Abstract

The second-harmonic generation (SHG) in finite Bragg stacks of alternating linear and nonlinear optical films is studied with the exact Green function under the assumption of perturbation theory. Various mechanisms of enhanced SHG are investigated analytically, and the scaling law with respect to the number N of stacking layers is derived for each mechanism. In particular, it is shown that there is a simple mechanism of the enhanced SHG having N 4 scaling, in which both the enhancement of the Green function and the phase matching condition peculiar to finite Bragg stacks are fulfilled simultaneously.

© 2005 Optical Society of America

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References

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Appl. Phys. Lett. (2)

G. T. Kiehne, A. E. Kryukov, and J. B. Ketterson, �??A numerical study of optical second-harmonic generation in a one-dimensional photonic structure,�?? Appl. Phys. Lett. 75, 1676�??1678 (1999).
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D�??Aguanno, and M. Scalora, �??Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,�?? Appl. Phys. Lett. 78, 3021�??3023 (2001).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. A (3)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J.W. Haus, �??Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,�?? Phys. Rev. A 56, 3166�??3174 (1997).
[CrossRef]

J.W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, �??Enhanced second-harmonic generation in media with a weak periodicity,�?? Phys. Rev. A 57, 2120�??2128 (1998).
[CrossRef]

M. S. Tomas, �??Green-function for multilayers - light-scattering in planar cavities,�?? Phys. Rev. A 51, 2545�??2559 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (1)

F. F. Ren, R. Li, C. Cheng, H. T.Wang, J. R. Qiu, J. H. Si, and K. Hirao, �??Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,�?? Phys. Rev. B 70, 245109 (2004).
[CrossRef]

Phys. Rev. E (1)

M. Centini, C. Sibilia, M. Scalora, G. D�??Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, �??Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,�?? Phys. Rev. E 60, 4891�??4898 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, �??Phase-matched frequency doubling at photonic band edges: Efficiency scaling as the fifth power of the length,�?? Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Other (3)

M. Bertolotti, C. M. Bowden, and C. Sibilia, eds., Nanoscale Linear and Nonlinear Optics, vol. 560 of AIP conference proceedings (American Institute of Physics, New York, 2001).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).

S. A. Akhmanov and R. V. Khohlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1973).

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Figures (8)

Fig. 1.
Fig. 1.

A schematic illustration of the system under study. The Bragg stack consists of alternating linear (dielectric function εA and thickness dA ) and nonlinear (dielectric function εB , thickness dB , and second-order susceptibility χ (2)) optical films. The fundamental harmonic wave with frequency ω is incident from the left of the Bragg stack and the second-harmonic wave with frequency 2ω is generated.

Fig. 2.
Fig. 2.

The photonic band structure of the Bragg stack under consideration. The Upper panel shows the real part of Bloch wave number kz , while the lower panel shows the imaginary part of kz . See text for the parameters used in the numerical calculation. The QPM points for the SHG are indicated by filled circles and dashed lines in the upper panel.

Fig. 3.
Fig. 3.

The transmittance of the fundamental harmonic (FH) wave (black line) and the SHG intensity (red line) in the Bragg stack having N = 21. The horizontal axis is the normalized frequency of the FH wave.

Fig. 4.
Fig. 4.

The SHG intensity in the Bragg stack having N = 101. The horizontal axis is again the normalized frequency of the FH wave.

Fig. 5.
Fig. 5.

(a) Peak frequency in the SHG spectrum versus N for the dominant SHG peak around ω = 0.524 in units of 2πc/a. The solid line is half of the band edge frequency ω = 1.0484. (b) A log-log plot of the peak SHG intensity versus N for the dominant SHG peak. The solid line represents a N 4 curve for comparison. (c) Peak frequencies in the SHG spectrum relevant to the QPM around ω = 0.9447 versus N. The solid line is the phase-marched frequency obtained from the photonic band structure given in Fig. 2. Open circle stands for the optimized N value which gives the maximum SHG intensity along the curve of each peak. (d) A log-log plot of peak SHG intensities relevant to the QPM versus N. The solid line represents a N 2 curve for comparison.

