Abstract

We investigate in detail the origin of multiple resonance peaks experimentally observed in an all-fiber acousto-optic tunable filter built with a photonic crystal fiber having slightly deformed air-hole structure and non-circular outer cladding. A model for the acousto-optic mode coupling in the PCF is formulated taking into account both the acoustic and the optical birefringence. All experimental results are in good agreement with numerical calculations.

© 2008 Optical Society of America

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  1. H. S. Kim, S. H. Yun, I. K. Hwang, and B. Y. Kim, "All-fiber acousto-optic tunable notch filter with electronically controllable spectral profile," Opt. Lett. 22, 1476-1478 (1997).
    [CrossRef]
  2. K. J. Lee, D. I. Yeom, and B. Y. Kim, "Narrowband, polarization insensitive all-fiber acousto-optic tunable bandpass filter," Opt. Express 15, 2987-2992 (2007).
    [CrossRef] [PubMed]
  3. B. Y. Kim, J. N. Blake, H. E. Engan, and H. J. Shaw, "All-fiber acousto-optic frequency shifter," Opt. Lett. 11, 389-391 (1986).
    [CrossRef] [PubMed]
  4. T. Jin, Q. Li, J. Zhao, K. Cheng, and X. Liu, "Ultra-Broad-Band AOTF Based on Cladding Etched Single-Mode Fiber," IEEE Photon. Technol. Lett. 14, 1133-1135 (2002).
    [CrossRef]
  5. D. I. Yeom, H. S. Kim, M. S. Kang, H. S. Park, and B. Y. Kim, "Narrow-Bandwidth All-Fiber Acoustooptic Tunable Filter With Low Polarization-Sensitivity," IEEE Photon. Technol. Lett. 17, 2646-2648 (2005).
    [CrossRef]
  6. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
    [CrossRef] [PubMed]
  7. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, "Highly birefringent photonic crystal fibers," Opt. Lett. 25, 1325-1327 (2000).
    [CrossRef]
  8. W. H. Reeves, J. C. Knight, P. J. Roberts, and P. St. J. Russell, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
    [PubMed]
  9. A. Diez, A. Birks, W. H. Reeves, B. J. Mangan, and P. St. J. Russell, "Excitation of cladding modes in photonic crystal fibers by flexural acoustic waves," Opt. Lett. 25, 1499-1501 (2000).
    [CrossRef]
  10. K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
    [CrossRef]
  11. D-I. Yeom, P. Steinvurzel, B. J. Eggleton, S. D. Lim, and B. Y. Kim, "Tunable acoustic gratings in solid-core photonic bandgap fiber," Opt. Express 15, 3513-3518 (2007).
    [CrossRef] [PubMed]
  12. M. W. Haakestad and H. E. Engan, "Acoustooptic properties of a weakly multimode solid core photonic crystal fiber," J. Lightwave Technol. 24, 838-845 (2006).
    [CrossRef]
  13. M. W. Haakestad and H. E. Engan, "Acoustooptic characterization of a birefringent two-mode photonic crystal fiber," Opt. Express 14, 7319-7328 (2006).
    [CrossRef] [PubMed]
  14. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
    [CrossRef] [PubMed]
  15. Y-J. Lee, D-S. Song, S-H. Kim, J. Huh, and Y-H. Lee, "Modal characteristics of photonic crystal fibers," J. Opt. Soc. Korea 7, 47-52 (2003).
    [CrossRef]
  16. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, "Mode classification and degeneracy in photonic crystal fibers," Opt. Express 11, 1310-1321 (2003).
    [CrossRef] [PubMed]
  17. N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lægsgaad, "Small-core photonic crystal fibers with weakly disordered air-hole claddings," J. Opt. A 6, 221-223 (2004).
    [CrossRef]
  18. H. E. Engan, B. Y. Kim, J. N. Blake, and H. J. Shaw, "Propagation and optical interaction of guided acoustic waves in two-mode optical fibers," J. Lightwave Technol. 6, 428-436 (1988).
    [CrossRef]
  19. P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides-I. Summary of results," IEEE Trans. Microwave Theory Tech. MTT- 23, 421-429 (1975).
    [CrossRef]
  20. B. Langli and K. Bløtekjær, "Effect of acoustic birefringence on acoustooptic interaction in birefrengent two-mode optical fibers," J. Lightwave Technol. 21, 528-535 (2003).
    [CrossRef]
  21. M. V. Andres, M. J. Tudor, and K. W. H. Foulds, "Analysis of an interferometric optical fibre detection technique applied to silicon vibrating sensors," Electron. Lett. 23, 774-775 (1987).
    [CrossRef]
  22. Amnon Yariv, "Coupled mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
    [CrossRef]

