Abstract

We demonstrate an all-fiber acousto-optic tunable bandpass filter exhibiting narrow optical bandwidth and negligible polarization dependence by employing a novel ultraviolet (UV)-induced core-mode blocker written in a high numerical aperture (NA) fiber. It was demonstrated that the device had the measured 3-dB optical bandwidth of 0.65 nm, the polarization-dependent center-wavelength splitting of 0.05 nm and the extinction ratio of -22dB at the wavelength around 1550 nm. The details of the transmission characteristics and the loss mechanism of the core-mode blocking element inscribed in the high NA fiber are discussed.

© 2007 Optical Society of America

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References

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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 (1997).
    [Crossref]
  2. H. S. Park, K. Y. Song, S. H. Yun, and B. Y. Kim, “All-fiber wavelength-tunable acoustooptic switches based on intermodal coupling in fibers,” J. Lightwave. Technol. 20,1864 (2002).
    [Crossref]
  3. S. H. Yun and H. S. Kim, “Resonance in fiber-based acoustooptic devices via acoustic radiation to air,” IEEE Photon. Technol. Lett. 16,147 (2004).
    [Crossref]
  4. M. S. Lee, I. K. Hwang, and B. Y. Kim, “Acousto-optic tunable bandpass Filter,” in Proceedings of IEEE OptoElectronics and Communications Conference, WG Fibre Devices (OECC/IOOC, Sydney, Australia, 2001), pp.324–325.
  5. D. A. Satorius, T. E. Dimmick, and G. L. Burdge, “Double-pass acoustooptic tunable bandpass filter with zero frequency shift and reduced polarization sensitivity,” IEEE Photon. Technol. Lett. 14,1324 (2002).
    [Crossref]
  6. P. Z. Dashti, C. -S. Kim, Q. Li, and H. P. Lee, “Demonstration of a Novel All-Fiber Bandpass Acousto-Optic Tunable Filter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper OFC2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2005-OFC2
  7. K. J. Lee, D. I. Yeom, and B. Y. Kim, “Narrow-bandwidth acousto-optic tunable bandpass filter using dispersion compensating fiber, in Proceedings of IEEE Pacific Rim Conference on Lasers and Electro-Optics (Toshi Center Kaikan, Tokyo, Japan, 2005), pp.1082–1083.
  8. 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 (2005).
    [Crossref]
  9. Y. G. Han, S. H. Kim, S. B. Lee, U. C. Paek, and Y. Chung, “Development of a novel core mode blocker with H2-loaded Ge-B co-doped fibers,” Electron. Lett. 39,1107 (2003).
    [Crossref]
  10. D. S. Starodubov, V. Grubsky, and J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10,1590 (1998).
    [Crossref]
  11. J. -L. Archambault, L. Reekie, and P. St. J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 29,453 (1993).
    [Crossref]
  12. P. St. J. Russell, J. -L. Archambault, and L. Reekie, “Fibre gratings,” Physics World, 10,41 (1993).
  13. K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23,125 (1993).
    [Crossref]
  14. M. Janos, J. Canning, and M. G. Sceats, “Incoherent scattering losses in optical fiber Bragg gratings,” Opt. Lett. 21,1827 (1996).
    [Crossref] [PubMed]
  15. H. Renner, D. Johlen, and E. Brinkmeyer “Modal field deformation and transition losses in UV side-written optical fibers,” Appl. Opt. 39,933 (2000).
    [Crossref]
  16. C. G. Askins and M. A. Putnam, “Photodarkening and photobleaching in fiber optic Bragg gratings,” J. Lightwave. Technol. 15,1363 (1997).
    [Crossref]

2005 (1)

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 (2005).
[Crossref]

2004 (1)

S. H. Yun and H. S. Kim, “Resonance in fiber-based acoustooptic devices via acoustic radiation to air,” IEEE Photon. Technol. Lett. 16,147 (2004).
[Crossref]

2003 (1)

Y. G. Han, S. H. Kim, S. B. Lee, U. C. Paek, and Y. Chung, “Development of a novel core mode blocker with H2-loaded Ge-B co-doped fibers,” Electron. Lett. 39,1107 (2003).
[Crossref]

2002 (2)

D. A. Satorius, T. E. Dimmick, and G. L. Burdge, “Double-pass acoustooptic tunable bandpass filter with zero frequency shift and reduced polarization sensitivity,” IEEE Photon. Technol. Lett. 14,1324 (2002).
[Crossref]