Fig. 6.
Fig. 6.

The transmittance of the FH wave (black line) and the SHG intensity (red line) of the defective Bragg stack having N = 23 and dD = 1.2a. The horizontal axis is again the normalized frequency of the FH wave.

Fig. 7.
Fig. 7.

The peak SHG intensities versus N for ω = ω D1 (black line with circles) and 2ω = ω D3 (red line with circles). The scaling functions derived in this paper are represented by green and blue lines, respectively.

Fig. 8.
Fig. 8.

(a) The electric field intensity of the FH wave at ω = ω D1 and N = 39. The intensity of the incident FH wave is normalized to be 1. The parallel vertical lines stand for the boundaries of the optical films. (b) The local density of states (LDOS) of photon at 2ω = ω D3 and N = 39. The LDOS is normalized by that in vacuum.

Tables (1)

Tables Icon

Table 1. The scalar form of the second-order susceptibility for all possible combination of

Equations (119)

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E x t = E L x t + E NL x t ,
E L x t = E x ω e iωt + c . c . ,
E NL x t = E x ; 2 ω e i 2 ωt + c . c . ,
P i ( x ; 2 ω ) = j , k χ ijk ( 2 ) ( x ) E j ( x ; ω ) E k ( x ; ω ) .
E ( x ; 2 ω ) = ( 2 ω c ) 2 d 3 x G ̂ ( x , x ; 2 ω ) P ( x ; 2 ω ) ,
× × G ̂ ( x , x ; ω ) q 2 ε ( x ) G ̂ ( x , x ; ω ) = îδ ( x x ) .
E ( i ) ( z ; ω ) = a ( i ) e ik i ( z z i ) + b ( i ) e ik i ( z z i ) ,
k i = { k A ( for odd i ) k B ( otherwise ) ,
k α = q α 2 k 2 , q α = ω c ε α ( α = A , B )
E ( z ; ω ) = { e ikz + r N e ikz for z < 0 t N e ik ( z D ) for z < D .
t N = det T N ( T N ) 22 ,
r N = ( T N ) 21 ( T N ) 22 ,
T N = T u T N 1 2 T l ,
a ( i + 2 ) b ( i + 2 ) = T a ( i ) b ( i ) ,
a ( i ) b ( i ) = T i 1 2 T l 1 r N .
a ( i + 1 ) b ( i + 1 ) = T AB a ( i ) b ( i ) .
T = U Λ U 1 , Λ = λ 1 0 0 λ 2 .
s = 1 k ( k y , k x , 0 ) ,
p i ± = 1 q i k ( ± k x k i , ± k y k i , k 2 ) .
E dip ( x ) = d 2 k ( 2 π ) 2 e i k · ( x x d ) E dip ( z ) ,
E dip ( x ) = ε ± e ik j z z d z ̂ p 0 z δ ( z z d ) ,
ε ± = ( ε ± ) s s + ( ε ± ) p p j ± ,
