Abstract

The performance of an acousto-optic deflector is studied for two-dimensional refractive index that varies as periodic and sinc functions in the transverse and longitudinal directions, respectively, with respect to the direction of light propagation. Phased array piezoelectric transducers can be operated at different phase shifts to produce a two-dimensionally inhomogeneous domain of phase grating in the acousto-optic media. Also this domain can be steered at different angles by selecting the phase shift appropriately. This mechanism of dynamically tilting the refractive index-modulated domain enables adjusting the incident angle of light on the phase grating plane without moving the light source. So the Bragg angle of incidence can be always achieved at any acoustic frequency, and consequently, the deflector can operate under the Bragg diffraction condition at the optimum diffraction efficiency. Analytic solutions are obtained for the Bragg diffraction of plane waves based on the second order coupled mode theory, and the diffraction efficiency is found to be unity for optimal index modulations at certain acoustic parameters.

© 2015 Optical Society of America

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References

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  1. G. D. Reddy, R. J. Cotton, A. S. Tolias, and P. Saggau, “Random-access multiphoton microscopy for fast three-dimensional imaging,” in Membrane Potential Imaging in the Nervous System and Heart, M. Canepari, D. Zecevic, and O. Bernus, ed. (Springer, 2015), Ch. 18.
  2. J.-C. Kastelik, S. Dupont, K. B. Yushkov, and J. Gazalet, “Frequency and angular bandwidth of acousto-optic deflectors with ultrasonic walk-off,” Ultrasonics 53(1), 219–224 (2013).
    [Crossref] [PubMed]
  3. W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. 14(3), 123–134 (1967).
    [Crossref]
  4. R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microw. Theory Tech. 18(8), 486–504 (1970).
    [Crossref]
  5. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
    [Crossref]
  6. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. A 63(1), 37–45 (1973).
    [Crossref]
  7. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56(11), 1502–1508 (1966).
    [Crossref]
  8. T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
    [Crossref]
  9. R. S. Chu and T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media,” J. Opt. Soc. Am. 66(3), 220–226 (1976).
    [Crossref]
  10. R. S. Chu and T. Tamir, “Diffraction of Gaussian beams by periodically modulated media for incidence close to a Bragg angle,” J. Opt. Soc. Am. 66(12), 1438–1440 (1976).
    [Crossref]
  11. R. S. Chu, J. A. Kong, and T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,” J. Opt. Soc. Am. 67(11), 1555–1561 (1977).
    [Crossref]
  12. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70(3), 300–304 (1980).
    [Crossref]
  13. N. Uchida and N. Niizeki, “Acoustooptic deflection materials and techniques,” Proc. IEEE 61(8), 1073–1092 (1973).
    [Crossref]
  14. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67(6), 825–829 (1977).
    [Crossref]
  15. J.-M. Andre, K. L. Guen, and P. Jonnard, “Rigorous coupled-wave theory for lossy volume grating in Laue geometry X-ray spectroscopy,” hal-01082017 (2014).
  16. L. Azar, Y. Shi, and S.-C. Wooh, “Beam focusing behavior of linear phased arrays,” NDT Int. 33(3), 189–198 (2000).
    [Crossref]
  17. X. Zhao and T. Gang, “Nonparaxial multi-Gaussian beam models and measurement models for phased array transducers,” Ultrasonics 49(1), 126–130 (2009).
    [Crossref] [PubMed]
  18. K. Nakahata and N. Kono, “3-D modelings of an ultrasonic phased array transducer and its radiation properties in solid,” in Ultrasonic Waves, A. A. dos Santos, Jr., ed. (InTech, 2012), p. 59–80.
  19. M. Gottlieb, L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Deflection (Marcel Dekker, 1983).
  20. S.-C. Wooh and Y. Shi, “Influence of phased array element size on beam steering behavior,” Ultrasonics 36(6), 737–749 (1998).
    [Crossref]
  21. S.-C. Wooh and Y. Shi, “Optimum beam steering of linear phased arrays,” Wave Motion 29(3), 245–265 (1999).
    [Crossref]
  22. S.-C. Wooh and Y. Shi, “A simulation Study of the beam steering characteristics for linear phased arrays,” JNE 18, 39–57 (1999).
  23. A. Korpel, Acousto-Optics (CRC Press, 1997).
  24. B. M. Watrasiewicz, “Some useful integrals of Si(x), Ci(x) and related integrals,” Opt. Acta (Lond.) 14(3), 317–322 (1967).
    [Crossref]
  25. F. G. Tricomi, Integral Equations (Dover, 1985).
  26. J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990).
  27. J. P. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, 1982).
  28. N. Uchida and Y. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40(12), 4692–4695 (1969).
    [Crossref]
  29. A. P. Goutzoulis and V. V. Kludzin, Principles of acousto-optics” in Design and Fabrication of Acousto-Optic Devices, A. P. Goutzoulis and D. R. Pape, ed. (Marcel Dekker, 1994).
  30. D. R. Pape, O. B. Gusev, S. V. Kulakov, and V. V. Momotok, “Design of acousto-optic deflectors” in Design and Fabrication of Acousto-Optic Devices, A. P. Goutzoulis and D. R. Pape, eds. (Marcel Dekker, 1994).

