Abstract

Transmission gratings that combine a large diffraction angle with a high diffraction efficiency and a low angular and wavelength dispersion could be used to collect sunlight in a light guide. In this paper we compare the diffractive properties of polarization gratings and classical surface-relief gratings and explore their possible use in solar concentrators. It is found that polarization gratings and surface-relief gratings have qualitatively comparable diffraction characteristics when their thickness parameters are within the same regime. Relatively large grating periods result in high diffraction efficiencies over a wide range of incident angles. For small grating periods the efficiency and the angular acceptance are decreased. Surface-relief gratings are preferred over polarization gratings as in-couplers for solar concentrators.

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References

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2010

2008

2007

X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. A 24, 866–881 (2007).
[CrossRef]

C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76, 043815 (2007).
[CrossRef]

R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76, 021701 (2007).
[CrossRef]

2006

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: Potential application for projection displays,” Mol. Cryst. Liq. Cryst. 451, 1–19 (2006).
[CrossRef]

C. Oh and M. J. Escuti, “Time-domain analysis of periodic anisotropic media at oblique incidence: an efficient FDTD implementation,” Opt. Express 14, 11870–11884 (2006).
[CrossRef] [PubMed]

2000

1999

1997

J. M. Shaw, J. D. Gelorme, N. C. LaBianca, W. E. Conley, and S. J. Holmes, “Negative photoresists for optical lithography,” IBM J. Res. Dev. 41, 81 –94 (1997).
[CrossRef]

1996

M. Schadt, H. Seiberle, and A. Schuster, “Optical patterning of multi-domain liquid-crystal displays with wide viewing angles,” Nature 381, 212–215 (1996).
[CrossRef]

1995

1992

M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-induced parallel alignment of liquid crystals by linearly polymerized photopolymers,” Jpn. J. Appl. Phys. 31, 2155–2164 (1992).
[CrossRef]

1984

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31, 579–588 (1984).
[CrossRef]

1982

1981

1980

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

1977

A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[CrossRef]

1969

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

1967

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

1935

C. V. Raman and N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves: Part I.” Proc. Indian Acad. Sci., Sect. A 2, 406–412 (1935).

Benitez, P.

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic Press, 2005), Chap. 13.

Bloss, W. H.

Castro, J. M.

Chigrinov, V.

M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-induced parallel alignment of liquid crystals by linearly polymerized photopolymers,” Jpn. J. Appl. Phys. 31, 2155–2164 (1992).
[CrossRef]

Conley, W. E.

J. M. Shaw, J. D. Gelorme, N. C. LaBianca, W. E. Conley, and S. J. Holmes, “Negative photoresists for optical lithography,” IBM J. Res. Dev. 41, 81 –94 (1997).
[CrossRef]

Cook, B. D.

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

de Boer, D. K. G.

de Gennes, P. G.

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1993).

Debije, M. G.

Escuti, M. J.

R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76, 021701 (2007).
[CrossRef]

C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76, 043815 (2007).
[CrossRef]

C. Oh and M. J. Escuti, “Time-domain analysis of periodic anisotropic media at oblique incidence: an efficient FDTD implementation,” Opt. Express 14, 11870–11884 (2006).
[CrossRef] [PubMed]

Gaylord, T.

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

Gaylord, T. K.

Gelorme, J. D.

J. M. Shaw, J. D. Gelorme, N. C. LaBianca, W. E. Conley, and S. J. Holmes, “Negative photoresists for optical lithography,” IBM J. Res. Dev. 41, 81 –94 (1997).
[CrossRef]

Goetzberger, A.

A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005).

Gori, F.

Grann, E. B.

Greubel, W.

A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[CrossRef]

Griesinger, M.

Hoeks, T. L.

Holmes, S. J.

J. M. Shaw, J. D. Gelorme, N. C. LaBianca, W. E. Conley, and S. J. Holmes, “Negative photoresists for optical lithography,” IBM J. Res. Dev. 41, 81 –94 (1997).
[CrossRef]

Klein, W. R.