( ε ± ) s = iq 2 2 k j s · p 0 , ( ε ± ) p = iq 2 2 k j p j ± · p 0 ,
E ( z ) = { t + ( j ) e ik j ( z D ) for z > D t ( j ) e ik j z for z < 0 ,
E ind ( z ) = c + ( j ) e ik j ( z z j ) + c ( j ) e ik j ( z z j ) ,
t + ( j ) = t + + ( j ) ε + e ik j z d + t + ( j ) ε e ik j z d ,
t ( j ) = t + ( j ) ε + e ik j z d + t ( j ) ε e ik j z d ,
c + ( j ) = c + + ( j ) ε + e ik j z d + c + ( j ) ε e ik j z d ,
c ( j ) = c + ( j ) ε + e ik j z d + c ( j ) ε e ik j z d ,
t + + ( j ) = det ( T u T N j 1 2 T BA ) ( T u T N 1 2 T l ) 22 ( T AB T j 2 2 T l ) 22 ,
t + ( j ) = det ( T u T N j 1 2 T BA ) ( T u T N 1 2 T l ) 22 ( T AB T j 2 2 T l ) 12 ,
t + ( j ) = 1 ( T u T N 1 2 T l ) 22 ( T u T N j 1 2 T BA ) 21 ,
t ( j ) = 1 ( T u T N 1 2 T l ) 22 ( T u T N j 1 2 T BA ) 22 ,
c + + ( j ) = ( T AB T j 2 2 T l ) 12 ( T u T N 1 2 T l ) 22 ( T u T N j 1 2 T BA ) 21 ,
c + ( j ) = ( T AB T j 2 2 T l ) 12 ( T u T N 1 2 T l ) 22 ( T u T N j 1 2 T BA ) 22 ,
c + ( j ) = ( T u T N j 1 2 T BA ) 21 ( T u T N 1 2 T l ) 22 ( T AB T j 2 2 T l ) 22 ,
c ( j ) = ( T u T N j 1 2 T BA ) 21 ( T u T N 1 2 T l ) 22 ( T AB T j 2 2 T l ) 12 .
G ̂ ( x , x ; ω ) = d k ( 2 π ) 2 e i k · ( x x ) G ̂ ( z , z ; ω ) ,
G ̂ ( z , z ; ω ) = e i k ( z D ) j 2 k j ( ( t + + ( j ) ) s s s e ik j ( z z j ) + ( t + ( j ) ) s s s e ik j ( z z j )
+ ( t + + ( j ) ) p p + p j + e ik j ( z z j ) + ( t + ( j ) ) p p + p j e ik j ( z z j ) ) ,
E ( z ; 2 ω ) = { e 2 ik ( z D ) T + for z > D e 2 ikz T for z < 0 ,
T ± = T 1 ± + Z 1 + T 2 ± + Z 2 + T 3 ± + Z 3 + T 1 ± Z 3 + T 2 ± Z 2 + T 3 ± Z 1 ,
T 1 αβ = j : even t αβ ( j ) ε β χ ( 2 ) ( a ( j ) ) 2 ,
T 2 αβ = j : even t αβ ( j ) ε β χ ( 2 ) ( a ( j ) b ( j ) + b ( j ) a ( j ) ) ,
T 3 αβ = j : even t αβ ( j ) ε β χ ( 2 ) ( b ( j ) ) 2 ( α , β = ± ) ,
Z 1 = d B 2 d B 2 dz e i ( k B ( 2 ω ) 2 k B ( ω ) ) z ,
Z 2 = d B 2 d B 2 dz e ik B ( 2 ω ) z ,
Z 3 = d B 2 d B 2 dz e i ( k B ( 2 ω ) + 2 k B ( ω ) ) z .
I ab = j : even ( λ b , ω 2 λ a , 2 ω ) j 2 2 ( a , b = 1,2 ) ,