2013 (1)

J.-C. Kastelik, S. Dupont, K. B. Yushkov, and J. Gazalet, “Frequency and angular bandwidth of acousto-optic deflectors with ultrasonic walk-off,” Ultrasonics 53(1), 219–224 (2013).
[Crossref] [PubMed]

2009 (1)

X. Zhao and T. Gang, “Nonparaxial multi-Gaussian beam models and measurement models for phased array transducers,” Ultrasonics 49(1), 126–130 (2009).
[Crossref] [PubMed]

2000 (1)

L. Azar, Y. Shi, and S.-C. Wooh, “Beam focusing behavior of linear phased arrays,” NDT Int. 33(3), 189–198 (2000).
[Crossref]

1999 (2)

S.-C. Wooh and Y. Shi, “Optimum beam steering of linear phased arrays,” Wave Motion 29(3), 245–265 (1999).
[Crossref]

S.-C. Wooh and Y. Shi, “A simulation Study of the beam steering characteristics for linear phased arrays,” JNE 18, 39–57 (1999).

1998 (1)

S.-C. Wooh and Y. Shi, “Influence of phased array element size on beam steering behavior,” Ultrasonics 36(6), 737–749 (1998).
[Crossref]

1982 (1)

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
[Crossref]

1980 (1)

1977 (2)

1976 (2)

1973 (2)

N. Uchida and N. Niizeki, “Acoustooptic deflection materials and techniques,” Proc. IEEE 61(8), 1073–1092 (1973).
[Crossref]

F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. A 63(1), 37–45 (1973).
[Crossref]

1970 (1)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microw. Theory Tech. 18(8), 486–504 (1970).
[Crossref]

1969 (2)

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

N. Uchida and Y. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40(12), 4692–4695 (1969).
[Crossref]

1967 (2)

B. M. Watrasiewicz, “Some useful integrals of Si(x), Ci(x) and related integrals,” Opt. Acta (Lond.) 14(3), 317–322 (1967).
[Crossref]

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. 14(3), 123–134 (1967).
[Crossref]

1966 (1)

Azar, L.

L. Azar, Y. Shi, and S.-C. Wooh, “Beam focusing behavior of linear phased arrays,” NDT Int. 33(3), 189–198 (2000).
[Crossref]

Burckhardt, C. B.

Chu, R. S.

Cook, B. D.

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. 14(3), 123–134 (1967).
[Crossref]

Dupont, S.

J.-C. Kastelik, S. Dupont, K. B. Yushkov, and J. Gazalet, “Frequency and angular bandwidth of acousto-optic deflectors with ultrasonic walk-off,” Ultrasonics 53(1), 219–224 (2013).
[Crossref] [PubMed]

Gang, T.

X. Zhao and T. Gang, “Nonparaxial multi-Gaussian beam models and measurement models for phased array transducers,” Ultrasonics 49(1), 126–130 (2009).
[Crossref] [PubMed]

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
[Crossref]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70(3), 300–304 (1980).
[Crossref]

Gazalet, J.

J.-C. Kastelik, S. Dupont, K. B. Yushkov, and J. Gazalet, “Frequency and angular bandwidth of acousto-optic deflectors with ultrasonic walk-off,” Ultrasonics 53(1), 219–224 (2013).
[Crossref] [PubMed]

Kaspar, F. G.

F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. A 63(1), 37–45 (1973).
[Crossref]

Kastelik, J.-C.

J.-C. Kastelik, S. Dupont, K. B. Yushkov, and J. Gazalet, “Frequency and angular bandwidth of acousto-optic deflectors with ultrasonic walk-off,” Ultrasonics 53(1), 219–224 (2013).
[Crossref] [PubMed]

Klein, W. R.

