Abstract

Coupled-wave analysis was applied to the problem of optical wave diffraction in a thick reflection-type phase grating in a condition of second-order Bragg regime. Within the approximation of two resonant waves in a medium with phase (refractive index) modulation, an analytical formula was derived for the diffraction efficiency. A shift of the maximum of a diffraction efficiency contour from the position determined by vector synchronism (second-order Bragg resonance) was detected.

© 2009 Optical Society of America

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References

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  1. M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
    [CrossRef]
  2. H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152-154 (1971).
    [CrossRef]
  3. J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
    [CrossRef]
  4. A. Saliminia, A. Villeneuve, T. V. Galstian, S. LaRoshelle, and K. Richardson, “First- and second-order Bragg gratings in single-mode planar waveguides of chalcogenide glasses,” J. Lightwave Technol. 17, 837-842 (1999).
    [CrossRef]
  5. R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353-362 (1976).
    [CrossRef]
  6. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811-818 (1981).
    [CrossRef]
  7. H. Wenzel, R. Güthel, and A. M. Shams-Zadeh-Amiri, “A comparative study of higher order Bragg gratings: coupled-mode theory versus mode expansion modeling,” IEEE J. Quantum Electron. 42, 64-70 (2006).
    [CrossRef]
  8. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2910-2947 (1969).
  9. R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).
    [CrossRef]
  10. G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1961).

2008 (1)

M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

2006 (1)

H. Wenzel, R. Güthel, and A. M. Shams-Zadeh-Amiri, “A comparative study of higher order Bragg gratings: coupled-mode theory versus mode expansion modeling,” IEEE J. Quantum Electron. 42, 64-70 (2006).
[CrossRef]

1999 (1)

1981 (1)

1976 (1)

1972 (1)

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
[CrossRef]

1971 (1)

H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152-154 (1971).
[CrossRef]

1970 (1)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2910-2947 (1969).

1961 (1)

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1961).

Abbate, G.

M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Alferness, R.

Bjorkholm, J. E.

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
[CrossRef]

Chu, R. S.

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).
[CrossRef]

Galstian, T. V.

Gaylord, T. K.

Güthel, R.

H. Wenzel, R. Güthel, and A. M. Shams-Zadeh-Amiri, “A comparative study of higher order Bragg gratings: coupled-mode theory versus mode expansion modeling,” IEEE J. Quantum Electron. 42, 64-70 (2006).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152-154 (1971).
[CrossRef]

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2910-2947 (1969).

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1961).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1961).

LaRoshelle, S.

Moharam, M. G.

Richardson, K.

Sakhno, O.

M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Saliminia, A.

Shams-Zadeh-Amiri, A. M.

H. Wenzel, R. Güthel, and A. M. Shams-Zadeh-Amiri, “A comparative study of higher order Bragg gratings: coupled-mode theory versus mode expansion modeling,” IEEE J. Quantum Electron. 42, 64-70 (2006).
[CrossRef]

Shank, C. V.

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
[CrossRef]

H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152-154 (1971).
[CrossRef]

Slussarenko, S. S.

M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Stumpe, J.

M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Tamir, T.

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).
[CrossRef]

Vasnetsov, M. V.

M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Villeneuve, A.

Wenzel, H.

H. Wenzel, R. Güthel, and A. M. Shams-Zadeh-Amiri, “A comparative study of higher order Bragg gratings: coupled-mode theory versus mode expansion modeling,” IEEE J. Quantum Electron. 42, 64-70 (2006).
[CrossRef]

Yu. Bazhenov, V.

M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Appl. Phys. Lett. (2)

H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152-154 (1971).
[CrossRef]

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2910-2947 (1969).

IEEE J. Quantum Electron. (1)

H. Wenzel, R. Güthel, and A. M. Shams-Zadeh-Amiri, “A comparative study of higher order Bragg gratings: coupled-mode theory versus mode expansion modeling,” IEEE J. Quantum Electron. 42, 64-70 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (2)

Mol. Cryst. Liq. Cryst. (1)

M. V. Vasnetsov, V. Yu. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Other (1)

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1961).