J a = j : even ( λ 1 , ω λ 2 , ω λ a , 2 ω ) j 2 2 ( a = 1,2 ) .
λ 1 = λ 2 * = e ik z ( ω ) a ,
λ 1 = 1 λ 2 = e Im ( k z ( ω ) ) a .
I 11 = I 22 * = j = 2,4 , , N 1 e i ( 2 k z ( ω ) k z ( 2 ω ) ) a j 2 2 ,
I 12 = I 21 * = j = 2,4 , , N 1 e i ( 2 k z ( ω ) + k z ( 2 ω ) ) a j 2 2 .
2 k z ( ω ) k z ( 2 ω ) = 2 π a Z .
2 k z ( ω ) + k z ( 2 ω ) = 2 π a Z .
J 1 = J 2 * = j = 2,4 , N 1 e ik z ( 2 ω ) a j 2 2 ,
k z ( 2 ω ) = 2 π a Z .
ρ N ( ω ) d arg ( t N ) .
e ik z ( ω ) a ( N 1 ) = ( T u U ) 22 ( U 1 T l ) 21 ( T u U ) 21 ( U 1 T l ) 11 ( e ( ω ) ) .
ω = ω c + αk z ( ω ) 2 .
ω FP = ω c + α ( ϕ ( ω c ) + 2 πM a ( N 1 ) ) 2 .
det U = iT + 1 ( Re T ) 2 Im T T 2 1 ,
Re T = ± 1 + ( ω ω c ) d Re T ω c + .
a ( i ) , b ( i ) U 1 ( ω FP ω c ) γ N .
( T u T N 1 2 T l ) 22 = e ik z ( 2 ω ) a N 1 2 det T l ( T u U ) 21 det U ( U 1 T l ) 21 .
k q ( t + ( j ) 2 + t ( j ) 2 ) = q Im [ p 0 * · ( E dip ( z d ) + E ind ( z d ) ) ]
= q 3 Im [ p 0 * G ̂ ( z d , z d ) p 0 ] ,
a ( D ) b ( D ) = T AD T N 3 4 T l 1 r N ,
T N = T u T N 3 4 T DA T AD T N 3 4 T l .
a ( D ) ( T AD U ) 12 e [ Im k z ( ω d ) ] a N 3 4 ( U 1 T l ) 21 ,
b ( D ) ( T AD U ) 22 e [ Im k z ( ω d ) ] a N 3 4 ( U 1 T l ) 21 ,
t + + ( D ) = det ( T u T N 3 4 T DA ) ( T N ) 22 ( T AD T N 3 4 T l ) 22 ,
t + ( D ) = det ( T u T N 3 4 T BA ) ( T N ) 22 ( T AD T N 3 4 T l ) 12 ,
t + ( D ) = 1 ( T N ) 22 ( T u T N 3 4 T DA ) 21 ,
t ( D ) = 1 ( T N ) 22 ( T u T N 3 4 T DA ) 22
( T N ) 22 det ( U 1 T l ) ( T u U ) 22 ( U 1 T DA T AD U ) 21 ( U 1 T l ) 12 ( U 1 T l ) 11 ( U 1 T l ) 21 .
ρ ( z ; 2 ω ) = 4 ω π c 2 ImTr G ̂ ( z , z ; 2 ω ) ,
ε B = n o 2 ( λ ) = 4.9048 + 0.11768 λ 2 0.04750 0.027169 λ 2 ,
χ zzz ( 2 ) = 34.4 [ pm V ] ,
χ zxx ( 2 ) = χ zyy ( 2 ) = 5.95 [ pm V ] ,
χ xxz ( 2 ) = χ yyz ( 2 ) = 5.95 [ pm V ] ,
χ yyy ( 2 ) = χ yxx ( 2 ) = χ xxy ( 2 ) = 3.07 [ pm V ] .
E ( x ; 3 ω ) = d 3 x ' G ̂ ( x , x ; 3 ω ) P ( x ; 3 ω ) ,
P i ( x ; 3 ω ) = j , k , l χ ijkl ( 3 ) E j ( x ; ω ) E k ( x ; ω ) E l ( x ; ω ) .
k z ( 3 ω ) ± 3 k z ( ω ) = 2 π a Z ,