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. 14(3), 123–134 (1967).
[Crossref]

Kogelnik, H.

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

Kong, J. A.

Magnusson, R.

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
[Crossref]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70(3), 300–304 (1980).
[Crossref]

Niizeki, N.

N. Uchida and N. Niizeki, “Acoustooptic deflection materials and techniques,” Proc. IEEE 61(8), 1073–1092 (1973).
[Crossref]

Ohmachi, Y.

N. Uchida and Y. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40(12), 4692–4695 (1969).
[Crossref]

Shi, Y.

L. Azar, Y. Shi, and S.-C. Wooh, “Beam focusing behavior of linear phased arrays,” NDT Int. 33(3), 189–198 (2000).
[Crossref]

S.-C. Wooh and Y. Shi, “A simulation Study of the beam steering characteristics for linear phased arrays,” JNE 18, 39–57 (1999).

S.-C. Wooh and Y. Shi, “Optimum beam steering of linear phased arrays,” Wave Motion 29(3), 245–265 (1999).
[Crossref]

S.-C. Wooh and Y. Shi, “Influence of phased array element size on beam steering behavior,” Ultrasonics 36(6), 737–749 (1998).
[Crossref]

Tamir, T.

Uchida, N.

N. Uchida and N. Niizeki, “Acoustooptic deflection materials and techniques,” Proc. IEEE 61(8), 1073–1092 (1973).
[Crossref]

N. Uchida and Y. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40(12), 4692–4695 (1969).
[Crossref]

Watrasiewicz, B. M.

B. M. Watrasiewicz, “Some useful integrals of Si(x), Ci(x) and related integrals,” Opt. Acta (Lond.) 14(3), 317–322 (1967).
[Crossref]

Wooh, S.-C.

L. Azar, Y. Shi, and S.-C. Wooh, “Beam focusing behavior of linear phased arrays,” NDT Int. 33(3), 189–198 (2000).
[Crossref]

S.-C. Wooh and Y. Shi, “A simulation Study of the beam steering characteristics for linear phased arrays,” JNE 18, 39–57 (1999).

S.-C. Wooh and Y. Shi, “Optimum beam steering of linear phased arrays,” Wave Motion 29(3), 245–265 (1999).
[Crossref]

S.-C. Wooh and Y. Shi, “Influence of phased array element size on beam steering behavior,” Ultrasonics 36(6), 737–749 (1998).
[Crossref]

Yushkov, K. B.

J.-C. Kastelik, S. Dupont, K. B. Yushkov, and J. Gazalet, “Frequency and angular bandwidth of acousto-optic deflectors with ultrasonic walk-off,” Ultrasonics 53(1), 219–224 (2013).
[Crossref] [PubMed]

Zhao, X.

X. Zhao and T. Gang, “Nonparaxial multi-Gaussian beam models and measurement models for phased array transducers,” Ultrasonics 49(1), 126–130 (2009).
[Crossref] [PubMed]

Appl. Phys. B (1)

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28(1), 1–14 (1982).
[Crossref]

Bell Syst. Tech. J. (1)

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

IEEE Trans. Microw. Theory Tech. (1)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microw. Theory Tech. 18(8), 486–504 (1970).
[Crossref]

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. 14(3), 123–134 (1967).
[Crossref]

J. Appl. Phys. (1)

N. Uchida and Y. Ohmachi, “Elastic and photoelastic properties of TeO2 single crystal,” J. Appl. Phys. 40(12), 4692–4695 (1969).
[Crossref]

J. Opt. Soc. Am. (6)

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

F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. A 63(1), 37–45 (1973).
[Crossref]

JNE (1)

S.-C. Wooh and Y. Shi, “A simulation Study of the beam steering characteristics for linear phased arrays,” JNE 18, 39–57 (1999).