2008

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

2007

2006

2005

D. I. Yeom, H. S. Kim, M. S. Kang, H. S. Park, and B. Y. Kim, "Narrow-Bandwidth All-Fiber Acoustooptic Tunable Filter With Low Polarization-Sensitivity," IEEE Photon. Technol. Lett. 17, 2646-2648 (2005).
[CrossRef]

2004

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lægsgaad, "Small-core photonic crystal fibers with weakly disordered air-hole claddings," J. Opt. A 6, 221-223 (2004).
[CrossRef]

2003

2002

W. H. Reeves, J. C. Knight, P. J. Roberts, and P. St. J. Russell, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
[PubMed]

T. Jin, Q. Li, J. Zhao, K. Cheng, and X. Liu, "Ultra-Broad-Band AOTF Based on Cladding Etched Single-Mode Fiber," IEEE Photon. Technol. Lett. 14, 1133-1135 (2002).
[CrossRef]

2001

2000

1997

1996

1988

H. E. Engan, B. Y. Kim, J. N. Blake, and H. J. Shaw, "Propagation and optical interaction of guided acoustic waves in two-mode optical fibers," J. Lightwave Technol. 6, 428-436 (1988).
[CrossRef]

1987

M. V. Andres, M. J. Tudor, and K. W. H. Foulds, "Analysis of an interferometric optical fibre detection technique applied to silicon vibrating sensors," Electron. Lett. 23, 774-775 (1987).
[CrossRef]

1986

1975

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides-I. Summary of results," IEEE Trans. Microwave Theory Tech. MTT- 23, 421-429 (1975).
[CrossRef]

1973

Amnon Yariv, "Coupled mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

Andres, M. V.

M. V. Andres, M. J. Tudor, and K. W. H. Foulds, "Analysis of an interferometric optical fibre detection technique applied to silicon vibrating sensors," Electron. Lett. 23, 774-775 (1987).
[CrossRef]

Arriaga, J.

Atkin, D. M.

Birks, A.

Birks, T. A.

Blake, J. N.

H. E. Engan, B. Y. Kim, J. N. Blake, and H. J. Shaw, "Propagation and optical interaction of guided acoustic waves in two-mode optical fibers," J. Lightwave Technol. 6, 428-436 (1988).
[CrossRef]

B. Y. Kim, J. N. Blake, H. E. Engan, and H. J. Shaw, "All-fiber acousto-optic frequency shifter," Opt. Lett. 11, 389-391 (1986).
[CrossRef] [PubMed]

Bløtekjær, K.

Cheng, K.

T. Jin, Q. Li, J. Zhao, K. Cheng, and X. Liu, "Ultra-Broad-Band AOTF Based on Cladding Etched Single-Mode Fiber," IEEE Photon. Technol. Lett. 14, 1133-1135 (2002).
[CrossRef]

Diez, A.

Eggleton, B. J.

Engan, H. E.

Folkenberg, J. R.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lægsgaad, "Small-core photonic crystal fibers with weakly disordered air-hole claddings," J. Opt. A 6, 221-223 (2004).
[CrossRef]

Foulds, K. W. H.

M. V. Andres, M. J. Tudor, and K. W. H. Foulds, "Analysis of an interferometric optical fibre detection technique applied to silicon vibrating sensors," Electron. Lett. 23, 774-775 (1987).
[CrossRef]

Guobin, R.

Haakestad, M. W.

Hansen, K. P.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lægsgaad, "Small-core photonic crystal fibers with weakly disordered air-hole claddings," J. Opt. A 6, 221-223 (2004).
[CrossRef]

Hong, K. S.

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

Huh, J.

Hwang, I. K.

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

H. S. Kim, S. H. Yun, I. K. Hwang, and B. Y. Kim, "All-fiber acousto-optic tunable notch filter with electronically controllable spectral profile," Opt. Lett. 22, 1476-1478 (1997).
[CrossRef]

Jin, T.

