Abstract

This work presents the characteristics and expected capabilities of an optical interconnect that uses a diffractive liquid crystal over silicon (LCOS) device as a routing element. Such an interconnect may be used in a neighborhood’s optical network to distribute high definition television, thus avoiding an electronic or optical transmitter for each user. The optimal characteristics of the LCOS device are calculated in terms of pixel number and silicon area and found to be feasible with today’s technology. Finally, its performance in terms of optical efficiency and number of output ports is evaluated and found suitable for a neighborhood with hundreds of households.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [PubMed]
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2010 (1)

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[CrossRef]

2009 (1)

2008 (1)

A. Georgiou, T. D. Wilkinson, N. Collings, and W. A. Crossland, “Algorithm for computing spot-generating holograms,” J. Opt. A Pure Appl. Opt. 10, 015306 (2008).
[CrossRef]

2006 (1)

R. James, M. C. Gardner, F. A. Fernández, and S. E. Day, “3D modelling of high resolution devices,” Mol. Cryst. Liq. Cryst. 450, 105–118 (2006).
[CrossRef]

2005 (2)

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

M. Komarcevic, I. G. Manolis, T. D. Wilkinson, and W. A. Crossland, “Polarization effects in reconfigurable liquid crystal phase holograms,” Opt. Commun. 244, 105–110 (2005).
[CrossRef]

2004 (1)

2003 (1)

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

2002 (1)

P. Chanclou, H. Ramanitra, P. Gravey, and M. Thual, “Design and performance of expanded mode fiber using microoptics,” J. Lightwave Technol. 20, 836–842 (2002).
[CrossRef]

2001 (2)

2000 (2)

1992 (1)

A. Kirk and T. Hall, “Design of binary computer generated holograms by simulated annealing: coding density and reconstruction error,” Opt. Commun. 94, 491–496 (1992).
[CrossRef]

1987 (1)

Abakoumov, D.

P. Evans, G. Baxter, H. Zhou, D. Abakoumov, S. Poole, and S. Frisken, “LCOS-based WSS with true integrated channel monitor for signal quality monitoring applications in ROADMS,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference (Digest) (Optical Society of America, 2008), pp. 1–3.
[PubMed]

Allebach, J. P.

Anderson, R. C.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Anisetti, M.

M. Anisetti, C. Ardagna, V. Bellandi, and E. Damiani, “Telecentric and achromatic F-theta scan lens system and method of use,” U.S. patent 5,404,247 (4 April 1995).

Apter, B.

Ardagna, C.

M. Anisetti, C. Ardagna, V. Bellandi, and E. Damiani, “Telecentric and achromatic F-theta scan lens system and method of use,” U.S. patent 5,404,247 (4 April 1995).

Bahat-Treidel, E.

Bala, K.

T. E. Stern, G. Ellinas, and K. Bala, Multiwavelength Optical Networks: Architectures, Design, and Control (Cambridge University, 2009).

Baxter, G.

P. Evans, G. Baxter, H. Zhou, D. Abakoumov, S. Poole, and S. Frisken, “LCOS-based WSS with true integrated channel monitor for signal quality monitoring applications in ROADMS,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference (Digest) (Optical Society of America, 2008), pp. 1–3.
[PubMed]

Beeckman, J.

Bellandi, V.

M. Anisetti, C. Ardagna, V. Bellandi, and E. Damiani, “Telecentric and achromatic F-theta scan lens system and method of use,” U.S. patent 5,404,247 (4 April 1995).

Blumenthal, D. J.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Bonas, I. G.

Bowers, J. E.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Chanclou, P.

P. Chanclou, H. Ramanitra, P. Gravey, and M. Thual, “Design and performance of expanded mode fiber using microoptics,” J. Lightwave Technol. 20, 836–842 (2002).
[CrossRef]

Chu, H. H.

Cizmar, T.

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[CrossRef]

Collings, N.

A. Georgiou, T. D. Wilkinson, N. Collings, and W. A. Crossland, “Algorithm for computing spot-generating holograms,” J. Opt. A Pure Appl. Opt. 10, 015306 (2008).
[CrossRef]

Crossland, A. W.

Crossland, W. A.