H. S. Park, K. Y. Song, S. H. Yun, and B. Y. Kim, “All-fiber wavelength-tunable acoustooptic switches based on intermodal coupling in fibers,” J. Lightwave. Technol. 20,1864 (2002).
[Crossref]

2000 (1)

1998 (1)

D. S. Starodubov, V. Grubsky, and J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10,1590 (1998).
[Crossref]

1997 (2)

C. G. Askins and M. A. Putnam, “Photodarkening and photobleaching in fiber optic Bragg gratings,” J. Lightwave. Technol. 15,1363 (1997).
[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 (1997).
[Crossref]

1996 (1)

1993 (3)

J. -L. Archambault, L. Reekie, and P. St. J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 29,453 (1993).
[Crossref]

P. St. J. Russell, J. -L. Archambault, and L. Reekie, “Fibre gratings,” Physics World, 10,41 (1993).

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23,125 (1993).
[Crossref]

Archambault, J. -L.

J. -L. Archambault, L. Reekie, and P. St. J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 29,453 (1993).
[Crossref]

P. St. J. Russell, J. -L. Archambault, and L. Reekie, “Fibre gratings,” Physics World, 10,41 (1993).

Askins, C. G.

C. G. Askins and M. A. Putnam, “Photodarkening and photobleaching in fiber optic Bragg gratings,” J. Lightwave. Technol. 15,1363 (1997).
[Crossref]

Bilodeau, F.

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23,125 (1993).
[Crossref]

Brinkmeyer, E.

Burdge, G. L.

D. A. Satorius, T. E. Dimmick, and G. L. Burdge, “Double-pass acoustooptic tunable bandpass filter with zero frequency shift and reduced polarization sensitivity,” IEEE Photon. Technol. Lett. 14,1324 (2002).
[Crossref]

Canning, J.

Chung, Y.

Y. G. Han, S. H. Kim, S. B. Lee, U. C. Paek, and Y. Chung, “Development of a novel core mode blocker with H2-loaded Ge-B co-doped fibers,” Electron. Lett. 39,1107 (2003).
[Crossref]

Dashti, P. Z.

P. Z. Dashti, C. -S. Kim, Q. Li, and H. P. Lee, “Demonstration of a Novel All-Fiber Bandpass Acousto-Optic Tunable Filter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper OFC2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2005-OFC2

Dimmick, T. E.

D. A. Satorius, T. E. Dimmick, and G. L. Burdge, “Double-pass acoustooptic tunable bandpass filter with zero frequency shift and reduced polarization sensitivity,” IEEE Photon. Technol. Lett. 14,1324 (2002).
[Crossref]

Feinberg, J.

D. S. Starodubov, V. Grubsky, and J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10,1590 (1998).
[Crossref]

Grubsky, V.

D. S. Starodubov, V. Grubsky, and J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10,1590 (1998).
[Crossref]

Han, Y. G.

Y. G. Han, S. H. Kim, S. B. Lee, U. C. Paek, and Y. Chung, “Development of a novel core mode blocker with H2-loaded Ge-B co-doped fibers,” Electron. Lett. 39,1107 (2003).
[Crossref]

Hill, K. O.

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23,125 (1993).
[Crossref]

Hwang, I. K.

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 (1997).
[Crossref]

M. S. Lee, I. K. Hwang, and B. Y. Kim, “Acousto-optic tunable bandpass Filter,” in Proceedings of IEEE OptoElectronics and Communications Conference, WG Fibre Devices (OECC/IOOC, Sydney, Australia, 2001), pp.324–325.

Janos, M.

Johlen, D.

Johnson, D. C.

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23,125 (1993).
[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 (2005).
[Crossref]

Kim, B. Y.

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 (2005).
[Crossref]

H. S. Park, K. Y. Song, S. H. Yun, and B. Y. Kim, “All-fiber wavelength-tunable acoustooptic switches based on intermodal coupling in fibers,” J. Lightwave. Technol. 20,1864 (2002).
[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 (1997).
[Crossref]

M. S. Lee, I. K. Hwang, and B. Y. Kim, “Acousto-optic tunable bandpass Filter,” in Proceedings of IEEE OptoElectronics and Communications Conference, WG Fibre Devices (OECC/IOOC, Sydney, Australia, 2001), pp.324–325.

K. J. Lee, D. I. Yeom, and B. Y. Kim, “Narrow-bandwidth acousto-optic tunable bandpass filter using dispersion compensating fiber, in Proceedings of IEEE Pacific Rim Conference on Lasers and Electro-Optics (Toshi Center Kaikan, Tokyo, Japan, 2005), pp.1082–1083.

Kim, C. -S.