T. Jin, Q. Li, J. Zhao, K. Cheng, and X. Liu, "Ultra-Broad-Band AOTF Based on Cladding Etched Single-Mode Fiber," IEEE Photon. Technol. Lett. 14, 1133-1135 (2002).
[CrossRef]

Jin, W.

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Ju, J.

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

Kang, M. S.

D. I. Yeom, H. S. Kim, M. S. Kang, H. S. Park, and B. Y. Kim, "Narrow-Bandwidth All-Fiber Acoustooptic Tunable Filter With Low Polarization-Sensitivity," IEEE Photon. Technol. Lett. 17, 2646-2648 (2005).
[CrossRef]

Kim, B. Y.

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

D-I. Yeom, P. Steinvurzel, B. J. Eggleton, S. D. Lim, and B. Y. Kim, "Tunable acoustic gratings in solid-core photonic bandgap fiber," Opt. Express 15, 3513-3518 (2007).
[CrossRef] [PubMed]

K. J. Lee, D. I. Yeom, and B. Y. Kim, "Narrowband, polarization insensitive all-fiber acousto-optic tunable bandpass filter," Opt. Express 15, 2987-2992 (2007).
[CrossRef] [PubMed]

D. I. Yeom, H. S. Kim, M. S. Kang, H. S. Park, and B. Y. Kim, "Narrow-Bandwidth All-Fiber Acoustooptic Tunable Filter With Low Polarization-Sensitivity," IEEE Photon. Technol. Lett. 17, 2646-2648 (2005).
[CrossRef]

H. S. Kim, S. H. Yun, I. K. Hwang, and B. Y. Kim, "All-fiber acousto-optic tunable notch filter with electronically controllable spectral profile," Opt. Lett. 22, 1476-1478 (1997).
[CrossRef]

H. E. Engan, B. Y. Kim, J. N. Blake, and H. J. Shaw, "Propagation and optical interaction of guided acoustic waves in two-mode optical fibers," J. Lightwave Technol. 6, 428-436 (1988).
[CrossRef]

B. Y. Kim, J. N. Blake, H. E. Engan, and H. J. Shaw, "All-fiber acousto-optic frequency shifter," Opt. Lett. 11, 389-391 (1986).
[CrossRef] [PubMed]

Kim, H. S.

D. I. Yeom, H. S. Kim, M. S. Kang, H. S. Park, and B. Y. Kim, "Narrow-Bandwidth All-Fiber Acoustooptic Tunable Filter With Low Polarization-Sensitivity," IEEE Photon. Technol. Lett. 17, 2646-2648 (2005).
[CrossRef]

H. S. Kim, S. H. Yun, I. K. Hwang, and B. Y. Kim, "All-fiber acousto-optic tunable notch filter with electronically controllable spectral profile," Opt. Lett. 22, 1476-1478 (1997).
[CrossRef]

Kim, S-H.

Knight, J. C.

Lægsgaad, J.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lægsgaad, "Small-core photonic crystal fibers with weakly disordered air-hole claddings," J. Opt. A 6, 221-223 (2004).
[CrossRef]

Langli, B.

Lee, K. J.

Lee, Y-H.

Lee, Y-J.

Li, Q.

T. Jin, Q. Li, J. Zhao, K. Cheng, and X. Liu, "Ultra-Broad-Band AOTF Based on Cladding Etched Single-Mode Fiber," IEEE Photon. Technol. Lett. 14, 1133-1135 (2002).
[CrossRef]

Lim, S. D.

Liu, X.

T. Jin, Q. Li, J. Zhao, K. Cheng, and X. Liu, "Ultra-Broad-Band AOTF Based on Cladding Etched Single-Mode Fiber," IEEE Photon. Technol. Lett. 14, 1133-1135 (2002).
[CrossRef]

Mangan, B. J.

McIsaac, P. R.

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides-I. Summary of results," IEEE Trans. Microwave Theory Tech. MTT- 23, 421-429 (1975).
[CrossRef]

Mortensen, N. A.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lægsgaad, "Small-core photonic crystal fibers with weakly disordered air-hole claddings," J. Opt. A 6, 221-223 (2004).
[CrossRef]

Nielsen, M. D.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lægsgaad, "Small-core photonic crystal fibers with weakly disordered air-hole claddings," J. Opt. A 6, 221-223 (2004).
[CrossRef]

Ortigosa-Blanch, A.