NDT Int. (1)

L. Azar, Y. Shi, and S.-C. Wooh, “Beam focusing behavior of linear phased arrays,” NDT Int. 33(3), 189–198 (2000).
[Crossref]

Opt. Acta (Lond.) (1)

B. M. Watrasiewicz, “Some useful integrals of Si(x), Ci(x) and related integrals,” Opt. Acta (Lond.) 14(3), 317–322 (1967).
[Crossref]

Proc. IEEE (1)

N. Uchida and N. Niizeki, “Acoustooptic deflection materials and techniques,” Proc. IEEE 61(8), 1073–1092 (1973).
[Crossref]

Ultrasonics (3)

J.-C. Kastelik, S. Dupont, K. B. Yushkov, and J. Gazalet, “Frequency and angular bandwidth of acousto-optic deflectors with ultrasonic walk-off,” Ultrasonics 53(1), 219–224 (2013).
[Crossref] [PubMed]

X. Zhao and T. Gang, “Nonparaxial multi-Gaussian beam models and measurement models for phased array transducers,” Ultrasonics 49(1), 126–130 (2009).
[Crossref] [PubMed]

S.-C. Wooh and Y. Shi, “Influence of phased array element size on beam steering behavior,” Ultrasonics 36(6), 737–749 (1998).
[Crossref]

Wave Motion (1)

S.-C. Wooh and Y. Shi, “Optimum beam steering of linear phased arrays,” Wave Motion 29(3), 245–265 (1999).
[Crossref]

Other (10)

A. Korpel, Acousto-Optics (CRC Press, 1997).

K. Nakahata and N. Kono, “3-D modelings of an ultrasonic phased array transducer and its radiation properties in solid,” in Ultrasonic Waves, A. A. dos Santos, Jr., ed. (InTech, 2012), p. 59–80.

M. Gottlieb, L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Deflection (Marcel Dekker, 1983).

F. G. Tricomi, Integral Equations (Dover, 1985).

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990).

J. P. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, 1982).

A. P. Goutzoulis and V. V. Kludzin, Principles of acousto-optics” in Design and Fabrication of Acousto-Optic Devices, A. P. Goutzoulis and D. R. Pape, ed. (Marcel Dekker, 1994).

D. R. Pape, O. B. Gusev, S. V. Kulakov, and V. V. Momotok, “Design of acousto-optic deflectors” in Design and Fabrication of Acousto-Optic Devices, A. P. Goutzoulis and D. R. Pape, eds. (Marcel Dekker, 1994).

G. D. Reddy, R. J. Cotton, A. S. Tolias, and P. Saggau, “Random-access multiphoton microscopy for fast three-dimensional imaging,” in Membrane Potential Imaging in the Nervous System and Heart, M. Canepari, D. Zecevic, and O. Bernus, ed. (Springer, 2015), Ch. 18.

J.-M. Andre, K. L. Guen, and P. Jonnard, “Rigorous coupled-wave theory for lossy volume grating in Laue geometry X-ray spectroscopy,” hal-01082017 (2014).

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Figures (7)

Fig. 1
Fig. 1 Difference in the refractive index profiles due to Fig. 1(a) static phased array transducers in conventional AOD and Fig. 1(b) dynamic phased array transducers in this study.

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Fig. 2
Fig. 2 Two-dimensional refractive index profile generated by a tilted lobe in an acousto-optic medium.

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Fig. 3
Fig. 3 Reflectance and Transmittance as a function of the index modulation strength Δn with L = 2.24 cm, Q = 4π and F = 75 MHz at Bragg incidence angle of 0.324°.

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Fig. 4
Fig. 4 Diffraction efficiency as a function of the incident angle θin with L = 2.24 cm, Q = 4π and F = 75 MHz for different index modulation strengths Δn.

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Fig. 5
Fig. 5 Diffraction efficiency as a function of the incident angle θin with Δn = 2.2 × 10−5, Q = 4π for different RF frequencies.

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Fig. 6
Fig. 6 Diffraction efficiency as a function of the incident angle θin for Q = 4π.

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Fig. 7
Fig. 7 Comparison of the diffraction efficiency obtained from different models.

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Equations (24)