P. Z. Dashti, C. -S. Kim, Q. Li, and H. P. Lee, “Demonstration of a Novel All-Fiber Bandpass Acousto-Optic Tunable Filter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper OFC2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2005-OFC2

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 (2005).
[Crossref]

S. H. Yun and H. S. Kim, “Resonance in fiber-based acoustooptic devices via acoustic radiation to air,” IEEE Photon. Technol. Lett. 16,147 (2004).
[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 (1997).
[Crossref]

Kim, S. H.

Y. G. Han, S. H. Kim, S. B. Lee, U. C. Paek, and Y. Chung, “Development of a novel core mode blocker with H2-loaded Ge-B co-doped fibers,” Electron. Lett. 39,1107 (2003).
[Crossref]

Lee, H. P.

P. Z. Dashti, C. -S. Kim, Q. Li, and H. P. Lee, “Demonstration of a Novel All-Fiber Bandpass Acousto-Optic Tunable Filter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper OFC2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2005-OFC2

Lee, K. J.

K. J. Lee, D. I. Yeom, and B. Y. Kim, “Narrow-bandwidth acousto-optic tunable bandpass filter using dispersion compensating fiber, in Proceedings of IEEE Pacific Rim Conference on Lasers and Electro-Optics (Toshi Center Kaikan, Tokyo, Japan, 2005), pp.1082–1083.

Lee, M. S.

M. S. Lee, I. K. Hwang, and B. Y. Kim, “Acousto-optic tunable bandpass Filter,” in Proceedings of IEEE OptoElectronics and Communications Conference, WG Fibre Devices (OECC/IOOC, Sydney, Australia, 2001), pp.324–325.

Lee, S. B.

Y. G. Han, S. H. Kim, S. B. Lee, U. C. Paek, and Y. Chung, “Development of a novel core mode blocker with H2-loaded Ge-B co-doped fibers,” Electron. Lett. 39,1107 (2003).
[Crossref]

Li, Q.

P. Z. Dashti, C. -S. Kim, Q. Li, and H. P. Lee, “Demonstration of a Novel All-Fiber Bandpass Acousto-Optic Tunable Filter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper OFC2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2005-OFC2

Malo, B.

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23,125 (1993).
[Crossref]

Paek, U. C.

Y. G. Han, S. H. Kim, S. B. Lee, U. C. Paek, and Y. Chung, “Development of a novel core mode blocker with H2-loaded Ge-B co-doped fibers,” Electron. Lett. 39,1107 (2003).
[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 (2005).
[Crossref]

H. S. Park, K. Y. Song, S. H. Yun, and B. Y. Kim, “All-fiber wavelength-tunable acoustooptic switches based on intermodal coupling in fibers,” J. Lightwave. Technol. 20,1864 (2002).
[Crossref]

Putnam, M. A.

C. G. Askins and M. A. Putnam, “Photodarkening and photobleaching in fiber optic Bragg gratings,” J. Lightwave. Technol. 15,1363 (1997).
[Crossref]

Reekie, L.

P. St. J. Russell, J. -L. Archambault, and L. Reekie, “Fibre gratings,” Physics World, 10,41 (1993).

J. -L. Archambault, L. Reekie, and P. St. J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 29,453 (1993).
[Crossref]

Renner, H.

Russell, P. St. J.

J. -L. Archambault, L. Reekie, and P. St. J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 29,453 (1993).
[Crossref]

P. St. J. Russell, J. -L. Archambault, and L. Reekie, “Fibre gratings,” Physics World, 10,41 (1993).

Satorius, D. A.

D. A. Satorius, T. E. Dimmick, and G. L. Burdge, “Double-pass acoustooptic tunable bandpass filter with zero frequency shift and reduced polarization sensitivity,” IEEE Photon. Technol. Lett. 14,1324 (2002).
[Crossref]

Sceats, M. G.

Song, K. Y.

H. S. Park, K. Y. Song, S. H. Yun, and B. Y. Kim, “All-fiber wavelength-tunable acoustooptic switches based on intermodal coupling in fibers,” J. Lightwave. Technol. 20,1864 (2002).
[Crossref]

Starodubov, D. S.

D. S. Starodubov, V. Grubsky, and J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10,1590 (1998).
[Crossref]

Yeom, D. I.

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 (2005).
[Crossref]

K. J. Lee, D. I. Yeom, and B. Y. Kim, “Narrow-bandwidth acousto-optic tunable bandpass filter using dispersion compensating fiber, in Proceedings of IEEE Pacific Rim Conference on Lasers and Electro-Optics (Toshi Center Kaikan, Tokyo, Japan, 2005), pp.1082–1083.