Park, H. C.

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

Park, H. S.

D. I. Yeom, H. S. Kim, M. S. Kang, H. S. Park, and B. Y. Kim, "Narrow-Bandwidth All-Fiber Acoustooptic Tunable Filter With Low Polarization-Sensitivity," IEEE Photon. Technol. Lett. 17, 2646-2648 (2005).
[CrossRef]

Reeves, W. H.

Roberts, P. J.

Russell, P. St. J.

Shaw, H. J.

H. E. Engan, B. Y. Kim, J. N. Blake, and H. J. Shaw, "Propagation and optical interaction of guided acoustic waves in two-mode optical fibers," J. Lightwave Technol. 6, 428-436 (1988).
[CrossRef]

B. Y. Kim, J. N. Blake, H. E. Engan, and H. J. Shaw, "All-fiber acousto-optic frequency shifter," Opt. Lett. 11, 389-391 (1986).
[CrossRef] [PubMed]

Shuisheng, J.

Shuqin, L.

Song, D-S.

Steinvurzel, P.

Tudor, M. J.

M. V. Andres, M. J. Tudor, and K. W. H. Foulds, "Analysis of an interferometric optical fibre detection technique applied to silicon vibrating sensors," Electron. Lett. 23, 774-775 (1987).
[CrossRef]

Wadsworth, W. J.

Yeom, D. I.

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

K. J. Lee, D. I. Yeom, and B. Y. Kim, "Narrowband, polarization insensitive all-fiber acousto-optic tunable bandpass filter," Opt. Express 15, 2987-2992 (2007).
[CrossRef] [PubMed]

D. I. Yeom, H. S. Kim, M. S. Kang, H. S. Park, and B. Y. Kim, "Narrow-Bandwidth All-Fiber Acoustooptic Tunable Filter With Low Polarization-Sensitivity," IEEE Photon. Technol. Lett. 17, 2646-2648 (2005).
[CrossRef]

Yeom, D-I.

Yun, S. H.

Zhao, J.

T. Jin, Q. Li, J. Zhao, K. Cheng, and X. Liu, "Ultra-Broad-Band AOTF Based on Cladding Etched Single-Mode Fiber," IEEE Photon. Technol. Lett. 14, 1133-1135 (2002).
[CrossRef]

Zhi, W.

Appl. Phys. Lett.

K. S. Hong, H. C. Park, I. K. Hwang, W. Jin, J. Ju, D. I. Yeom, and B. Y. Kim "1000 nm tunable acousto-optic filter based on photonic crystal fiber," Appl. Phys. Lett. 92, 031110 (2008).
[CrossRef]

Electron. Lett.

M. V. Andres, M. J. Tudor, and K. W. H. Foulds, "Analysis of an interferometric optical fibre detection technique applied to silicon vibrating sensors," Electron. Lett. 23, 774-775 (1987).
[CrossRef]

IEEE J. Quantum Electron.

Amnon Yariv, "Coupled mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

IEEE Photon. Technol. Lett.

T. Jin, Q. Li, J. Zhao, K. Cheng, and X. Liu, "Ultra-Broad-Band AOTF Based on Cladding Etched Single-Mode Fiber," IEEE Photon. Technol. Lett. 14, 1133-1135 (2002).
[CrossRef]

D. I. Yeom, H. S. Kim, M. S. Kang, H. S. Park, and B. Y. Kim, "Narrow-Bandwidth All-Fiber Acoustooptic Tunable Filter With Low Polarization-Sensitivity," IEEE Photon. Technol. Lett. 17, 2646-2648 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. MTT

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides-I. Summary of results," IEEE Trans. Microwave Theory Tech. MTT- 23, 421-429 (1975).
[CrossRef]

J. Lightwave Technol.

J. Opt. A

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lægsgaad, "Small-core photonic crystal fibers with weakly disordered air-hole claddings," J. Opt. A 6, 221-223 (2004).
[CrossRef]

J. Opt. Soc. Korea

Opt. Express

Opt. Lett.

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Figures (9)

Fig. 1.
Fig. 1.

SEM images of cleaved PCF end face

Fig. 2.
Fig. 2.

