Abstract

The profile reshaping for a Gaussian beam reflected from a symmetrical metal-cladding waveguide is studied numerically and experimentally. For the thick guiding film, the distortion of a beam with very narrow frequency distribution is remarkable when the center wavelength of the beam locates in the resonant dip. This result compels us to question the validity of the method of using a position sensitive detector to measure the larger Goos–Hänchen (GH) shift. Therefore, a CCD is introduced to observe and record in real time the profile of the reflected beam, and the experiment conforms well to the simulations. The shaping technique developed in this paper promises much potential for optics devices based on the GH effect.

© 2011 Optical Society of America

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  1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalflexion,” Ann. Phys. 436, 333-346 (1947).
    [CrossRef]
  2. F. Goos and H. Hänchen, “Neumessung des strahlversetzugseffektes bei totalreflexion,” Ann. Phys. 440, 251-252 (1949).
    [CrossRef]
  3. J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50, 1664-1667(1983).
    [CrossRef]
  4. X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372-374(2004).
    [CrossRef]
  5. L. Chen, Z. Cao, F. Ou, H. Li, Q. Shen, and H. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. 32, 1432-1434 (2007).
    [CrossRef] [PubMed]
  6. J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14, 3024-3029 (2006).
    [CrossRef] [PubMed]
  7. L. Wang, H. Chen, and S. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30, 2936-2938 (2005).
    [CrossRef] [PubMed]
  8. X. Chen and C. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617 (2004).
    [CrossRef]
  9. T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841-2843 (2000).
    [CrossRef]
  10. Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
    [CrossRef]
  11. X. Yin and L. Hessenlink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
    [CrossRef]
  12. K. Artman, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437, 87-102 (1948).
    [CrossRef]
  13. X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
    [CrossRef]
  14. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330-7338 (2000).
    [CrossRef]
  15. P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
    [CrossRef]
  16. H. Lu, Z. Cao, H. Li, and Q. Shen, “Study of ultrahigh-order modes in a symmetrical metal-cladding optical waveguide,” Appl. Phys. Lett. 85, 4579-4581 (2004).
    [CrossRef]
  17. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1-69 (2006).
    [CrossRef]
  18. W. P. Chen and J. M. Chen, “Use of surface plasma waves for determination of the thickness and optical constants of thin metallic films,” J. Opt. Soc. Am. 71, 189-191 (1981).
    [CrossRef]
  19. P. K. Tien and R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325-1337(1970).
    [CrossRef]

2010

X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
[CrossRef]

2008

Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
[CrossRef]

2007

2006

J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14, 3024-3029 (2006).
[CrossRef] [PubMed]

X. Yin and L. Hessenlink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1-69 (2006).
[CrossRef]

2005

2004

X. Chen and C. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617 (2004).
[CrossRef]

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372-374(2004).
[CrossRef]

H. Lu, Z. Cao, H. Li, and Q. Shen, “Study of ultrahigh-order modes in a symmetrical metal-cladding optical waveguide,” Appl. Phys. Lett. 85, 4579-4581 (2004).
[CrossRef]

2000

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330-7338 (2000).
[CrossRef]

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841-2843 (2000).
[CrossRef]

1988

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

1983

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50, 1664-1667(1983).
[CrossRef]

1981

1970

1949

F. Goos and H. Hänchen, “Neumessung des strahlversetzugseffektes bei totalreflexion,” Ann. Phys. 440, 251-252 (1949).
[CrossRef]

1948

K. Artman, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437, 87-102 (1948).
[CrossRef]

1947

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalflexion,” Ann. Phys. 436, 333-346 (1947).
[CrossRef]

Artman, K.

K. Artman, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437, 87-102 (1948).
[CrossRef]

Bado, P.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

Birman, J. L.

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50, 1664-1667(1983).
[CrossRef]

Cao, Z.

X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
[CrossRef]

Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
[CrossRef]

L. Chen, Z. Cao, F. Ou, H. Li, Q. Shen, and H. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. 32, 1432-1434 (2007).
[CrossRef] [PubMed]

H. Lu, Z. Cao, H. Li, and Q. Shen, “Study of ultrahigh-order modes in a symmetrical metal-cladding optical waveguide,” Appl. Phys. Lett. 85, 4579-4581 (2004).
[CrossRef]

Chen, H.

Chen, J. M.

Chen, L.

Chen, W. P.

Chen, X.

X. Chen and C. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617 (2004).
[CrossRef]

Fang, N.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372-374(2004).
[CrossRef]

Goos, F.

F. Goos and H. Hänchen, “Neumessung des strahlversetzugseffektes bei totalreflexion,” Ann. Phys. 440, 251-252 (1949).
[CrossRef]

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalflexion,” Ann. Phys. 436, 333-346 (1947).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, “Neumessung des strahlversetzugseffektes bei totalreflexion,” Ann. Phys. 440, 251-252 (1949).
[CrossRef]

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalflexion,” Ann. Phys. 436, 333-346 (1947).
[CrossRef]

He, J.

He, S.

He, Y.

Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
[CrossRef]

Hesselink, L.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372-374(2004).
[CrossRef]

Hessenlink, L.

X. Yin and L. Hessenlink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330-7338 (2000).
[CrossRef]

Lai, H. M.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330-7338 (2000).
[CrossRef]

Li, C.

X. Chen and C. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617 (2004).
[CrossRef]

Li, H.

Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
[CrossRef]

L. Chen, Z. Cao, F. Ou, H. Li, Q. Shen, and H. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. 32, 1432-1434 (2007).
[CrossRef] [PubMed]

H. Lu, Z. Cao, H. Li, and Q. Shen, “Study of ultrahigh-order modes in a symmetrical metal-cladding optical waveguide,” Appl. Phys. Lett. 85, 4579-4581 (2004).
[CrossRef]

Li, T.

