn. J. Appl. Phys. 42, L373–L375 (2003).
    [CrossRef]
  10. L. Sun, Z.-H. Lv, W. Wu, W.-T. Liu, and J.-M. Yuan, “Double-grating polarizer for terahertz radiation with high extinction ratio,” Appl. Opt. 49, 2066-2071 (2010).
    [CrossRef] [PubMed]
  11. S. K. Awasthi, A. Srivastava, U. Malaviya, and S. P. Ojha, “Wide-angle, broadband plate polarizer in Terahertz frequency region,” Solid State Commun. 148, 506–509 (2008).
    [CrossRef]
  12. I. Yamada, K. Takano, M. Hangyo, M. Saito, and W. Watanabe, “Terahertz wire-grid polarizer with micrometer-pitch Al gratings,” Opt. Lett. 34, 274–276 (2009).
    [CrossRef] [PubMed]
  13. L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
    [CrossRef] [PubMed]
  14. J.-S. Li, D.-G. Xu, and J.-Q. Yao, “Compact terahertz wave polarizing beam splitter,” Appl. Opt. 49(24), 4494–4497 (2010).
    [CrossRef] [PubMed]
  15. J.-B. Masson, M.-P. Sauviat, J.-L. Martin, and G. Gallot, “Ionic contrast terahertz near field imaging of axonal water fluxes,” Proc. Natl. Acad. Sci. U.S.A. 103(13), 4808–4812 (2006).
    [CrossRef] [PubMed]
  16. R. mendis and D. Mittleman, “A 2-D artificial dielectric with 0 ≤ n < 1 for the terahertz region,” IEEE Trans. Microwave Theory Tech. 58(7), 1993–1998 (2010).
    [CrossRef]
  17. A. V. Spivey and S. Cundiff, “Brewster’s angle attenuator for terahertz pulses,” Appl. Opt. 41, 2408–2412 (2002).
  18. G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
    [CrossRef] [PubMed]
  19. M. Born and E. Wolf, Principle of Optics , 7th Ed. (Cambridge University Press, 1999).
  20. D. Brewster, A Treatise on Optics (MacMillan and Co., 1831).
  21. E. Hecht, Optics , 2nd ed. (Addison-Wesley, 1990).

2010

2009

I. Yamada, K. Takano, M. Hangyo, M. Saito, and W. Watanabe, “Terahertz wire-grid polarizer with micrometer-pitch Al gratings,” Opt. Lett. 34, 274–276 (2009).
[CrossRef] [PubMed]

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

I. Yamada, K. Takano, M. Hangyo, M. Saito, and W. Watanabe, “Terahertz wire-grid polarizers with micrometer-pitch Al gratings,” Opt. Lett. 34(3), 274–276 (2009).
[CrossRef] [PubMed]

H. Dong, Y. Gong, V. Paulose, and M. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282, 3671–3675 (2009).
[CrossRef]

2008

C.-F. Hsieh, Y.-C. Lai, R.-P. Pan, and C.-L. Pan, “Polarizing terahertz waves with nematic liquid crystals,” Opt. Lett. 33(11), 1174–1176 (2008).
[CrossRef] [PubMed]

S. K. Awasthi, A. Srivastava, U. Malaviya, and S. P. Ojha, “Wide-angle, broadband plate polarizer in Terahertz frequency region,” Solid State Commun. 148, 506–509 (2008).
[CrossRef]

2006

J.-B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31(2), 265–267 (2006).
[CrossRef] [PubMed]

J.-B. Masson, M.-P. Sauviat, J.-L. Martin, and G. Gallot, “Ionic contrast terahertz near field imaging of axonal water fluxes,” Proc. Natl. Acad. Sci. U.S.A. 103(13), 4808–4812 (2006).
[CrossRef] [PubMed]

2003

T. Kondo, T. Nagashima, and M. Hangyo, “Fabrication of wire-grid-type polarizers for THz region using a general-purpose color printer,” Jpn. J. Appl. Phys. 42, L373–L375 (2003).
[CrossRef]

D. Mittleman, Sensing with Terahertz Radiation (Springer, 2003).

2002

A. V. Spivey and S. Cundiff, “Brewster’s angle attenuator for terahertz pulses,” Appl. Opt. 41, 2408–2412 (2002).

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
[CrossRef] [PubMed]

1999

M. Born and E. Wolf, Principle of Optics , 7th Ed. (Cambridge University Press, 1999).

1996

Q. Wu and X. C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68(12), 1604–1606 (1996).
[CrossRef]

1990

1977

1831

D. Brewster, A Treatise on Optics (MacMillan and Co., 1831).

Awasthi, S. K.