Calculated electric field distributions of (a) the LP01 modes, (b) the even LP11 modes, and (c) the odd LP11 modes.

Fig. 3.
Fig. 3.

(a). Microscopic images of near fields of the even LP11 mode and the odd LP11 modes at 633 nm after alignment of the major and minor axes of the cladding ellipse along the laboratory frame, respectively. (b) Diagram of the angular misalignment between the acoustic and optical axes for modeling of AO mode coupling in the fiber.

Fig. 4.
Fig. 4.

Calculated transmission spectra when an acoustic grating suffers from (a) cosine amplitude modulation and (b) sine amplitude modulation, respectively.

Fig. 5.
Fig. 5.

A schematic of the AOTF (SMF: single mode fiber, MS: mode stripper, OSA: optical spectrum analyzer).

Fig. 6.
Fig. 6.

Measured transmission spectra when an acoustic vibration direction is parallel to (a) major axis, and (b) minor axis. (Doted line: calculation).

Fig. 7.
Fig. 7.

Measured and calculated optical beatlength between the LP01 and the even LP11 modes as a function of optical wavelength.

Fig. 8.
Fig. 8.

Measured transmission spectra when initial acoustic vibration angles θ are (a) π/6, (b) π/4, and (c) π/3. (Doted line: calculation).

Fig. 9.
Fig. 9.

Measured transmission spectra when initial acoustic vibration angles θ are (a) 2π/3, (b) 3π/4, and (c) 5π/6. (Doted line: calculation).

Equations (7)

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

κ ( z ) = ( κ x ( even ) ( z ) κ y ( odd ) ( z ) ) = κ 0 ( sin θ sin α e i 2 π Λ s z + cos θ cos α e i 2 π Λ l z sin θ cos α e i 2 π Λ s z cos θ sin α e i 2 π Λ l z )
κ ( z ) = ( κ x ( even ) ( z ) κ y ( odd ) ( z ) ) = κ 0 ( ± cos α e i 2 π Λ l z sin α e i 2 π Λ l z )
κ ( z ) = ( κ x ( even ) ( z ) κ y ( odd ) ( z ) ) = κ 0 ( ± sin α e i 2 π Λ s z cos α e i 2 π Λ s z )
κ ( z ) = ( κ x ( even ) ( z ) κ y ( odd ) ( z ) ) = κ 0 2 ( e i 2 π Λ s z + e i 2 π Λ l z e i 2 π Λ s z e i 2 π Λ l z ) = κ 0 e i k ave z ( cos ( k mod z ) i sin ( k mod z ) )
κ θ = 0 ( z ) = ( κ x ( even ) ( z ) κ y ( odd ) ( z ) ) = κ 0 ( 3 2 e i 2 π Λ l z 1 2 e i 2 π Λ l z ) κ θ = π 2 ( z ) = ( κ x ( even ) ( z ) κ y ( odd ) ( z ) ) = κ 0 ( 1 2 e i 2 π Λ s z 3 2 e i 2 π Λ s z )
κ θ = π 6 = κ 0 ( 1 4 e i 2 π Λ s z + 3 4 e i 2 π Λ l z 3 4 e i 2 π Λ s z 3 4 e i 2 π Λ l z ) κ θ = π 4 = κ 0 ( 2 4 e i 2 π Λ s z + 6 4 e i 2 π Λ l z 6 4 e i 2 π Λ s z 2 4 e i 2 π Λ l z ) κ θ = π 3 = κ 0 ( 3 4 e i 2 π Λ s z + 3 4 e i 2 π Λ l z 3 4 e i 2 π Λ s z 1 4 e i 2 π Λ l z )
κ θ = 2 π 3 = κ 0 ( 3 4 e i 2 π Λ s z - 3 4 e i 2 π Λ l z 3 4 e i 2 π Λ s z 1 4 e i 2 π Λ l z ) κ θ = 3 π 4 = κ 0 ( 2 4 e i 2 π Λ s z 6 4 e i 2 π Λ l z 6 4 e i 2 π Λ s z + 2 4 e i 2 π Λ l z ) κ θ = 5 π 6 = κ 0 ( 1 4 e i 2 π Λ s z 3 4 e i 2 π Λ l z 3 4 e i 2 π Λ s z + 3 4 e i 2 π Λ l z )

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