Abstract

Beam splitting of low-contrast rectangular gratings under second Bragg angle incidence is studied. The grating period is between λ and 2λ. The diffraction behaviors of the three transmitted propagating orders are illustrated by analyzing the first three propagating grating modes. From a simplified modal approach, the design conditions of gratings as a high-efficiency element with most of its energy concentrated in the 2nd transmitted order (90%) and of gratings as a 1×2 beam splitter with a total efficiency over 90% are derived. The grating parameters for achieving exactly the splitting pattern by use of rigorous coupled-wave analysis verified the design method. A 1×3 beam splitter is also demonstrated. Moreover, the polarization-dependent diffraction behaviors are investigated, which suggest the possibility of designing polarization-selective elements under such a configuration. The proposed concept of using the second Bragg angle should be helpful for developing new grating-based devices.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. S. Wang, C. Zhou, Y. Zhang, and H. Ru, “Deep-etched high-density fused-silica transmission gratings with high efficiency at a wavelength of 1550nm,” Appl. Opt. 45, 2567-2571 (2006).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2007 (4)

2006 (2)

A. Drauschke, “Analysis of nearly depth-independent transmission of lamellar gratings in zeroth diffraction order in TM polarization,” J. Opt. A, Pure Appl. Opt. 8, 511-517 (2006).
[CrossRef]

S. Wang, C. Zhou, Y. Zhang, and H. Ru, “Deep-etched high-density fused-silica transmission gratings with high efficiency at a wavelength of 1550nm,” Appl. Opt. 45, 2567-2571 (2006).
[CrossRef] [PubMed]

2005 (4)

2004 (2)

C. F. R. Mateus, C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518-520 (2004).
[CrossRef]

D. Delbeke, R. Baets, and P. Muys, “Polarization-selective beam splitter based on a highly efficient simple binary diffraction grating,” Appl. Opt. 43, 6157-6165 (2004).
[CrossRef] [PubMed]

2003 (1)

1999 (1)

P. Lalanne, J. Hazart, P. Chavel, E. Cambril, and H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A, Pure Appl. Opt. 1, 215-219 (1999).
[CrossRef]

1997 (1)

1996 (1)

1992 (1)

1984 (1)

1983 (1)

1982 (1)

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Baets, R.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Cambril, E.

P. Lalanne, J. Hazart, P. Chavel, E. Cambril, and H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A, Pure Appl. Opt. 1, 215-219 (1999).
[CrossRef]

Case, S. K.

Chang, C. J.

C. F. R. Mateus, C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518-520 (2004).
[CrossRef]

Chavel, P.

P. Lalanne, J. Hazart, P. Chavel, E. Cambril, and H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A, Pure Appl. Opt. 1, 215-219 (1999).
[CrossRef]

Clausnitzer, T.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Delbeke, D.

Deng, Y.

C. F. R. Mateus, C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518-520 (2004).
[CrossRef]

Drauschke, A.

A. Drauschke, “Analysis of nearly depth-independent transmission of lamellar gratings in zeroth diffraction order in TM polarization,” J. Opt. A, Pure Appl. Opt. 8, 511-517 (2006).
[CrossRef]

Enger, R. C.

Fahr, S.

Feng, J.

Fuchs, H.-J.

Garnet, E.

Gaylord, T. K.

Hagness, S.

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed.(Artech, 2000).

Hazart, J.

P. Lalanne, J. Hazart, P. Chavel, E. Cambril, and H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A, Pure Appl. Opt. 1, 215-219 (1999).
[CrossRef]

Huang, C. Y.

C. F. R. Mateus, C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518-520 (2004).
[CrossRef]

Iwata, K.

Jupé, M.

Kämpfe, T.

Kikuta, H.

Kley, E.-B.

Lalanne, P.

P. Lalanne, J. Hazart, P. Chavel, E. Cambril, and H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A, Pure Appl. Opt. 1, 215-219 (1999).
[CrossRef]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

Launois, H.

P. Lalanne, J. Hazart, P. Chavel, E. Cambril, and H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A, Pure Appl. Opt. 1, 215-219 (1999).
[CrossRef]

Limpert, J.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Miller, J. M.

Moharam, M. G.

Morris, G. M.

Muys, P.

Neureuther, A. R.

C. F. R. Mateus, C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518-520 (2004).
[CrossRef]

Noponen, E.

Ohira, Y.

Parriaux, O.

Peschel, U.

R. Mateus, C. F.

C. F. R. Mateus, C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518-520 (2004).
[CrossRef]

Ristau, D.

Ru, H.

Taflove, A.

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed.(Artech, 2000).

Taghizadeh, M. R.

Tishchenko, A.

Tishchenko, A. V.

Tünnermann, A.

Turunen, J.

Vasara, A.