S. K. Awasthi, A. Srivastava, U. Malaviya, and S. P. Ojha, “Wide-angle, broadband plate polarizer in Terahertz frequency region,” Solid State Commun. 148, 506–509 (2008).
[CrossRef]

Booshehri, L.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principle of Optics , 7th Ed. (Cambridge University Press, 1999).

Brewster, D.

D. Brewster, A Treatise on Optics (MacMillan and Co., 1831).

Carr, G. L.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
[CrossRef] [PubMed]

Costley, A. E.

Cundiff, S.

Dong, H.

H. Dong, Y. Gong, V. Paulose, and M. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282, 3671–3675 (2009).
[CrossRef]

Fattinger, C.

Gallot, G.

J.-B. Masson, M.-P. Sauviat, J.-L. Martin, and G. Gallot, “Ionic contrast terahertz near field imaging of axonal water fluxes,” Proc. Natl. Acad. Sci. U.S.A. 103(13), 4808–4812 (2006).
[CrossRef] [PubMed]

J.-B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31(2), 265–267 (2006).
[CrossRef] [PubMed]

Gong, Y.

H. Dong, Y. Gong, V. Paulose, and M. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282, 3671–3675 (2009).
[CrossRef]

Grischkowsky, D.

Hangyo, M.

Hauge, R.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Hecht, E.

E. Hecht, Optics , 2nd ed. (Addison-Wesley, 1990).

Hilton, D.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Hong, M.

H. Dong, Y. Gong, V. Paulose, and M. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282, 3671–3675 (2009).
[CrossRef]

Hsieh, C.-F.

Hursey, K. H.

Jordan, K.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
[CrossRef] [PubMed]

Kawayama, I.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Keiding, S. R.

Kondo, T.

T. Kondo, T. Nagashima, and M. Hangyo, “Fabrication of wire-grid-type polarizers for THz region using a general-purpose color printer,” Jpn. J. Appl. Phys. 42, L373–L375 (2003).
[CrossRef]

Kono, J.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Lai, Y.-C.

Li, J.-S.

Liu, W.-T.

Lv, Z.-H.

Malaviya, U.

S. K. Awasthi, A. Srivastava, U. Malaviya, and S. P. Ojha, “Wide-angle, broadband plate polarizer in Terahertz frequency region,” Solid State Commun. 148, 506–509 (2008).
[CrossRef]

Martin, J.-L.

J.-B. Masson, M.-P. Sauviat, J.-L. Martin, and G. Gallot, “Ionic contrast terahertz near field imaging of axonal water fluxes,” Proc. Natl. Acad. Sci. U.S.A. 103(13), 4808–4812 (2006).
[CrossRef] [PubMed]

Martin, M. C.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
[CrossRef] [PubMed]

Masson, J.-B.

J.-B. Masson, M.-P. Sauviat, J.-L. Martin, and G. Gallot, “Ionic contrast terahertz near field imaging of axonal water fluxes,” Proc. Natl. Acad. Sci. U.S.A. 103(13), 4808–4812 (2006).
[CrossRef] [PubMed]

J.-B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31(2), 265–267 (2006).
[CrossRef] [PubMed]

McKinney, W. R.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
[CrossRef] [PubMed]

mendis, R.

R. mendis and D. Mittleman, “A 2-D artificial dielectric with 0 ≤ n < 1 for the terahertz region,” IEEE Trans. Microwave Theory Tech. 58(7), 1993–1998 (2010).
[CrossRef]

Mittleman, D.

