Abstract

The apodized transmitting volume Bragg grating (TVBG) with the amplitude of the refractive index modulation (RIM) dropped to zero at both the surfaces is discussed. The proposed method for apodization is based on the recording of two identical uniform TVBGs with slightly different directions of their Bragg wave vectors in a single wafer. As a result, slow sinusoidal moiré modulation occurs in the direction perpendicular to the averaged direction of the two wave vectors of Bragg modulation. The cutting of a specimen at the planes of the two closest zeros of the moiré envelope produces the apodized TVBG. The longitudinal sinusoidal semi-period profile of RIM requires its specific maximum value to achieve 100% diffraction efficiency at the exact Bragg condition. Such an apodized TVBG demonstrates a significant suppression of the sidelobes in the diffraction efficiency depending on the angular or the wavelength detuning, in comparison to the diffraction operation of a uniform TVBG.

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

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References

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  1. O. Efimov, L. Glebov, and V. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett. 25(23), 1693–1695 (2000).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  5. X. Zhang, X. Yuan, S. Wu, J. Feng, K. Zou, and G. Zhang, “Two-dimensional angular filtering by volume Bragg gratings in photothermorefractive glass,” Opt. Lett. 36(11), 2167–2169 (2011).
    [Crossref]
  6. S. Wu, X. Yuan, X. Zhang, J. Feng, K. Zou, and G. Zhang, “Broadband angular filtering with a volume Bragg grating and a surface grating pair,” Opt. Lett. 39(14), 4068–4071 (2014).
    [Crossref]
  7. A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33(4), 384–386 (2008).
    [Crossref]
  8. Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42(2), 308–312 (2010).
    [Crossref]
  9. L. Siiman, J. Lumeau, L. Canioni, and L. Glebov, “Ultrashort laser pulse diffraction by transmitting volume Bragg gratings in photo-thermo-refractive glass,” Opt. Lett. 34(17), 2572–2574 (2009).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  13. I. Divliansky, F. Kompan, E. Hale, M. Segall, A. Schülzgen, and L. Glebov, “Wavefront shaping optical elements recorded in photo-thermo-refractive glass,” Appl. Opt. 58(13), D61–D67 (2019).
    [Crossref]
  14. A. Badalyan, R. Hovsepyan, P. Mantashyan, V. Mekhitaryan, and R. Drampyan, “Nondestructive readout of holograms recorded by Bessel beam technique in LiNbO3:Fe and LiNbO3:Fe:Cu crystals,” Eur. Phys. J. D 68(4), 82 (2014).
    [Crossref]
  15. A. J. Asuncion and R. A. Guerrero, “Generating superimposed Bessel beams with a volume holographic axicon,” Appl. Opt. 56(14), 4206–4212 (2017).
    [Crossref]
  16. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
    [Crossref]
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    [Crossref]
  19. J. M. Tsui, C. Thompson, V. Mehta, J. M. Roth, V. I. Smirnov, and L. B. Glebov, “Coupled-wave analysis of apodized volume gratings,” Opt. Express 12(26), 6642–6653 (2004).
    [Crossref]
  20. V. Smirnov, J. Lumeau, S. Mokhov, B. Zeldovich, and L. Glebov, “Ultranarrow bandwidth moiré reflecting Bragg gratings recorded in photo-thermo-refractive glass,” Opt. Lett. 35(4), 592–594 (2010).
    [Crossref]
  21. S. Mokhov, D. Ott, I. Divliansky, B. Zeldovich, and L. Glebov, “Moiré volume Bragg grating filter with tunable bandwidth,” Opt. Express 22(17), 20375–20386 (2014).
    [Crossref]
  22. F. Gao, X. Yuan, and X. Zhang, “Sidelobes suppression in angular filtering with volume Bragg gratings combination,” Chin. Opt. Lett. 14(6), 060502 (2016).
    [Crossref]

2019 (2)

2018 (1)