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

Kogelnik, H.

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

Komanduri, R. K.

R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76, 021701 (2007).
[CrossRef]

Kostuk, R. K.

Kozinkov, V.

M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-induced parallel alignment of liquid crystals by linearly polymerized photopolymers,” Jpn. J. Appl. Phys. 31, 2155–2164 (1992).
[CrossRef]

LaBianca, N. C.

J. M. Shaw, J. D. Gelorme, N. C. LaBianca, W. E. Conley, and S. J. Holmes, “Negative photoresists for optical lithography,” IBM J. Res. Dev. 41, 81 –94 (1997).
[CrossRef]

Magnusson, R.

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

Minano, J. C.

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic Press, 2005), Chap. 13.

Moharam, M.

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

Moharam, M. G.

Myer, B.

Nagendra Nath, N. S.

C. V. Raman and N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves: Part I.” Proc. Indian Acad. Sci., Sect. A 2, 406–412 (1935).

Nikolova, L.

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31, 579–588 (1984).
[CrossRef]

Oh, C.

C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76, 043815 (2007).
[CrossRef]

C. Oh and M. J. Escuti, “Time-domain analysis of periodic anisotropic media at oblique incidence: an efficient FDTD implementation,” Opt. Express 14, 11870–11884 (2006).
[CrossRef] [PubMed]

Park, B.

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: Potential application for projection displays,” Mol. Cryst. Liq. Cryst. 451, 1–19 (2006).
[CrossRef]

Pommet, D. A.

Prost, J.

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1993).

Raman, C. V.

C. V. Raman and N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves: Part I.” Proc. Indian Acad. Sci., Sect. A 2, 406–412 (1935).

Reinhardt, E. R.

Richards, B. S.

Rowan, B. C.

Sarkissian, H.

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: Potential application for projection displays,” Mol. Cryst. Liq. Cryst. 451, 1–19 (2006).
[CrossRef]

Schadt, M.

M. Schadt, H. Seiberle, and A. Schuster, “Optical patterning of multi-domain liquid-crystal displays with wide viewing angles,” Nature 381, 212–215 (1996).
[CrossRef]

M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-induced parallel alignment of liquid crystals by linearly polymerized photopolymers,” Jpn. J. Appl. Phys. 31, 2155–2164 (1992).
[CrossRef]

Schmitt, K.

M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-induced parallel alignment of liquid crystals by linearly polymerized photopolymers,” Jpn. J. Appl. Phys. 31, 2155–2164 (1992).
[CrossRef]

Schuster, A.

M. Schadt, H. Seiberle, and A. Schuster, “Optical patterning of multi-domain liquid-crystal displays with wide viewing angles,” Nature 381, 212–215 (1996).
[CrossRef]

Seiberle, H.

M. Schadt, H. Seiberle, and A. Schuster, “Optical patterning of multi-domain liquid-crystal displays with wide viewing angles,” Nature 381, 212–215 (1996).
[CrossRef]

Shaw, J. M.

J. M. Shaw, J. D. Gelorme, N. C. LaBianca, W. E. Conley, and S. J. Holmes, “Negative photoresists for optical lithography,” IBM J. Res. Dev. 41, 81 –94 (1997).
[CrossRef]

Swanson, R. M.

R. M. Swanson, “Photovoltaic concentrators,” in Handbook of Photovoltaic Science and Engineering , A. Luque and S. Hegedus, eds. (Wily & Sons, 2003).

Tabirian, N.

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: Potential application for projection displays,” Mol. Cryst. Liq. Cryst. 451, 1–19 (2006).
[CrossRef]

Tervo, J.

Todorov, T.

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31, 579–588 (1984).
[CrossRef]

Turunen, J.

Urbach, H. P.

van Heesch, C. M.

Verbunt, P. P. C.

Wachters, A. J.

Wachters, A. J. H.

Wei, X.

Welford, W. T.

Winston, R.

Xu, M.