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Figures (8)

Fig. 1
Fig. 1

(a) Scheme and a vector diagram of Bragg (first-order) resonance diffraction. Wave vectors of the incident wave U and reflected wave V form a symmetric triangle with grating vector K. (b) Same for the second-order Bragg resonance. The wave vectors of the incident and the reflected waves are coupled via two grating vectors K. A wave vector that belongs to an intermediate diffraction order W is also shown.

Fig. 2
Fig. 2

Diagram shows a geometric relation between σ x and σ z components of wave vector σ of a field in a modulated medium in the approximation of three coupled waves [Eq. (4) at the limit κ 0 ]. Nonresonant orders of diffraction are not shown. First-order Bragg resonance is depicted with dashed arrows and second-order Bragg resonance is shown with solid arrows.

Fig. 3
Fig. 3

(a) Fragment of the dispersion curve around the second-order Bragg resonance position. The thick curve shows the approximation according to Eq. (10). Three roots R 1 3 are indicated by dots. The thin curve shows the limit for κ 0 [Eqs. (5b, 5c)], intersecting circles). (b) Same fragment of the dispersion curve, calculated in accordance with an exact solution of Eq. (4). Parameters of calculation are κ 2 = 0.013 , k 2 = 1 , and K 2 = 0.75 .

Fig. 4
Fig. 4

Calculated contour of the diffraction efficiency for second-order Bragg resonance. The vertical line indicates an exact geometric second-order Bragg resonance position at Δ σ x = 0 . Parameters of calculation are k 2 = 1 , K 2 = 0.75 , κ 2 = 0.013 , and d = 25 Λ .

Fig. 5
Fig. 5

Calculated contour of diffraction efficiency for nearly counterpropagating waves. Two separate symmetric maxima correspond to the positive and the negative angles of incidence. The vertical lines indicate the symmetric positions of second-order Bragg resonances. Parameters of calculation are k 2 = 1 , K 2 = 0.95 , κ 2 = 0.013 , and d = 25 Λ .

Fig. 6
Fig. 6

Transformation of the diffraction efficiency contour with increasing the ratio K k . Contours 1–3 correspond to the ratios ( K k ) 2 = 0.975 , 1, and 1.025, respectively. Other parameters are the same as in Fig. 5.

Fig. 7
Fig. 7

Six wave vectors that correspond to the solution of the dispersion Eq. (A3) (exact second-order Bragg resonance).

Fig. 8
Fig. 8

Vector diagram with four wave vectors of the partial waves that belong to the solution for the propagating wave U.

Equations (54)