S. Mokhov, D. Ott, V. Smirnov, I. Divliansky, B. Zeldovich, and L. Glebov, “Moiré apodized reflective VBG,” Opt. Eng. 57(03), 1 (2018).
[Crossref]

2017 (1)

2016 (1)

2015 (1)

M. SeGall, I. Divliansky, C. Jollivet, A. Schülzgen, and L. Glebov, “Holographically encoded volume phase masks,” Opt. Eng. 54(7), 076104 (2015).
[Crossref]

2014 (5)

2011 (2)

2010 (2)

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42(2), 308–312 (2010).
[Crossref]

V. Smirnov, J. Lumeau, S. Mokhov, B. Zeldovich, and L. Glebov, “Ultranarrow bandwidth moiré reflecting Bragg gratings recorded in photo-thermo-refractive glass,” Opt. Lett. 35(4), 592–594 (2010).
[Crossref]

2009 (1)

2008 (1)

2004 (1)

2003 (1)

2000 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Anderson, B.

Andrusyak, O.

Arain, M. A.

Asuncion, A. J.

Badalyan, A.

A. Badalyan, R. Hovsepyan, P. Mantashyan, V. Mekhitaryan, and R. Drampyan, “Nondestructive readout of holograms recorded by Bessel beam technique in LiNbO3:Fe and LiNbO3:Fe:Cu crystals,” Eur. Phys. J. D 68(4), 82 (2014).
[Crossref]

Canioni, L.

Ciapurin, I.

Dawson, J. W.

Divliansky, I.

Drachenberg, D.

Drampyan, R.

A. Badalyan, R. Hovsepyan, P. Mantashyan, V. Mekhitaryan, and R. Drampyan, “Nondestructive readout of holograms recorded by Bessel beam technique in LiNbO3:Fe and LiNbO3:Fe:Cu crystals,” Eur. Phys. J. D 68(4), 82 (2014).
[Crossref]

Efimov, O.

Feng, J.

Gao, F.

Glebov, L.

I. Divliansky, F. Kompan, E. Hale, M. Segall, A. Schülzgen, and L. Glebov, “Wavefront shaping optical elements recorded in photo-thermo-refractive glass,” Appl. Opt. 58(13), D61–D67 (2019).
[Crossref]

S. Mokhov, D. Ott, V. Smirnov, I. Divliansky, B. Zeldovich, and L. Glebov, “Moiré apodized reflective VBG,” Opt. Eng. 57(03), 1 (2018).
[Crossref]

M. SeGall, I. Divliansky, C. Jollivet, A. Schülzgen, and L. Glebov, “Holographically encoded volume phase masks,” Opt. Eng. 54(7), 076104 (2015).
[Crossref]

B. Anderson, G. Venus, D. Ott, I. Divliansky, J. W. Dawson, D. Drachenberg, M. Messerly, P. Pax, J. Tassano, and L. Glebov, “Fundamental mode operation of a ribbon fiber laser by way of volume Bragg gratings,” Opt. Lett. 39(22), 6498–6500 (2014).
[Crossref]

S. Mokhov, D. Ott, I. Divliansky, B. Zeldovich, and L. Glebov, “Moiré volume Bragg grating filter with tunable bandwidth,” Opt. Express 22(17), 20375–20386 (2014).
[Crossref]

V. Smirnov, J. Lumeau, S. Mokhov, B. Zeldovich, and L. Glebov, “Ultranarrow bandwidth moiré reflecting Bragg gratings recorded in photo-thermo-refractive glass,” Opt. Lett. 35(4), 592–594 (2010).
[Crossref]

L. Siiman, J. Lumeau, L. Canioni, and L. Glebov, “Ultrashort laser pulse diffraction by transmitting volume Bragg gratings in photo-thermo-refractive glass,” Opt. Lett. 34(17), 2572–2574 (2009).
[Crossref]

A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33(4), 384–386 (2008).
[Crossref]

O. Efimov, L. Glebov, and V. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett. 25(23), 1693–1695 (2000).
[Crossref]

Glebov, L. B.