Wang, B.

Wang, S.

Yokomori, K.

Zellmer, H.

Zhang, Y.

Zhou, C.

Zöllner, K.

Appl. Opt. (10)

R. C. Enger and S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220-3228 (1983).
[CrossRef] [PubMed]

K. Yokomori, “Dielectric surface-relief gratings with high diffraction efficiencies,” Appl. Opt. 23, 2303-2310 (1984).
[CrossRef] [PubMed]

H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566-1572 (1997).
[CrossRef] [PubMed]

T. Clausnitzer, J. Limpert, K. Zöllner, H. Zellmer, H.-J. Fuchs, E.-B. Kley, A. Tünnermann, M. Jupé, and D. Ristau, “Highly-efficient transmission gratings in fused silica for chirped pulse amplification systems,” Appl. Opt. 42, 6934-6938 (2003).
[CrossRef] [PubMed]

D. Delbeke, R. Baets, and P. Muys, “Polarization-selective beam splitter based on a highly efficient simple binary diffraction grating,” Appl. Opt. 43, 6157-6165 (2004).
[CrossRef] [PubMed]

S. Wang, C. Zhou, H. Ru, and Y. Zhang, “Optimized condition for etching fused-silica phase gratings with inductively coupled plasma technology,” Appl. Opt. 44, 4429-4434 (2005).
[CrossRef] [PubMed]

S. Wang, C. Zhou, Y. Zhang, and H. Ru, “Deep-etched high-density fused-silica transmission gratings with high efficiency at a wavelength of 1550nm,” Appl. Opt. 45, 2567-2571 (2006).
[CrossRef] [PubMed]

T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, and O. Parriaux, “Investigation of the polarization-dependent diffraction of deep dielectric rectangular gratings illuminated in Littrow mounting,” Appl. Opt. 46, 819-826 (2007).
[CrossRef] [PubMed]

S. Fahr, T. Clausnitzer, E.-B. Kley, and A. Tünnermann, “Reflective diffractive beam splitter for laser interferometers,” Appl. Opt. 46, 6092-6095 (2007).
[CrossRef] [PubMed]

E. Garnet, A. V. Tishchenko, and O. Parriaux, “Cancellation of the zeroth order in a phase mask by mode interplay in a high index contrast binary grating,” Appl. Opt. 46, 6719-6726 (2007).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

C. F. R. Mateus, C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518-520 (2004).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

P. Lalanne, J. Hazart, P. Chavel, E. Cambril, and H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A, Pure Appl. Opt. 1, 215-219 (1999).
[CrossRef]

A. Drauschke, “Analysis of nearly depth-independent transmission of lamellar gratings in zeroth diffraction order in TM polarization,” J. Opt. A, Pure Appl. Opt. 8, 511-517 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309-330 (2005).
[CrossRef]

Other (1)

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed.(Artech, 2000).

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Figures (11)

Fig. 1
Fig. 1

Schematic of a low-contrast grating under second Bragg angle incidence.

Fig. 2
Fig. 2

Plot of F ( n e f f 2 ) for a grating in Fig. 1 with period d = 1.5 λ , duty cycle f = 0.5 and TE polarization. The straight line (blue online) indicates the incident condition of cos ( k x d ) = 1 . The three black crossing points indicate the excited three propagating grating modes.

Fig. 3
Fig. 3

Mode profiles of the three propagating modes for TE polarization ( E y ) for the same grating as is used in Fig. 2.

Fig. 4
Fig. 4

Difference of effective mode indices versus grating period d with duty cycle f = 0.5 . The cross point indicates the period that satisfies the design condition of Eq. (12b).

Fig. 5
Fig. 5

Diffraction efficiency of the three transmitted orders versus depth for a grating with period d = 1.296 λ and duty cycle of f = 0.6 for TE polarization; a high-efficiency element with depth h = 1.927 λ .

Fig. 6
Fig. 6

Diffraction efficiency versus depth for a grating with period d = 1.4457 λ and duty cycle f = 0.5 for TM polarization; a 1 × 2 beam splitter with depth h = 2.28 λ .

Fig. 7
Fig. 7

Near-field distribution of amplitude E y for TE polarization for the 1 × 2 beam splitter with period d = 1.2923 λ , duty cycle f = 0.4 , and depth h = 1.8595 λ . The solid white curve indicates the grating profile.

Fig. 8
Fig. 8

Diffraction efficiency versus depth for a grating with period d = 1.0784 λ and duty cycle f = 0.5 for TE polarization; a 1 × 3 beam splitter with depth h = 0.92 λ .

Fig. 9
Fig. 9

Effective indices of the propagating grating modes versus period d for gratings with duty cycle f = 0.5 for both TE and TM polarizations. The arrow T indicates the intersection point with d = 1.1094 λ .