R. mendis and D. Mittleman, “A 2-D artificial dielectric with 0 ≤ n < 1 for the terahertz region,” IEEE Trans. Microwave Theory Tech. 58(7), 1993–1998 (2010).
[CrossRef]

D. Mittleman, Sensing with Terahertz Radiation (Springer, 2003).

Nagashima, T.

T. Kondo, T. Nagashima, and M. Hangyo, “Fabrication of wire-grid-type polarizers for THz region using a general-purpose color printer,” Jpn. J. Appl. Phys. 42, L373–L375 (2003).
[CrossRef]

Neil, G. R.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
[CrossRef] [PubMed]

Neill, G. F.

Ojha, S. P.

S. K. Awasthi, A. Srivastava, U. Malaviya, and S. P. Ojha, “Wide-angle, broadband plate polarizer in Terahertz frequency region,” Solid State Commun. 148, 506–509 (2008).
[CrossRef]

Pan, C.-L.

Pan, R.-P.

Paulose, V.

H. Dong, Y. Gong, V. Paulose, and M. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282, 3671–3675 (2009).
[CrossRef]

Pint, C.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Ren, L.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Rice, W.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Saito, M.

Sauviat, M.-P.

J.-B. Masson, M.-P. Sauviat, J.-L. Martin, and G. Gallot, “Ionic contrast terahertz near field imaging of axonal water fluxes,” Proc. Natl. Acad. Sci. U.S.A. 103(13), 4808–4812 (2006).
[CrossRef] [PubMed]

Spivey, A. V.

Srivastava, A.

S. K. Awasthi, A. Srivastava, U. Malaviya, and S. P. Ojha, “Wide-angle, broadband plate polarizer in Terahertz frequency region,” Solid State Commun. 148, 506–509 (2008).
[CrossRef]

Sun, L.

Takano, K.

Takeya, K.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Tonouchi, M.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

van Exter, M.

Wang, X.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Ward, J. M.

Watanabe, W.

Williams, G. P.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principle of Optics , 7th Ed. (Cambridge University Press, 1999).

Wu, Q.

Q. Wu and X. C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68(12), 1604–1606 (1996).
[CrossRef]

Wu, W.

Xu, D.-G.

Yamada, I.

Yao, J.-Q.

Yuan, J.-M.

Zhang, X. C.

Q. Wu and X. C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68(12), 1604–1606 (1996).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

Q. Wu and X. C. Zhang, “Ultrafast electro-optic field sensors,” Appl. Phys. Lett. 68(12), 1604–1606 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

R. mendis and D. Mittleman, “A 2-D artificial dielectric with 0 ≤ n < 1 for the terahertz region,” IEEE Trans. Microwave Theory Tech. 58(7), 1993–1998 (2010).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

T. Kondo, T. Nagashima, and M. Hangyo, “Fabrication of wire-grid-type polarizers for THz region using a general-purpose color printer,” Jpn. J. Appl. Phys. 42, L373–L375 (2003).
[CrossRef]

Nano Lett.

L. Ren, C. Pint, L. Booshehri, W. Rice, X. Wang, D. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, R. Hauge, and J. Kono, “Carbon nanotube terahertz polarizer,” Nano Lett. 9, 2610–2613 (2009).
[CrossRef] [PubMed]

Nature

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, “High-power terahertz radiation from relativistic electrons,” Nature 420, 153–156 (2002).
[CrossRef] [PubMed]

Opt. Commun.

H. Dong, Y. Gong, V. Paulose, and M. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282, 3671–3675 (2009).
[CrossRef]

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.A.

J.-B. Masson, M.-P. Sauviat, J.-L. Martin, and G. Gallot, “Ionic contrast terahertz near field imaging of axonal water fluxes,” Proc. Natl. Acad. Sci. U.S.A. 103(13), 4808–4812 (2006).
[CrossRef] [PubMed]

Solid State Commun.

S. K. Awasthi, A. Srivastava, U. Malaviya, and S. P. Ojha, “Wide-angle, broadband plate polarizer in Terahertz frequency region,” Solid State Commun. 148, 506–509 (2008).
[CrossRef]

Other

D. Mittleman, Sensing with Terahertz Radiation (Springer, 2003).

M. Born and E. Wolf, Principle of Optics , 7th Ed. (Cambridge University Press, 1999).

D. Brewster, A Treatise on Optics (MacMillan and Co., 1831).

E. Hecht, Optics , 2nd ed. (Addison-Wesley, 1990).

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Figures (10)

Fig. 1
Fig. 1

Schematic of the transmission through parallel plates, for s- and p-polarizations.