Guerrero, R. A.

Hale, E.

He, Y.

Hovsepyan, R.

A. Badalyan, R. Hovsepyan, P. Mantashyan, V. Mekhitaryan, and R. Drampyan, “Nondestructive readout of holograms recorded by Bessel beam technique in LiNbO3:Fe and LiNbO3:Fe:Cu crystals,” Eur. Phys. J. D 68(4), 82 (2014).
[Crossref]

Jollivet, C.

M. SeGall, I. Divliansky, C. Jollivet, A. Schülzgen, and L. Glebov, “Holographically encoded volume phase masks,” Opt. Eng. 54(7), 076104 (2015).
[Crossref]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings, 2nd ed., (Academic Press, Burlington, MA, 2009), pp. 189–216.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Kompan, F.

Lei, S.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42(2), 308–312 (2010).
[Crossref]

Lumeau, J.

Mantashyan, P.

A. Badalyan, R. Hovsepyan, P. Mantashyan, V. Mekhitaryan, and R. Drampyan, “Nondestructive readout of holograms recorded by Bessel beam technique in LiNbO3:Fe and LiNbO3:Fe:Cu crystals,” Eur. Phys. J. D 68(4), 82 (2014).
[Crossref]

Mehta, V.

Mekhitaryan, V.

A. Badalyan, R. Hovsepyan, P. Mantashyan, V. Mekhitaryan, and R. Drampyan, “Nondestructive readout of holograms recorded by Bessel beam technique in LiNbO3:Fe and LiNbO3:Fe:Cu crystals,” Eur. Phys. J. D 68(4), 82 (2014).
[Crossref]

Messerly, M.

Mokhov, S.

Ott, D.

Pax, P.

Riza, N. A.

Roth, J. M.

Schülzgen, A.

I. Divliansky, F. Kompan, E. Hale, M. Segall, A. Schülzgen, and L. Glebov, “Wavefront shaping optical elements recorded in photo-thermo-refractive glass,” Appl. Opt. 58(13), D61–D67 (2019).
[Crossref]

M. SeGall, I. Divliansky, C. Jollivet, A. Schülzgen, and L. Glebov, “Holographically encoded volume phase masks,” Opt. Eng. 54(7), 076104 (2015).
[Crossref]

Segall, M.

I. Divliansky, F. Kompan, E. Hale, M. Segall, A. Schülzgen, and L. Glebov, “Wavefront shaping optical elements recorded in photo-thermo-refractive glass,” Appl. Opt. 58(13), D61–D67 (2019).
[Crossref]

M. SeGall, I. Divliansky, C. Jollivet, A. Schülzgen, and L. Glebov, “Holographically encoded volume phase masks,” Opt. Eng. 54(7), 076104 (2015).
[Crossref]

Sevian, A.

Shang-hong, Z.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42(2), 308–312 (2010).
[Crossref]

Shen, B.

Sheng-bao, Z.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42(2), 308–312 (2010).
[Crossref]

Siiman, L.

Smirnov, V.

Smirnov, V. I.

Tan, J.

Tassano, J.

Thompson, C.

Tsui, J. M.

Venus, G.

Wang, X.

Wu, S.

Xing-chun, C.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42(2), 308–312 (2010).
[Crossref]

Xiong, B.

Yaqoob, Z.

Yuan, X.

Zeldovich, B.

Zhang, G.

Zhang, X.

Zheng, G.

Zhuo-liang, W.