k z ( 3 ω ) ± k z ( ω ) = 2 π a Z .
T 1 + + = C χ ˜ 1 + + ( 2 ) ( G 1 F 2 + G 2 F 1 ) ,
T 2 + + = 2 C χ ˜ 2 + + ( 2 ) ( G 1 F 4 + G 2 F 3 ) ,
T 3 + + = C χ ˜ 3 + + ( 2 ) ( G 1 F 6 + G 2 F 5 ) ,
T 1 + = C χ ˜ 1 + ( 2 ) ( G 3 F 2 + G 4 F 1 ) ,
T 2 + = 2 C χ ˜ 2 + ( 2 ) ( G 3 F 4 + G 4 F 3 ) ,
T 3 + = C χ ˜ 3 + ( 2 ) ( G 3 F 6 + G 4 F 5 ) ,
T 1 + = C χ ˜ 1 + ( 2 ) ( G 5 F 1 + G 6 F 2 ) ,
T 2 + = 2 C χ ˜ 2 + ( 2 ) ( G 5 F 3 + G 6 F 4 ) ,
T 3 + = C χ ˜ 3 + ( 2 ) ( G 5 F 5 + G 6 F 6 ) ,
T 1 = C χ ˜ 1 ( 2 ) ( G 7 F 1 + G 8 F 2 ) ,
T 2 = 2 C χ ˜ 2 ( 2 ) ( G 7 F 3 + G 8 F 4 ) ,
T 3 = C χ ˜ 3 ( 2 ) ( G 7 F 5 + G 8 F 6 ) .
C = i ( 2 ω c ) 2 2 k B ( 2 ω ) det ( T u , 2 ω T 2 ω N 3 2 T BA , 2 ω ) ( T u , 2 ω T 2 ω N 1 2 T l , 2 ω ) 22 ,
C = i ( 2 ω c ) 2 2 k B ( 2 ω ) 1 ( T u , 2 ω T 2 ω N 1 2 T l , 2 ω ) 22 ,
G 1 = ( T AB , 2 ω U 2 ω ) 21 ( U 2 ω 1 T l , 2 ω ) 12 ,
G 2 = ( T AB , 2 ω U 2 ω ) 22 ( U 2 ω 1 T l , 2 ω ) 22 ,
G 3 = ( T AB , 2 ω U 2 ω ) 11 ( U 2 ω 1 T l , 2 ω ) 12 ,
G 4 = ( T AB , 2 ω U 2 ω ) 12 ( U 2 ω 1 T l , 2 ω ) 22 ,
G 5 = ( T u , 2 ω U 2 ω ) 21 λ 1,2 ω N 3 2 ( U 2 ω 1 T BA , 2 ω ) 11 ,
G 6 = ( T u , 2 ω U 2 ω ) 22 λ 2,2 ω N 3 2 ( U 2 ω 1 T BA , 2 ω ) 21 ,
G 5 = ( T u , 2 ω U 2 ω ) 21 λ 1,2 ω N 3 2 ( U 2 ω 1 T BA , 2 ω ) 12 ,
G 6 = ( T u , 2 ω U 2 ω ) 22 λ 2,2 ω N 3 2 ( U 2 ω 1 T BA , 2 ω ) 22 ,
F 1 = f 1 2 I 11 + 2 f 1 f 2 J 1 + f 2 2 I 12 ,
F 2 = f 1 2 I 21 + 2 f 1 f 2 J 2 + f 2 2 I 22 ,
F 3 = f 1 f 3 I 11 + ( f 1 f 4 + f 2 f 3 ) J 1 + f 2 f 4 I 12 ,
F 4 = f 1 f 3 I 21 + ( f 1 f 4 + f 2 f 3 ) J 2 + f 2 f 4 I 22 ,
F 5 = f 3 2 I 11 + 2 f 3 f 4 J 1 + f 4 2 I 12 ,
F 6 = f 3 2 I 21 + 2 f 3 f 4 J 2 + f 4 2 I 22 ,
f 1 = ( T AB , ω U ω ) 11 { ( U ω 1 T l , ω ) 11 + r N ( U ω 1 T l , ω ) 12 } ,
f 2 = ( T AB , ω U ω ) 12 { ( U ω 1 T l , ω ) 21 + r N ( U ω 1 T l , ω ) 22 } ,
f 3 = ( T AB , ω U ω ) 21 { ( U ω 1 T l , ω ) 11 + r N ( U ω 1 T l , ω ) 12 } ,
f 4 = ( T AB , ω U ω ) 22 { ( U ω 1 T l , ω ) 21 + r N ( U ω 1 T l , ω ) 22 } ,

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