Fig. 2
Fig. 2

Fresnel reflection coefficients for a single air-silicon interface (black) and transmission coefficients through a full silicon wafer (red), both for s- (solid) and p-polarization (dashed). The lines are theoretical calculations, and the dots are the corresponding experimental terahertz measurements.

Fig. 3
Fig. 3

Theoretical propagation of a terahertz pulse (black) through a silicon plate, for p-polarization (red) and s-polarization (blue), with a plate thickness e = 540μm and an incidence angle at Brewster’s angle, for n = 3.41.

Fig. 4
Fig. 4

Number of wafer required to get an amplitude extinction ratio of at least 102 (20 dB amplitude, 40 dB intensity) versus the refractive index of the wafer.

Fig. 5
Fig. 5

Experimental setup. Tx: photoconductive emitter; OAPM: off-axis parabolic mirror; P: polarizer; Si-BS: silicon beam splitter; Rx and Rs: receivers.

Fig. 6
Fig. 6

Experimental terahertz waveforms of s- (black) and p- (red) polarization components after the propagation through a silicon wafer for two incidence angles of 30° (A) and Brewster’s angle at 73° (B), for experimental data (dots) and theoretical calculation using Eq. (3) (lines).

Fig. 7
Fig. 7

Picture of the 4-wafer silicon polarizer.

Fig. 8
Fig. 8

Parallel (A) and orthogonal (B) components of the THz pulse after propagation through the 4-wafer silicon polarizer at several orientations. Reference pulse (thin black line) is shifted by +20 ps. Orientations α are 0° (thick black), 45° (red), 90° (blue) and 135° (green).

Fig. 9
Fig. 9

Parallel E || (black) and orthogonal E (red) components of the THz pulse after propagation through the 4-wafer silicon polarizer. Inset is an expansion of E .

Fig. 10
Fig. 10

Detected maximum amplitude for various angle of the 4-wafer silicon polarizer mounted on a rotation stage, for parallel E || (black) and orthogonal E (red) electric field components. Dots are experimental data and solid lines are theoretical calculations.

Equations (11)

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

t 12 s ( θ 1 , θ 2 ) = 2 cos θ 1 sin θ 2 sin ( θ 1 + θ 2 ) r 12 s ( θ 1 , θ 2 ) = sin ( θ 1 θ 2 ) sin ( θ 1 + θ 2 )
t 12 p ( θ 1 , θ 2 ) = 2 cos θ 1 sin θ 2 sin ( θ 1 + θ 2 ) cos ( θ 1 θ 2 ) r 12 p ( θ 1 , θ 2 ) = tan ( θ 1 θ 2 ) tan ( θ 1 + θ 2 )
T = n t 12 ( θ 1 , θ 2 ) t 21 ( θ 2 , θ 1 ) e i β e i α 1 + r 12 ( θ 1 , θ 2 ) r 21 ( θ 2 , θ 1 ) e 2 i β with { α = 2 π λ cos ( θ 1 θ 2 ) cos θ 2 e β = 2 π λ cos ( θ 2 ) n e .
θ B = arctan ( n ) ,
Δ t = e c cos θ 2 [ n cos ( θ 1 θ 2 ) ] ,
w = e sin ( θ 1 θ 2 ) cos ( θ 2 ) = e [ sin θ 1 cos 2 θ 1 n 2 sin 2 θ 1 ]
d = n e n 2 sin 2 θ 1 .
ρ = T p T s .
ρ k = 1 / T s k
Δ = h sin 2 θ cos θ and τ = 2 L c with L = h cos θ .
[ E | | E ] = [ cos α sin α sin α cos α ] [ 1 0 0 0 ] [ cos α sin α sin α cos α ] [ E 0 0 ] = E 0 2 [ 1 + cos ( 2 α ) sin ( 2 α ) ]

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