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42(2), 308–312 (2010).
[Crossref]

Zou, K.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Chin. Opt. Lett. (3)

Eur. Phys. J. D (1)

A. Badalyan, R. Hovsepyan, P. Mantashyan, V. Mekhitaryan, and R. Drampyan, “Nondestructive readout of holograms recorded by Bessel beam technique in LiNbO3:Fe and LiNbO3:Fe:Cu crystals,” Eur. Phys. J. D 68(4), 82 (2014).
[Crossref]

Opt. Eng. (2)

M. SeGall, I. Divliansky, C. Jollivet, A. Schülzgen, and L. Glebov, “Holographically encoded volume phase masks,” Opt. Eng. 54(7), 076104 (2015).
[Crossref]

S. Mokhov, D. Ott, V. Smirnov, I. Divliansky, B. Zeldovich, and L. Glebov, “Moiré apodized reflective VBG,” Opt. Eng. 57(03), 1 (2018).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (1)

Z. Sheng-bao, Z. Shang-hong, C. Xing-chun, W. Zhuo-liang, and S. Lei, “Spectral beam combining of fiber lasers based on a transmitting volume Bragg grating,” Opt. Laser Technol. 42(2), 308–312 (2010).
[Crossref]

Opt. Lett. (8)

L. Siiman, J. Lumeau, L. Canioni, and L. Glebov, “Ultrashort laser pulse diffraction by transmitting volume Bragg gratings in photo-thermo-refractive glass,” Opt. Lett. 34(17), 2572–2574 (2009).
[Crossref]

X. Zhang, X. Yuan, J. Feng, F. Gao, B. Xiong, and K. Zou, “Optimization of spatial filter with volume Bragg gratings in photo-thermo-refractive glass,” Opt. Lett. 39(3), 663–665 (2014).
[Crossref]

X. Zhang, X. Yuan, S. Wu, J. Feng, K. Zou, and G. Zhang, “Two-dimensional angular filtering by volume Bragg gratings in photothermorefractive glass,” Opt. Lett. 36(11), 2167–2169 (2011).
[Crossref]

S. Wu, X. Yuan, X. Zhang, J. Feng, K. Zou, and G. Zhang, “Broadband angular filtering with a volume Bragg grating and a surface grating pair,” Opt. Lett. 39(14), 4068–4071 (2014).
[Crossref]

A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33(4), 384–386 (2008).
[Crossref]

O. Efimov, L. Glebov, and V. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett. 25(23), 1693–1695 (2000).
[Crossref]

B. Anderson, G. Venus, D. Ott, I. Divliansky, J. W. Dawson, D. Drachenberg, M. Messerly, P. Pax, J. Tassano, and L. Glebov, “Fundamental mode operation of a ribbon fiber laser by way of volume Bragg gratings,” Opt. Lett. 39(22), 6498–6500 (2014).
[Crossref]

V. Smirnov, J. Lumeau, S. Mokhov, B. Zeldovich, and L. Glebov, “Ultranarrow bandwidth moiré reflecting Bragg gratings recorded in photo-thermo-refractive glass,” Opt. Lett. 35(4), 592–594 (2010).
[Crossref]

Other (1)

R. Kashyap, Fiber Bragg Gratings, 2nd ed., (Academic Press, Burlington, MA, 2009), pp. 189–216.

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Figures (4)

Fig. 1.
Fig. 1. (a) Uniform reflecting VBG; (b) Uniform transmitting VBG.
Fig. 2.
Fig. 2. (a) Moiré modulation along the Bragg modulation for creation of an apodized reflecting VBG; (b) Moiré modulation perpendicular to the Bragg modulation for creation of an apodized transmitting VBG.
Fig. 3.
Fig. 3. Diffraction efficiencies: η of moiré apodized TVBG (solid line) and ηu of uniform TVBG (dashed line).
Fig. 4.
Fig. 4. Dependence of diffraction efficiency η on dimensionless detuning Φ for different types of TVBGs: apodized (thick line), uniform (thin line), and apodized with 5% phase mismatch in apodization profile (dashed line).