A. Georgiou, T. D. Wilkinson, N. Collings, and W. A. Crossland, “Algorithm for computing spot-generating holograms,” J. Opt. A Pure Appl. Opt. 10, 015306 (2008).
[CrossRef]

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

M. Komarcevic, I. G. Manolis, T. D. Wilkinson, and W. A. Crossland, “Polarization effects in reconfigurable liquid crystal phase holograms,” Opt. Commun. 244, 105–110 (2005).
[CrossRef]

C. A. T. H. Tee, W. A. Crossland, T. D. Wilkinson, and A. B. Davey, “Binary phase modulation using electrically addressed transmissive and silicon backplane spatial light modulators,” Opt. Eng. 39, 2527–2534 (2000).
[CrossRef]

W. A. Crossland, I. G. Manolis, M. M. Redmond, K. L. Tan, T. D. Wilkinson, M. J. Holmes, T. R. Parker, H. H. Chu, J. Croucher, V. A. Handerek, S. T. Warr, B. Robertson, I. G. Bonas, R. Franklin, C. Stace, H. J. White, R. A. Woolley, and G. Henshall, “Holographic optical switching: the ROSES demonstrator,” J. Lightwave Technol. 18, 1845–1854 (2000).
[CrossRef]

A. Georgiou, M. Komarcevic, and W. A. Crossland, “Noise suppression in liquid crystal beam steering devices,” presented at SPIE Great Lakes Photonics Symposium, 12–16 June, 2006, Dayton, Ohio, USA.

Croucher, J.

Damiani, E.

M. Anisetti, C. Ardagna, V. Bellandi, and E. Damiani, “Telecentric and achromatic F-theta scan lens system and method of use,” U.S. patent 5,404,247 (4 April 1995).

Davey, A. B.

C. A. T. H. Tee, W. A. Crossland, T. D. Wilkinson, and A. B. Davey, “Binary phase modulation using electrically addressed transmissive and silicon backplane spatial light modulators,” Opt. Eng. 39, 2527–2534 (2000).
[CrossRef]

Day, S. E.

R. James, M. C. Gardner, F. A. Fernández, and S. E. Day, “3D modelling of high resolution devices,” Mol. Cryst. Liq. Cryst. 450, 105–118 (2006).
[CrossRef]

Dholakia, K.

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[CrossRef]

Efron, U.

Ellinas, G.

T. E. Stern, G. Ellinas, and K. Bala, Multiwavelength Optical Networks: Architectures, Design, and Control (Cambridge University, 2009).

Evans, P.

P. Evans, G. Baxter, H. Zhou, D. Abakoumov, S. Poole, and S. Frisken, “LCOS-based WSS with true integrated channel monitor for signal quality monitoring applications in ROADMS,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference (Digest) (Optical Society of America, 2008), pp. 1–3.
[PubMed]

Fernandez, F. A.

Fernández, F. A.

R. James, M. C. Gardner, F. A. Fernández, and S. E. Day, “3D modelling of high resolution devices,” Mol. Cryst. Liq. Cryst. 450, 105–118 (2006).
[CrossRef]

Franklin, R.

Frisken, S.

P. Evans, G. Baxter, H. Zhou, D. Abakoumov, S. Poole, and S. Frisken, “LCOS-based WSS with true integrated channel monitor for signal quality monitoring applications in ROADMS,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference (Digest) (Optical Society of America, 2008), pp. 1–3.
[PubMed]

Gardner, M. C.

R. James, M. C. Gardner, F. A. Fernández, and S. E. Day, “3D modelling of high resolution devices,” Mol. Cryst. Liq. Cryst. 450, 105–118 (2006).
[CrossRef]

Georgiou, A.

A. Georgiou, T. D. Wilkinson, N. Collings, and W. A. Crossland, “Algorithm for computing spot-generating holograms,” J. Opt. A Pure Appl. Opt. 10, 015306 (2008).
[CrossRef]

A. Georgiou, M. Komarcevic, and W. A. Crossland, “Noise suppression in liquid crystal beam steering devices,” presented at SPIE Great Lakes Photonics Symposium, 12–16 June, 2006, Dayton, Ohio, USA.

Georgiou, A. G.

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

Gravey, P.

P. Chanclou, H. Ramanitra, P. Gravey, and M. Thual, “Design and performance of expanded mode fiber using microoptics,” J. Lightwave Technol. 20, 836–842 (2002).
[CrossRef]

Hall, T.

A. Kirk and T. Hall, “Design of binary computer generated holograms by simulated annealing: coding density and reconstruction error,” Opt. Commun. 94, 491–496 (1992).
[CrossRef]

Handerek, V. A.