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E ( x , z ) = [ U ( z ) exp ( i K z ) + W ( z ) + V ( z ) exp ( i K z ) ] exp ( i σ x x ) ,
2 E ( x , z ) x 2 + 2 E ( x , z ) z 2 = k 0 2 ε ( z ) ,
{ [ k 2 σ x 2 ( σ z + K ) 2 ] u = κ w [ k 2 σ x 2 σ z 2 ] w = κ ( u + v ) [ k 2 σ x 2 ( σ z K ) 2 ] v = κ w } .
( k 2 σ 2 ) ( k 2 σ 2 K 2 ) 2 4 σ z 2 K 2 = 2 κ 2 ( k 2 σ 2 K 2 ) ,
( σ 2 k 2 + K 2 ) 2 = 4 σ z 2 K 2 ,
( σ z ± K ) 2 + σ x 2 = k 2 ,
σ 2 = k 2 .
( k 2 σ x 2 ) ( k 2 σ x 2 K 2 ) 2 = 2 κ 2 ( k 2 σ x 2 K 2 ) .
R 1 : σ x = ( k 2 K 2 ) 1 2 = σ x 0 ,
R 2 : σ x = [ k 2 K 2 2 ( K 2 4 + 2 κ 2 ) 1 2 ] 1 2 σ x 0 κ 2 K 2 σ x 0 ,
R 3 : σ x = [ k 2 K 2 2 + ( K 2 4 + 2 κ 2 ) 1 2 ] 1 2 k + κ 2 k 2 K .
σ x 2 σ x 0 2 2 σ x 0 Δ σ x .
σ z 2 = σ x 0 2 ( Δ σ x ) 2 K 2 + κ 2 σ x 0 Δ σ x K 4 ,
( σ x 0 Δ σ x K + κ 2 2 K 3 ) 2 σ z 2 = κ 4 4 K 6 .
U ( z ) = u 1 exp ( i σ z 1 z ) + u 2 exp ( i σ z 2 z )
V ( z ) = v 1 exp ( i σ z 1 z ) + v 2 exp ( i σ z 2 z )
σ z 1 , 2 = ± ( σ x 0 2 ( Δ σ x ) 2 K 2 + κ 2 σ x 0 Δ σ x K 4 ) 1 2 ,
U ( z = 0 ) = u 1 + u 2 = U .
V ( z = d ) = v 1 exp ( i σ z 1 d ) + v 2 exp ( i σ z 2 d ) = 0 .
η = V ( z = 0 ) 2 U 2 ,
v 1 , 2 = u 1 , 2 k 2 σ x 2 ( σ z 1 , 2 + K ) 2 k 2 σ x 2 ( σ z 1 , 2 K ) 2 ,
v 1 , 2 = u 1 , 2 σ x 0 Δ σ x + K σ z 1 , 2 σ x 0 Δ σ x K σ z 1 , 2 = u 1 , 2 q 1 , 2 ,
v 1 + v 2 = i U 2 sin ( σ z 1 d ) q 1 exp ( i σ z 1 d ) q 2 exp ( i σ z 1 d ) = i U κ 2 K 3 sin ( σ z 1 d ) 2 i ( σ x 0 Δ σ x K + κ 2 2 K 3 ) sin ( σ z 1 d ) 2 σ z 1 cos ( σ z 1 d ) ,
η = 1 1 + 4 K 6 κ 4 σ z 1 2 sin 2 ( σ z 1 d ) .
v 1 + v 2 = i U κ 2 K 3 sinh ( σ z 1 d ) 2 i ( σ x 0 Δ σ x K + κ 2 2 K 3 ) sinh ( σ z 1 d ) 2 σ z 1 cosh ( σ z 1 d ) ,
η = 1 1 + 4 K 6 κ 4 σ z 1 2 sinh 2 ( σ z 1 d ) .
η = ( 1 + 4 K 6 κ 4 d 2 ) 1 .
σ z 1 = i κ 2 2 K 3 ,
η = tanh 2 ( κ 2 2 K 3 d ) .
σ z 2 = ( σ x 2 σ x 0 2 ) 2 + 2 κ 2 K 2 ( σ x 2 σ x 0 2 ) 4 K 2 ,
w = u σ x 0 2 σ x 2 σ z 2 2 K σ z κ ,
w u κ ( κ 2 K 2 + κ 4 4 K 6 i κ 2 K 2 ) u κ 1 i K 2 .
ξ 3 + a ξ 2 + b ξ + c = 0 ,
a = K 2 ,
b = ( 4 σ z 2 K 2 + 2 κ 2 ) ,
c = 4 σ z 2 K 4 .
ξ 1 = 2 Q 1 2 cos ( t 2 π 3 ) a 3 ( R 1 branch ) ,
ξ 2 = 2 Q 1 2 cos ( t + 2 π 3 ) a 3 ( R 2 branch ) ,
ξ 3 = 2 Q 1 2 cos t a 3 ( R 3 branch ) ,
t = 1 3 arccos P Q 3 2 ,
P = ( 2 a 3 9 a b + 27 c ) 54 ,
Q = ( a 2 3 a b ) 9 .
σ x j = ± ( σ x 0 2 σ z 2 ξ i ) 1 2 ( i = 1 , 2 , 3 , j = 1 6 ) .
a = 5 K 2 ,
b = 4 K 4 + 4 K 2 ( σ x 2 σ x 0 2 ) 2 κ 2 ,
c = 4 K 4 ( σ x 2 σ x 0 2 ) .
σ z j = ± ( σ x 0 2 σ x 2 ξ i ) 1 2 ( i = 1 , 2 , 3 , j = 1 6 )
( K 2 σ z 2 ) ( σ z 4 4 K 2 σ z 2 ) = 2 κ 2 σ z 2
σ z 2 = 0 .
( K 2 σ z 2 ) ( σ z 2 4 K 2 ) = 2 κ 2
σ z 2 = 5 K 2 2 ± 9 K 4 4 + 2 κ 2 .
σ z 1 , 2 = 0 ,
σ z 3 , 4 ± ( K κ 2 3 K 3 ) ,
σ z 5 , 6 ± ( 2 K + κ 2 3 K 3 ) .

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