Equations (37)

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n ( r ) = n 0 + n 1 ( r ) cos ( Q r + γ ) , | n 1 | << n 0 .
× E = i ω μ 0 H , × H = i ω ε 0 ε r ( r ) E , ε r ( r ) = n 0 2 + n 0 n 1 ( r ) ( e i Q r + i γ + e i Q r i γ ) .
2 E + k 2 ( 1 + n 0 1 n 1 ( r ) ( e i Q r + i γ + e i Q r i γ ) ) E = 0 , k = n 0 ω / n 0 ω c c = 2 π n 0 / 2 π n 0 λ λ .
E ( r ) = A ( r ) e i k A r + B ( r ) e i k B r , k A,B = u A,B k , | u A,B | = 1.
2 ( A ( r ) e i k A r ) = ( 2 i ( k A A ) k 2 A ) e i k A r .
2 i ( k A A ) + k 2 n 0 1 n 1 ( r ) e i γ e i ( Q + k B k A ) r B = 0.
{ ( u A A ) = i κ ( r ) e 2 i D r B ( r ) , ( u B B ) = i κ ( r ) e 2 i D r A ( r ) , κ TE ( r ) = e i γ π n 1 ( r ) / π n 1 ( r ) λ λ , D = 1 2 ( k A k B Q ) .
κ TM ( r ) = κ TE ( r ) cos ( 2 θ ) , θ = ( k A , Q ) .
cos ( Q 1 r ) + cos ( Q 2 r ) = 2 cos ( Mr ) cos ( Qr ) , Q = 1 2 ( Q 1 + Q 2 ) , M = 1 2 ( Q 1 Q 2 ) .
n ( r ) = n 0 + n 1 ( z ) cos ( Q x ) , Q = 2 π / 2 π Λ Λ , 0 z l .
n 0 sin θ = sin θ air , sin θ B,air = n 0 sin θ B , u A = k A / k A k k = ( u A , x , u A , z ) = ( cos θ , sin θ ) , u B = k B / k B k k = ( u B , x , u B , z ) = ( cos θ B , sin θ B ) .
D = 0 k B = k A Q .
Q = 2 k res sin θ 0 , k res = 2 π n 0 / 2 π n 0 λ res λ res , Λ = λ res / λ res ( 2 n 0 sin θ 0 ) ( 2 n 0 sin θ 0 ) .
D x = 0 sin θ B = Q / Q k k sin θ .
{ cos θ d d z A = i κ ( z ) e 2 i D z z B ( z ) , cos θ B d d z B = i κ ( z ) e 2 i D z z A ( z ) , κ TE ( z ) = e i γ π n 1 ( z ) / π n 1 ( z ) λ λ , D z = π n 0 λ 1 ( cos θ cos θ B ) .
A = a e i D z z + i γ / γ 2 2 , B = b e i D z z i γ / γ 2 2 , d d z ( a b ) = ( i D z i κ / i κ cos θ 0 cos θ 0 i κ / i κ cos θ 0 cos θ 0 i D z ) ( a b ) , κ TE ( z ) = π n 1 ( z ) / π n 1 ( z ) λ res λ res , κ TM = κ TE cos ( 2 θ 0 ) .
n 1 ( z ) = n ¯ 1 = const , ( a ( l ) b ( l ) ) = ( p q q p ) ( a ( 0 ) b ( 0 ) ) , p = cos G + i Φ G 1 sin G , q = i S u G 1 sin G , | p | 2 + | q | 2 = 1 , G = S u 2 + Φ 2 , Φ = D z l = π n 0 λ 1 ( cos θ cos θ B ) l , S u,TE = π n ¯ 1 l / π n ¯ 1 l ( λ res cos θ 0 ) ( λ res cos θ 0 ) , S u,TM = S u,TE cos ( 2 θ 0 ) .
Φ = D z l = 2 π l λ res tan θ 0 ( sin θ air , 0 λ res Δ λ cos θ air , 0 Δ θ air ) , Δ λ = λ λ res , Δ θ air = θ air θ air , 0 .