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n II ( x,z )= n 2 ( λ 0 )+Δncos( 2π Λ z )( sin( bx ) bx )
sin(bL/2) bL/2 =1/2
δn=− 1 2 n 2 3 ( λ 0 )pe
[ ∂ 2 ∂ x 2 + ∂ 2 ∂ z 2 + k 0 2 n II 2 ( x,z ) ] E II,y ( x,z )=0
E II,y ( x,z )= ∑ m=−∞ ∞ E m ( x ) e imπ/2 e i κ mz z
κ mz = k 0z +mKcos θ ˜
d 2 E m ( x ) d x 2 +( k 2 2 − κ mz 2 ) E m ( x )+i Δn n 2 ( sin( bx ) bx ) k 2 2 ( E m+1 − E m−1 )=0
d 2 E 0 d x 2 + ϕ 1 2 L 2 E 0 =i ϕ ϕ 1 L 2 ( sin( bx ) bx ) E −1
d 2 E −1 d x 2 + ϕ 1 2 β 2 L 2 E −1 =−i ϕ ϕ 1 L 2 ( sin( bx ) bx ) E 0
ϕ 1 = k 0 n 2 Lcos θ 2
ϕ= k 0 ΔnL cos θ 2
β= 1− Q ϕ 1 ( 1−α )
Q= K 2 L n 2 k 0 cos θ 2
α= 2 k 0 sin θ in K
E 0 (x)= A 1 e i ϕ 1 L x + A 2 e −i ϕ 1 L x + iϕ 4bL B 1 ∑ j=1 4 [ 1−2 δ j1 −2 δ j2 ] e i ( −1 ) j−1 ϕ 1 L x [ Si( b 1j x )−iCi( b 1j x ) ] + iϕ 4bL B 2 ∑ j=1 4 [ 1−2 δ j1 −2 δ j2 ] e i ( −1 ) j−1 ϕ 1 L x [ Si( b 2j x )−iCi( b 2j x ) ] + ϕ 2 4β L 2 ∑ j=1 2 ( −1 ) j e i (−1) j−1 ϕ 1 L x ∫ 0 x e i (−1) 2−j ϕ 1 L x ″ sin( b x ″ ) b x ″ ×[ ∑ p=1 2 (−1) p e i (−1) p−1 ϕ 1 β L x ″ ∫ 0 x ″ e i (−1) 2−p ϕ 1 β L x ′ sin( b x ′ ) b x ′ E 0 ( x ′ ) d x ′ ]d x ″
E −1 (x)= B 1 e i ϕ 1 β L x + B 2 e −i ϕ 1 β L x + iϕ 4bLβ A 1 ∑ j=1 4 [ 1−2 δ j1 −2 δ j2 ] e i ( −1 ) j−1 ϕ 1 β L x [ Si( a 1j x )−iCi( a 1j x ) ] + iϕ 4bLβ A 2 ∑ j=1 4 [ 1−2 δ j1 −2 δ j2 ] e i ( −1 ) j−1 ϕ 1 β L x [ Si( a 2j x )−iCi( a 2j x ) ] + ϕ 2 4β L 2 ∑ j=1 2 ( −1 ) j e i ( −1 ) j−1 ϕ 1 β L x ∫ 0 x e i ( −1 ) 2−j ϕ 1 β L x ″ sin( b x ″ ) b x ″ ×[ ∑ p=1 2 (−1) p e i ( −1 ) p−1 ϕ 1 L x ″ ∫ 0 x ″ e i ( −1 ) 2−p ϕ 1 L x ′ sin( b x ′ ) b x ′ E −1 ( x ′ ) d x ′ ]d x ″
b 11 = a 24 =b− ϕ 1 L ( 1−β ), b 21 = a 21 =b− ϕ 1 L ( 1+β ) b 12 = a 12 =−b+ ϕ 1 L ( 1+β ), b 22 = a 13 =−b+ ϕ 1 L ( 1−β ) b 13 = a 22 =−b− ϕ 1 L ( 1−β ), b 23 = a 23 =−b− ϕ 1 L ( 1+β ) b 14 = a 14 =b+ ϕ 1 L ( 1+β ), b 24 = a 11 =b+ ϕ 1 L ( 1−β )
E I,y (x,z)= G m e i k in,mx x e i k in,mz z + r 0 e −i κ 1,0x x e i κ 0z z + r −1 e −i κ 1,−1x x e i κ −1z z
E II,y (x,z)= E 0 (x) e i κ 0z z −i E −1 (x) e i κ −1z z
E III,y (x,z)= t 0 e i κ 3,0x x e i κ 0z z + t −1 e i κ 3,−1x x e i κ −1z z
κ r,mx = k r 2 − κ mz 2
At x=−L/2, E I,y ( x,z )= E II,y ( x,z ) and ∂ E I,y ∂x = ∂ E II,y ∂x
At x=L/2, E II,y ( x,z )= E III,y ( x,z ) and ∂ E II,y ∂x = ∂ E III,y ∂x
( κ 1,0x k 1x ) | r 0 | 2 +( κ 1,−1x k 1x ) | r −1 | 2 +( κ 3,0x k 1x ) | t 0 | 2 +( κ 3,−1x k 1x ) | t −1 | 2 =1

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