Zeldovich, B.

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: Potential application for projection displays,” Mol. Cryst. Liq. Cryst. 451, 1–19 (2006).
[CrossRef]

Zhang, D.

Appl. Opt.

Appl. Phys.

A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[CrossRef]

Bell Syst. Tech. J.

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

IBM J. Res. Dev.

J. M. Shaw, J. D. Gelorme, N. C. LaBianca, W. E. Conley, and S. J. Holmes, “Negative photoresists for optical lithography,” IBM J. Res. Dev. 41, 81 –94 (1997).
[CrossRef]

IEEE Trans. Sonics Ultrason.

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

J. Mod. Opt.

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31, 579–588 (1984).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-induced parallel alignment of liquid crystals by linearly polymerized photopolymers,” Jpn. J. Appl. Phys. 31, 2155–2164 (1992).
[CrossRef]

Mol. Cryst. Liq. Cryst.

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: Potential application for projection displays,” Mol. Cryst. Liq. Cryst. 451, 1–19 (2006).
[CrossRef]

Nature

M. Schadt, H. Seiberle, and A. Schuster, “Optical patterning of multi-domain liquid-crystal displays with wide viewing angles,” Nature 381, 212–215 (1996).
[CrossRef]

Opt. Commun.

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. Moharam, T. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76, 043815 (2007).
[CrossRef]

Phys. Rev. E

R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76, 021701 (2007).
[CrossRef]

Proc. Indian Acad. Sci., Sect. A

C. V. Raman and N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves: Part I.” Proc. Indian Acad. Sci., Sect. A 2, 406–412 (1935).

Other

Grating Solver Development Co., http://www.gsolver.com/ .

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1993).

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005).

R. M. Swanson, “Photovoltaic concentrators,” in Handbook of Photovoltaic Science and Engineering , A. Luque and S. Hegedus, eds. (Wily & Sons, 2003).

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic Press, 2005), Chap. 13.

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Figures (14)

Fig. 1
Fig. 1

Sketch of a grating for in-coupling applications. Here, the +1st order (m = 1) is transmitted into TIR for all angels of incidence, while the 0th order (m = 0) is lost. The −1st order (m = −1) is transmitted into TIR for small angles of incidence and is lost for larger values of θ.

Fig. 2
Fig. 2

Calculated transmitted diffraction efficiency as a function of the grating thickness d at normal incidence for a thin surface-relief grating in SU-8: Λ = 15μm, λ = 633 nm, s-polarization. Solid lines show the RCWA results, dotted lines the corresponding Fraunhofer approximation (FA). (a) Sinusoidal surface-relief profile. (b) Rectangular surface-relief profile.

Fig. 3
Fig. 3

Calculated (RCWA) transmitted diffraction efficiency as a function of the angle of incidence for a thin surface-relief grating in SU-8: Λ = 15μm, λ = 633 nm, s-polarization. (a) Sinusoidal surface-relief profile with d = 821 nm. (b) Rectangular surface-relief profile with d = 536 nm.

Fig. 4
Fig. 4

Confocal microscopy image (a) and corresponding cross section (b) of a thin surface-relief grating in SU-8: Λ ≈ 15.6μm, d ≈ 0.6μm.

Fig. 5
Fig. 5

Measured (a) and calculated (RCWA) (b) transmitted diffraction efficiency as a function of the angle of incidence for an experimental thin surface-relief profile in SU-8: Λ ≈ 15.6μm, d ≈ 0.6μm, λ = 633 nm, s-polarized light.

Fig. 6
Fig. 6

Calculated (RCWA) transmitted diffraction efficiency as a function of the grating thickness d at normal incidence for a non-thin surface-relief grating in SU-8: Λ = 600 nm, λ = 633 nm, s-polarization. (a) Sinusoidal surface-relief profile. (b) Rectangular surface-relief profile.

