Scaling law of enhanced second-harmonic generation in finite Bragg stacks

Tetsuyuki Ochiai; Kazuaki Sakoda

doi:10.1364/OPEX.13.009094

Scaling law of enhanced second-harmonic generation in finite Bragg stacks

Tetsuyuki Ochiai and Kazuaki Sakoda

Optics Express
Vol. 13,
Issue 22,
pp. 9094-9114
(2005)
•https://doi.org/10.1364/OPEX.13.009094

Get Citation
Copy Citation Text
Tetsuyuki Ochiai and Kazuaki Sakoda, "Scaling law of enhanced second-harmonic generation in finite Bragg stacks," Opt. Express 13, 9094-9114 (2005)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (398 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Research Papers
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Multiharmonic generation (190.4160)
- Bragg reflectors (230.1480)
- Multiple scattering (290.4210)

About this Article
History
- Original Manuscript: August 4, 2005
- Revised Manuscript: October 25, 2005
- Manuscript Accepted: October 25, 2005
- Published: October 31, 2005

Accessible

Open Access

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.

Full Article | PDF Article

OSA Recommended Articles

Defective photonic crystals with greatly enhanced second-harmonic generation

B. Shi, Z. M. Jiang, and Xun Wang
Opt. Lett. 26(15) 1194-1196 (2001)

Magnetization-induced second- and third-harmonic generation in magnetophotonic crystals

Oleg A. Aktsipetrov, Tatyana V. Dolgova, Andrey A. Fedyanin, Tatyana V. Murzina, Mitsuteru Inoue, Kazuhiro Nishimura, and Hironaga Uchida
J. Opt. Soc. Am. B 22(1) 176-186 (2005)

Simultaneous second- and third-harmonic generation in one-dimensional photonic crystals

Vladimir V. Konotop and Vladimir Kuzmiak
J. Opt. Soc. Am. B 16(9) 1370-1376 (1999)

References

View by:
Article Order
|
Year
|
Author
|
Publication

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 ).
J. Trull , R. Vilaseca , J. Martorell , and R. Corbalan , “ Second-harmonic generation in local modes of a truncated periodic structure ,” Opt. Lett. 20 , 1746 – 1748 ( 1995 ).
[Crossref] [PubMed]
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]
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]
G. D’Aguanno , M. Centini , C. Sibilia , M. Bertolotti , M. Scalora , M. J. Bloemer , and C. M. Bowden , “ Enhancement of χ (2) cascading processes in one-dimensional photonic bandgap structures ,” Opt. Lett. 24 , 1663 – 1665 ( 1999 ).
[Crossref]
M. Centini , C. Sibilia , M. Scalora , G. D’Aguanno , M. Bertolotti , M. J. Bloemer , C. M. Bowden , and I. Nefe-dov , “ Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions ,” Phys. Rev. E 60 , 4891 – 4898 ( 1999 ).
[Crossref]
C. D. Angelis , F. Gringoli , M. Midrio , D. Modotto , J. S. Aitchison , and G. F. Nalesso , “ Conversion efficiency for second-harmonic generation in photonic crystals ,” J. Opt. Soc. Am. B 18 , 348 – 351 ( 2001 ).
[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]
B. Shi , Z. M. Jiang , and X. Wang , “ Defective photonic crystals with greatly enhanced second-harmonic generation ,” Opt. Lett. 26 , 1194 – 1196 ( 2001 ).
[Crossref]
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]
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]
J. E. Sipe , “ New green-function formalism for surface optics ,” J. Opt. Soc. Am. B 4 , 481 – 489 ( 1987 ).
[Crossref]
M. S. Tomas , “ Green-function for multilayers - light-scattering in planar cavities ,” Phys. Rev. A 51 , 2545 – 2559 ( 1995 ).
[Crossref] [PubMed]
N. Hashizume , M. Ohashi , T. Kondo , and R. Ito , “ Optical harmonic-generation in multilayered structures - a comprehensive analysis ,” J. Opt. Soc. Am. B 12 , 1894 – 1904 ( 1995 ).
[Crossref]
S. A. Akhmanov and R. V. Khohlov , Problems of Nonlinear Optics ( Gordon and Breach, New York , 1973 ).