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).

Helkey, R.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Henshall, G.

Holmes, M. J.

Ilias, M. G.

James, R.

P. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17, 10895–10909 (2009).
[CrossRef] [PubMed]

R. James, M. C. Gardner, F. A. Fernández, and S. E. Day, “3D modelling of high resolution devices,” Mol. Cryst. Liq. Cryst. 450, 105–118 (2006).
[CrossRef]

Jerphagnon, O.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Kaman, V.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Keating, A.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Kirk, A.

A. Kirk and T. Hall, “Design of binary computer generated holograms by simulated annealing: coding density and reconstruction error,” Opt. Commun. 94, 491–496 (1992).
[CrossRef]

Komarcevic, M.

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

M. Komarcevic, I. G. Manolis, T. D. Wilkinson, and W. A. Crossland, “Polarization effects in reconfigurable liquid crystal phase holograms,” Opt. Commun. 244, 105–110 (2005).
[CrossRef]

A. Georgiou, M. Komarcevic, and W. A. Crossland, “Noise suppression in liquid crystal beam steering devices,” presented at SPIE Great Lakes Photonics Symposium, 12–16 June, 2006, Dayton, Ohio, USA.

Liu, B.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Manolis, I. G.

Mazilu, M.

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[CrossRef]

Neyts, K.

Parker, T. R.

Poole, S.

P. Evans, G. Baxter, H. Zhou, D. Abakoumov, S. Poole, and S. Frisken, “LCOS-based WSS with true integrated channel monitor for signal quality monitoring applications in ROADMS,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference (Digest) (Optical Society of America, 2008), pp. 1–3.
[PubMed]

Poulsen, H. N.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Pusarla, C.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Ramanitra, H.

P. Chanclou, H. Ramanitra, P. Gravey, and M. Thual, “Design and performance of expanded mode fiber using microoptics,” J. Lightwave Technol. 20, 836–842 (2002).
[CrossRef]

Redmond, M. M.

Robertson, B.

Sechrist, J. R.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Seldowitz, M. A.

Stace, C.

Stern, T. E.

T. E. Stern, G. Ellinas, and K. Bala, Multiwavelength Optical Networks: Architectures, Design, and Control (Cambridge University, 2009).

Sweeney, D. W.

Tan, K. L.

Tee, C. A. T. H.

C. A. T. H. Tee, W. A. Crossland, T. D. Wilkinson, and A. B. Davey, “Binary phase modulation using electrically addressed transmissive and silicon backplane spatial light modulators,” Opt. Eng. 39, 2527–2534 (2000).
[CrossRef]

Thual, M.

P. Chanclou, H. Ramanitra, P. Gravey, and M. Thual, “Design and performance of expanded mode fiber using microoptics,” J. Lightwave Technol. 20, 836–842 (2002).
[CrossRef]

Vanbrabant, P.

Warr, S. T.

White, H. J.

Wilkinson, T. D.

A. Georgiou, T. D. Wilkinson, N. Collings, and W. A. Crossland, “Algorithm for computing spot-generating holograms,” J. Opt. A Pure Appl. Opt. 10, 015306 (2008).
[CrossRef]

M. Komarcevic, I. G. Manolis, T. D. Wilkinson, and W. A. Crossland, “Polarization effects in reconfigurable liquid crystal phase holograms,” Opt. Commun. 244, 105–110 (2005).
[CrossRef]

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

K. L. Tan, S. T. Warr, M. G. Ilias, T. D. Wilkinson, M. M. Redmond, A. W. Crossland, and B. Robertson, “Dynamic holography for optical interconnections. I. Noise floor of low-crosstalk holographic switches,” J. Opt. Soc. Am. A 18, 195–204(2001).
[CrossRef]

K. L. Tan, S. T. Warr, M. G. Ilias, T. D. Wilkinson, M. M. Redmond, A. W. Crossland, and B. Robertson, “Dynamic holography for optical interconnections. II. Routing holograms with predictable location and intensity of each diffraction order,” J. Opt. Soc. Am. A 18, 205–215 (2001).
[CrossRef]

W. A. Crossland, I. G. Manolis, M. M. Redmond, K. L. Tan, T. D. Wilkinson, M. J. Holmes, T. R. Parker, H. H. Chu, J. Croucher, V. A. Handerek, S. T. Warr, B. Robertson, I. G. Bonas, R. Franklin, C. Stace, H. J. White, R. A. Woolley, and G. Henshall, “Holographic optical switching: the ROSES demonstrator,” J. Lightwave Technol. 18, 1845–1854 (2000).
[CrossRef]

C. A. T. H. Tee, W. A. Crossland, T. D. Wilkinson, and A. B. Davey, “Binary phase modulation using electrically addressed transmissive and silicon backplane spatial light modulators,” Opt. Eng. 39, 2527–2534 (2000).
[CrossRef]

Woolley, R. A.