Yun, S. H.

S. H. Yun and H. S. Kim, “Resonance in fiber-based acoustooptic devices via acoustic radiation to air,” IEEE Photon. Technol. Lett. 16,147 (2004).
[Crossref]

H. S. Park, K. Y. Song, S. H. Yun, and B. Y. Kim, “All-fiber wavelength-tunable acoustooptic switches based on intermodal coupling in fibers,” J. Lightwave. Technol. 20,1864 (2002).
[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 (1997).
[Crossref]

Annu. Rev. Mater. Sci. (1)

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23,125 (1993).
[Crossref]

Appl. Opt. (1)

Electron. Lett. (2)

J. -L. Archambault, L. Reekie, and P. St. J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 29,453 (1993).
[Crossref]

Y. G. Han, S. H. Kim, S. B. Lee, U. C. Paek, and Y. Chung, “Development of a novel core mode blocker with H2-loaded Ge-B co-doped fibers,” Electron. Lett. 39,1107 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (4)

D. S. Starodubov, V. Grubsky, and J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10,1590 (1998).
[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 (2005).
[Crossref]

S. H. Yun and H. S. Kim, “Resonance in fiber-based acoustooptic devices via acoustic radiation to air,” IEEE Photon. Technol. Lett. 16,147 (2004).
[Crossref]

D. A. Satorius, T. E. Dimmick, and G. L. Burdge, “Double-pass acoustooptic tunable bandpass filter with zero frequency shift and reduced polarization sensitivity,” IEEE Photon. Technol. Lett. 14,1324 (2002).
[Crossref]

J. Lightwave. Technol. (2)

H. S. Park, K. Y. Song, S. H. Yun, and B. Y. Kim, “All-fiber wavelength-tunable acoustooptic switches based on intermodal coupling in fibers,” J. Lightwave. Technol. 20,1864 (2002).
[Crossref]

C. G. Askins and M. A. Putnam, “Photodarkening and photobleaching in fiber optic Bragg gratings,” J. Lightwave. Technol. 15,1363 (1997).
[Crossref]

Opt. Lett. (2)

Physics World (1)

P. St. J. Russell, J. -L. Archambault, and L. Reekie, “Fibre gratings,” Physics World, 10,41 (1993).

Other (3)

P. Z. Dashti, C. -S. Kim, Q. Li, and H. P. Lee, “Demonstration of a Novel All-Fiber Bandpass Acousto-Optic Tunable Filter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper OFC2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2005-OFC2

K. J. Lee, D. I. Yeom, and B. Y. Kim, “Narrow-bandwidth acousto-optic tunable bandpass filter using dispersion compensating fiber, in Proceedings of IEEE Pacific Rim Conference on Lasers and Electro-Optics (Toshi Center Kaikan, Tokyo, Japan, 2005), pp.1082–1083.

M. S. Lee, I. K. Hwang, and B. Y. Kim, “Acousto-optic tunable bandpass Filter,” in Proceedings of IEEE OptoElectronics and Communications Conference, WG Fibre Devices (OECC/IOOC, Sydney, Australia, 2001), pp.324–325.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of an acousto-optic tunable bandpass filter.

Fig. 2.
Fig. 2.

Microscopic image of the damage track at the core-cladding boundary of the UV-exposed DCF.

Fig. 3.
Fig. 3.

Core mode loss of the fabricated core mode blocker.

Fig. 4.
Fig. 4.

The spectral characteristic of the fabricated core mode blocker irradiated from the side with (a) parallel and (b) perpendicularly polarized UV light to the longitudinal axis of the DCF.

Fig. 5.
Fig. 5.

Experimental configuration for measuring the power of side-scattered light from the core mode blocker. ASE, amplified spontaneous emission.

Fig. 6.
Fig. 6.

Polar plot of scattered light from the core mode blocker. The large arrow at the left indicates the direction of incident UV irradiation.

Fig. 7.
Fig. 7.

(a). Measured transmission spectrum of the acousto-optic tunable bandpass filter for the acoustic frequency of 2.636 MHz and (b) Center wavelength of the acousto-optic tunable bandpass filter as a function of the acoustic frequency. The Inset shows optical transmission spectra for two orthogonal input polarizations.

Equations (2)

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P ( Z , Φ ) = p ( z , φ ) G ( Z z , Φ φ ) dzdφ
P ( Z , Φ ) = Z Z + l p ( z ) dz . Φ θ 2 Φ + θ 2 p ( φ )

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