B ( l ) A ( 0 ) = e i Φ i γ q , η u = | B ( l ) | 2 | A ( 0 ) | 2 = | q | 2 = S u 2 S u 2 + Φ 2 sin 2 S u 2 + Φ 2 .
η u = 1 S u = π / π 2 2 n ¯ 1 , TE = λ res cos θ 0 / λ res cos θ 0 ( 2 l ) ( 2 l ) .
n 1 ( z ) = N 1 sin ( π z / π z l l ) .
D z = 0 : ( a ( l ) b ( l ) ) = ( cos S i sin S i sin S cos S ) ( a ( 0 ) b ( 0 ) ) , S TE = π λ res cos θ 0 0 l n 1 ( z ) d z = 2 N 1 l λ res cos θ 0 , S TM = S TE cos ( 2 θ 0 ) .
η ( S , Φ ) = | B ( l ) | 2 / | B ( l ) | 2 | A ( 0 ) | 2 | A ( 0 ) | 2 ; Φ = 0 : B ( l ) = i sin S 0 e i γ A ( 0 ) , η 0 = sin 2 S 0 2 .
η 0 = 1 S 0 = π / π 2 2 N 1 , TE = π λ res cos θ 0 / π λ res cos θ 0 ( 4 l ) , ( 4 l ) , N 1 , TM = N 1 , TE / N 1 , TE cos ( 2 θ 0 ) cos ( 2 θ 0 ) .
sin ( θ air , 0 + δ θ B,air ) = 2 sin θ air , 0 ( 1 + δ λ / δ λ λ res λ res ) sin θ air , 0 δ θ B,air / δ θ B,air δ λ δ λ = 2 tan θ air , 0 / 2 tan θ air , 0 λ res λ res .
A ( z ) = A ( 0 ) : d d z B = i 2 e i γ π l 1 S sin ( π z / π z l l ) e 2 i D z z A ( 0 ) , ξ = z / z l l : B ( l ) / B ( l ) A ( 0 ) A ( 0 ) = i 2 e i γ π S 0 1 sin ( π ξ ) e 2 i Φ ξ d ξ = i e i γ π 2 S e i Φ cos Φ / cos Φ ( 4 Φ 2 π 2 ) ( 4 Φ 2 π 2 ) , η = | B ( l ) | 2 / | B ( l ) | 2 | A ( 0 ) | | A ( 0 ) | 2 = π 4 S 2 cos 2 Φ / cos 2 Φ ( 4 Φ 2 π 2 ) 2 ( 4 Φ 2 π 2 ) 2 , η = 1 2 π 4 S 2 / 1 2 π 4 S 2 ( 4 Φ 2 π 2 ) 2 ( 4 Φ 2 π 2 ) 2 .
Φ >> S u : η u = S u 2 / S u 2 Φ 2 Φ 2 sin 2 Φ , η u = 1 2 S u 2 / S u 2 Φ 2 Φ 2 .
H = e y H , H k A , k B , Q , Q = e x Q .
2 H = ε r × ( i ω ε 0 E ) i ω ε 0 ε r × E = g × ( × H ) + ω 2 c 2 ε r H , g = ε r 1 ε r .
g = e x g , g = i n 0 1 n 1 ( r ) ( e i Q r + i γ e i Q r i γ ) Q .
g × ( × H ) = ( g ) H = e y g H / H x x .
2 H + k 2 ( 1 + n 0 1 n 1 ( r ) ( e i Q r + i γ + e i Q r i γ ) ) H g H / H x x = 0.
H ( r ) = A m ( r ) e i k A r + B m ( r ) e i k B r .
k A k B Q 0 : k A , x = k cos θ , θ = ( k A , Q ) , Q 2 k cos θ , k B , x = k cos θ B , θ B θ .
H / H x x = i k cos θ ( A m ( r ) e i k A r B m ( r ) e i k B r ) .
2 i ( k A A m ) cos ( 2 θ ) k 2 n 0 1 n 1 ( r ) e i γ B m ( r ) e i ( k B + Q k A ) r = 0.
{ ( u A A m ) = i κ ( r ) e 2 i D r B m ( r ) , ( u B B m ) = i κ ( r ) e 2 i D r A m ( r ) , κ TM ( r ) = e i γ cos ( 2 θ ) π n 1 ( r ) / π n 1 ( r ) λ λ , D = 1 2 ( k A k B Q ) .

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