X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
[CrossRef]

Liu, X.

X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
[CrossRef]

Liu, Z.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372-374(2004).
[CrossRef]

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330-7338 (2000).
[CrossRef]

Lu, H.

H. Lu, Z. Cao, H. Li, and Q. Shen, “Study of ultrahigh-order modes in a symmetrical metal-cladding optical waveguide,” Appl. Phys. Lett. 85, 4579-4581 (2004).
[CrossRef]

Maine, P.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

Mourou, G.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

Ou, F.

Pattanayak, D. N.

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50, 1664-1667(1983).
[CrossRef]

Pessot, M.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

Puri, A.

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50, 1664-1667(1983).
[CrossRef]

Qiao, H.

Qiao, Z.

X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
[CrossRef]

Sakata, T.

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841-2843 (2000).
[CrossRef]

Shen, Q.

Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
[CrossRef]

L. Chen, Z. Cao, F. Ou, H. Li, Q. Shen, and H. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. 32, 1432-1434 (2007).
[CrossRef] [PubMed]

H. Lu, Z. Cao, H. Li, and Q. Shen, “Study of ultrahigh-order modes in a symmetrical metal-cladding optical waveguide,” Appl. Phys. Lett. 85, 4579-4581 (2004).
[CrossRef]

Shimokawa, F.

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841-2843 (2000).
[CrossRef]

Strickland, D.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

Tien, P. K.

Togo, H.

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841-2843 (2000).
[CrossRef]

Ulrich, R.

Wang, L.

Wang, Y.

Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
[CrossRef]

Winful, H. G.

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1-69 (2006).
[CrossRef]

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330-7338 (2000).
[CrossRef]

Yang, Q.

X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
[CrossRef]

Yi, J.

Yin, X.

X. Yin and L. Hessenlink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372-374(2004).
[CrossRef]

Yu, T.

Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
[CrossRef]

Zhang, X.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372-374(2004).
[CrossRef]

Zhu, P.

X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
[CrossRef]

Zhu, S.

Ann. Phys.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalflexion,” Ann. Phys. 436, 333-346 (1947).
[CrossRef]

F. Goos and H. Hänchen, “Neumessung des strahlversetzugseffektes bei totalreflexion,” Ann. Phys. 440, 251-252 (1949).
[CrossRef]

K. Artman, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437, 87-102 (1948).
[CrossRef]

Appl. Phys. Lett.

H. Lu, Z. Cao, H. Li, and Q. Shen, “Study of ultrahigh-order modes in a symmetrical metal-cladding optical waveguide,” Appl. Phys. Lett. 85, 4579-4581 (2004).
[CrossRef]

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372-374(2004).
[CrossRef]

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841-2843 (2000).
[CrossRef]

Y. Wang, H. Li, Z. Cao, T. Yu, Q. Shen, and Y. He, “Oscillating wave sensor based on the Goos-Hänchen effect,” Appl. Phys. Lett. 92, 061117 (2008).
[CrossRef]

X. Yin and L. Hessenlink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

IEEE J. Quantum Electron.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

X. Liu, Q. Yang, Z. Qiao, T. Li, P. Zhu, and Z. Cao, “Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system,” Opt. Commun. 283, 2681-2685 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1-69 (2006).
[CrossRef]

Phys. Rev. E

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330-7338 (2000).
[CrossRef]

X. Chen and C. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617 (2004).
[CrossRef]

Phys. Rev. Lett.

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50, 1664-1667(1983).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Configuration of the SMCW. The parameters are as follows: ε 1 = 2.25 , ε 2 = ε 4 = 28 + 1.8 i , ε 3 = 1 , d 2 = 25 nm , and d 3 = 0.5 mm .

Fig. 2
Fig. 2

Reflectivity versus wavelength in the SMCW and the thickness of the guiding film are d 3 = 0.1 mm , 0.3 mm , and 0.5 mm , respectively. Other parameters are the same as in Fig. 1. The incident angles are selected such that the reflectivity of λ = 858.94 nm is minimal. The shaded area corresponds to the resonant dip for the case of d 3 = 0.5 mm .

Fig. 3
Fig. 3

Profile of the Gaussian incident beam with the waist width ω 0 = 800 μm (black solid curve). The center wavelength of the incident beam is tunable. The profiles of the corresponding reflected beam form the case d 3 = 0.5 mm in Fig. 2 (dashed, dashed–dotted, and dotted curves). Perpendicular dashed lines are the magnitudes of the GH shift.

Fig. 4
Fig. 4

Schematic drawing of the experimental setup.

Fig. 5
Fig. 5

Cross-section intensity profiles of the reflected beam.

Equations (5)

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

r 1234 = r 12 + r 12 r 23 r 34 exp ( 2 i k 3 z d 3 ) + [ r 23 + r 34 exp ( 2 i k 3 z d 3 ) ] exp ( 2 i k 2 z d 2 ) 1 + r 23 r 34 exp ( 2 i k 3 z d 3 ) + r 12 [ r 23 + r 34 exp ( 2 i k 3 z d 3 ) ] exp ( 2 i k 2 z d 2 ) ,
r i j = { k i z / ε i k j z / ε j k i z / ε i + k j z / ε j for TM polarization k i z k j z k i z + k j z for TE polarization ,
ψ i ( x , z = 0 ) = 1 2 π A ( k x ) exp ( i k x x ) d k x ,
ψ r ( x , z = 0 ) = 1 2 π r 1234 ( k x ) A ( k x ) exp ( i k x x ) d k x .
L = 2 Im ( Δ β rad ) Im ( β 0 ) 2 Im ( Δ β rad ) 2 cos θ r ,

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