Xu, Y.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Yuan, S.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Zheng, X.

X. Zheng, V. Kaman, S. Yuan, Y. Xu, O. Jerphagnon, A. Keating, R. C. Anderson, H. N. Poulsen, B. Liu, J. R. Sechrist, C. Pusarla, R. Helkey, D. J. Blumenthal, and J. E. Bowers, “Three-dimensional MEMS photonic cross-connect switch design and performance,” IEEE J. Sel. Top. Quantum Electron. 9, 571–578(2003).
[CrossRef]

Zhou, H.

P. Evans, G. Baxter, H. Zhou, D. Abakoumov, S. Poole, and S. Frisken, “LCOS-based WSS with true integrated channel monitor for signal quality monitoring applications in ROADMS,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference (Digest) (Optical Society of America, 2008), pp. 1–3.
[PubMed]

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Figures (13)

Fig. 1
Fig. 1

Position of a holographic switch in an optical network.

Fig. 2
Fig. 2

Optical arrangement of a holographic interconnect.

Fig. 3
Fig. 3

Suggested arrangement of the input and output fibers (here S max = 192 ).

Fig. 4
Fig. 4

Telecentric F-theta lens will focus the beam at normal angle to the focal plane, its position will be proportional to the input field angle, and the focal plane is flat.

Fig. 5
Fig. 5

Apodization losses as a function of the beam width.

Fig. 6
Fig. 6

Dimensions of an LCOS device designed for holographic optical interconnects.

Fig. 7
Fig. 7

Hologram diffraction efficiency versus number of output users. Efficiency increases, thus making holographic interconnects best suited for networks with many users.

Fig. 8
Fig. 8

Sinc envelope formed at the output plane due to the square pixel shape. Keeping fibers near the center reduces attenuation.

Fig. 9
Fig. 9

Two-dimensional sinc envelope attenuation as a contour map.

Fig. 10
Fig. 10

Acceptable loss due to the sinc envelope as a function of useful area fraction. The two stars correspond to the proposed points of operation for the LOIS device.

Fig. 11
Fig. 11

Diffraction efficiency of a phase-quantized hologram. From the top, efficiency for a single spot, a 100 spot, and a 10 spot generating hologram.

Fig. 12
Fig. 12

Fringing fields in an LCOS device between pixels smooths the phase transitions.

Fig. 13
Fig. 13

Diffraction efficiency of a blazed grating with and without fringing fields for a device with the characteristics of LOIS.

Tables (2)

Tables Icon

Table 1 Characteristics of the Commercial LCOS Chip Together with LOIS, the Proposed LCOS Chip

Tables Icon

Table 2 Total Losses for the LOIS Device Taking Two Scenarios: When Total Loss Is 5.2 dB ( α = 0.59 ) and When Total Loss Is 3.2 dB ( α = 0.17 )

Equations (13)

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

2 w d = 4 λ π f 2 w i ,
P d = 1 w d 2 π L / 2 x = + L / 2 L / 2 y = + L / 2 e ( x 2 + y 2 ) / 2 w d 2 d x d y ,
F = ( Δ g Δ ) 2 ,
F = ( 1 g L N ) 2 .
η sinc = F sinc 2 ( u K ) sinc 2 ( v K ) ,
K = π Δ Δ g ,
η pq = [ π / p θ = π / p cos ( θ ) d θ ] / [ π / p θ = π / p 1 d θ ] ,
= sinc ( π p ) ,
d = 1 2 ϕ 2 π λ Δ n ,
u max = ± λ f 2 Δ or u max = ± 0.5 ( N λ f L ) ,
S max = α ( N ( λ f ) L D Fo ) 2 .
2 w o = 4 λ π f 2 w d d Fo λ f π 4 2 w d d Fo ,
S max = α ( π 4 ) 2 ( d Fo D Fo ) 2 ( 2 w d L ) 2 N 2 .

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