Fig. 7
Fig. 7

Calculated (RCWA) transmitted diffraction efficiency as a function of the angle of incidence for a non-thin surface-relief grating in SU-8: Λ = 600 nm, λ = 633 nm, s-polarization. Left panel: sinusoidal surface-relief profile with d = 410 nm. Right panel: rectangular surface-relief profile with d = 351 nm.

Fig. 8
Fig. 8

SEM images of the top view (a) and cross section (b) of a non-thin surface-relief grating in SU-8: Λ ≈ 0.59μm, d ≈ 0.4μm.

Fig. 9
Fig. 9

Measured (a) and calculated (RCWA) (b) transmitted diffraction efficiency as a function of the angle of incidence for an experimental non-thin surface-relief grating in SU-8: Λ ≈ 0.59μm, d ≈ 0.4μm, λ = 633 nm, s-polarization.

Fig. 10
Fig. 10

Calculated (RCWA) transmitted diffraction efficiency as a function of the grating thickness d at normal incidence for a small-period surface-relief grating in a material with a high refractive index: n = 2.49, Λ = 600 nm, λ = 633 nm, s-polarization. (a) Sinusoidal surface-relief profile. (b) Rectangular surface-relief profile.

Fig. 11
Fig. 11

Calculated (RCWA) transmitted diffraction efficiency as a function of the angle of incidence for s-polarized light for a small-period surface-relief grating in a material with a high refractive index: n = 2.49, Λ = 600 nm, λ = 633 nm. (a) Sinusoidal surface-relief profile with d = 289 nm. (b) Rectangular surface-relief profile with d = 262 nm.

Fig. 12
Fig. 12

Polarization microscopy images of thin polarization gratings. (a) Polarization grating made with ZLI-2222-000-EB019802 TJ-98-009 (Merck). (b) Polarization grating made with E7 (Merck). In (b) we observe a periodically recurring misalignment. The director indicated in the picture may be Λ/2 off.

Fig. 13
Fig. 13

Measured diffraction efficiency as a function of the angle of incidence for an experimental thin polarization grating: ne ≈ 1.61, no ≈ 1.50, Λ ≈ 15μm, d ≈ 3μm, λ = 633 nm, s-polarized light.

Fig. 14
Fig. 14

Calculated (FEM) diffraction efficiency as a function of the angle of incidence for a non-thin polarization grating (λ = 633 nm, s-polarized light). (a) Polarization grating with Λ = 678 nm, d = 875 nm and Δn = 0.2. (b) Polarization grating with Λ = 600 nm, d = 323 nm and Δn = 0.59.

Equations (12)

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

E m = 1 Λ 0 Λ E in exp [ 2 π i ( m x / Λ + d n ( x ) / λ ) ] d x ,
η m = J m 2 ( π Δ n d / λ ) ,
η m = sinc 2 ( m π / 2 ) cos 2 ( π Δ n d / λ m π / 2 )
= { cos 2 ( π d Δ n / λ , for m = 0 ; 4 / ( m π ) 2 sin 2 ( π d Δ n / λ ) , for m = 2 k + 1 ; 0 , for m = 2 k , k 0 ,
Q = 2 π d λ n ¯ Λ 2 ,
ρ = 2 λ 2 n ¯ Δ n Λ 2 ,
d = { 2.4 λ / ( π Δ n ) ( sinusoidal profile ) ; λ / ( 2 Δ n ) ( rectangular profile ) ,
η 0 = cos 2 ( π Δ n d / λ ) ,
η ± 1 = 1 S 3 ' 2 sin 2 ( π Δ n d / λ ) ,
n = (cos θ n cos ϕ n , cos θ n sin ϕ n , sin θ n ) = ( cos ( π x / Λ ) , sin ( π x / Λ ) , 0 ) ,
F = 1 2 K 1 ( n ) 2 + 1 2 K 2 ( n × n ) 2 + 1 2 K 2 ( n × × n ) 2
= 1 2 ( π Λ ) 2 [ K 1 sin 2 ( π x / Λ ) + K 3 cos 2 ( π x / Λ ) ] ,

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