Fig. 10
Fig. 10

Diffraction efficiency versus depth for TM polarization for gratings with period d = 1.1094 λ and duty cycle f = 0.5 , where n 1 e f f = n 2 e f f .

Fig. 11
Fig. 11

Effective indices of the propagating grating modes versus duty cycle f for gratings with period d = 1.5 λ for both TE and TM polarizations. Two arrows: T1, the first intersection point; T2, the second intersection point.

Tables (1)

Tables Icon

Table 1 Overlap Integrals between the First Five Grating Modes and Diffraction Orders for a Grating with Period d = 1.5 λ and Duty Cycle f = 0.5 under Second Bragg Angle Incidence for TE Polarization

Equations (33)

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

F ( n e f f 2 ) = cos ( k r f d ) cos ( k g ( 1 f ) d ) k r 2 + k g 2 2 k g k r sin ( k r f d ) sin ( k g ( 1 f ) d ) = cos ( k x d ) ,
k x = 2 n 1 π λ sin θ , k r = 2 π λ n 1 2 n e f f 2 ,
k g = 2 π λ n 2 2 n e f f 2 , f = b d ,
E y n ( x ) u m ( x ) = 0 d E y n ( x ) u m ( x ) d x 2 0 d E y n ( x ) 2 d x 0 d u m ( x ) 2 d x ,
E y i n ( x ) = e i k x x = a 0 w 0 ( x ) + a 1 w 1 ( x ) + a 2 w 2 ( x ) = u 0 ( x ) + u 1 ( x ) + u 2 ( x )
0 d w m ( x ) 2 d x = 1 , a m = 0 d w m ( x ) e i k x x d x ,
u m ( x ) = a m w m ( x ) , m = 0 , 1 , 2 ,
E y ( x , h ) = u 0 ( x ) exp ( i k n 0 e f f h ) + u 1 ( x ) exp ( i k n 1 e f f h ) + u 2 ( x ) exp ( i k n 2 e f f h ) = E 2 exp ( + i k x x ) + E 1 + E 0 exp ( i k x x ) ,
0 d ( u 0 ( x ) + u 2 ( x ) ) d x = 0 ,
E 1 2 = 1 d 2 0 d ( u 0 ( x ) exp ( i k ( n 0 e f f n 2 e f f ) h ) + u 2 ( x ) ) d x 2 .
η 1 = E 1 2 = 4 A 2 sin 2 ( k ( n 0 e f f n 2 e f f ) h 2 ) ,
A = 1 d 0 d u 0 ( x ) d x .
0 d ( ( u 0 ( x ) + u 2 ( x ) ) cos ( k x x ) + i u 1 ( x ) sin ( k x x ) ) d x d = 1 ,
0 d ( ( u 0 ( x ) + u 2 ( x ) ) cos ( k x x ) i u 1 ( x ) sin ( k x x ) ) d x = 0 ,
E 0 2 = 0 d ( ( u 0 ( x ) e i k ( n 0 e f f n 2 e f f ) h + u 2 ( x ) ) cos ( k x x ) + i u 1 ( x ) e i k ( n 1 e f f n 2 e f f ) h sin ( k x x ) ) d x 2 d 2 ,
E 2 2 = 0 d ( ( u 0 ( x ) e i k ( n 0 e f f n 2 e f f ) h + u 2 ( x ) ) cos ( k x x ) i u 1 ( x ) e i k ( n 1 e f f n 2 e f f ) h sin ( k x x ) ) d x 2 d 2 .
k ( n 0 e f f n 2 e f f ) h = 2 n π , n = 1 , 2 , 3 ,
η 0 = cos 2 ( k ( n 1 e f f n 2 e f f ) h 2 ) ,
η 2 = sin 2 ( k ( n 1 e f f n 2 e f f ) h 2 ) .
k ( n 0 e f f n 2 e f f ) h = 2 n π ,
k ( n 1 e f f n 2 e f f ) h = ( 2 m 1 ) π ,
m , n = 1 , 2 , 3 .
h = λ n 0 e f f n 2 e f f ,
n 0 e f f n 2 e f f = 2 ( n 1 e f f n 2 e f f ) .
k ( n 0 e f f n 2 e f f ) h = 2 n π ,
k ( n 1 e f f n 2 e f f ) h = ( 2 m 1 ) 2 π ,
m , n = 1 , 2 , 3 .
h = λ n 0 e f f n 2 e f f ,
n 0 e f f n 2 e f f = 4 ( n 1 e f f n 2 e f f ) .
k r f d + k g ( 1 f ) d = 2 π .
n 1 e f f = n 2 e f f = n 1 n 2 n 1 2 + n 2 2 .
k r f d = π ,
k g ( 1 f ) d = π .

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