2004 (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]

2002 (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]

2001 (3)

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]

C. D. Angelis , F. Gringoli , M. Midrio , D. Modotto , J. S. Aitchison , and G. F. Nalesso , “ Conversion efficiency for second-harmonic generation in photonic crystals ,” J. Opt. Soc. Am. B 18 , 348 – 351 ( 2001 ).
[Crossref]

B. Shi , Z. M. Jiang , and X. Wang , “ Defective photonic crystals with greatly enhanced second-harmonic generation ,” Opt. Lett. 26 , 1194 – 1196 ( 2001 ).
[Crossref]

1999 (3)

G. D’Aguanno , M. Centini , C. Sibilia , M. Bertolotti , M. Scalora , M. J. Bloemer , and C. M. Bowden , “ Enhancement of χ (2) cascading processes in one-dimensional photonic bandgap structures ,” Opt. Lett. 24 , 1663 – 1665 ( 1999 ).
[Crossref]

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]

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

1998 (1)

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]

1997 (1)

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]

1995 (3)

M. S. Tomas , “ Green-function for multilayers - light-scattering in planar cavities ,” Phys. Rev. A 51 , 2545 – 2559 ( 1995 ).
[Crossref] [PubMed]

N. Hashizume , M. Ohashi , T. Kondo , and R. Ito , “ Optical harmonic-generation in multilayered structures - a comprehensive analysis ,” J. Opt. Soc. Am. B 12 , 1894 – 1904 ( 1995 ).
[Crossref]

J. Trull , R. Vilaseca , J. Martorell , and R. Corbalan , “ Second-harmonic generation in local modes of a truncated periodic structure ,” Opt. Lett. 20 , 1746 – 1748 ( 1995 ).
[Crossref] [PubMed]

1987 (1)

J. E. Sipe , “ New green-function formalism for surface optics ,” J. Opt. Soc. Am. B 4 , 481 – 489 ( 1987 ).
[Crossref]

Abram, I.

Aitchison, J. S.

Akhmanov, S. A.

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

Angelis, C. D.

Bertolotti, M.

Bloemer, M. J.

Bowden, C. M.

Centini, M.

Cheng, C.

Cole, J. D.

Corbalan, R.

D’Aguanno, G.

Dowling, J. P.

Dumeige, Y.

Gringoli, F.

Hashizume, N.

Haus, J. W.

Hirao, K.

Ito, R.

Jiang, Z. M.

B. Shi , Z. M. Jiang , and X. Wang , “ Defective photonic crystals with greatly enhanced second-harmonic generation ,” Opt. Lett. 26 , 1194 – 1196 ( 2001 ).
[Crossref]

Kalocsai, A. G.

Ketterson, J. B.

Khohlov, R. V.

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

Kiehne, G. T.

Kondo, T.

Kryukov, A. E.

Levenson, A.

Levenson, J. A.

Li, R.

Manka, A. S.

Martorell, J.

Meriadec, C.

Midrio, M.

Modotto, D.

Monnier, P.

Nalesso, G. F.

Nefe-dov, I.

Ohashi, M.

Qiu, J. R.

Ren, F. F.

Sagnes, I.

Sakoda, K.

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

Sauvage, S.

Scalora, M.

Shi, B.

B. Shi , Z. M. Jiang , and X. Wang , “ Defective photonic crystals with greatly enhanced second-harmonic generation ,” Opt. Lett. 26 , 1194 – 1196 ( 2001 ).
[Crossref]

Si, J. H.

Sibilia, C.

Sipe, J. E.

J. E. Sipe , “ New green-function formalism for surface optics ,” J. Opt. Soc. Am. B 4 , 481 – 489 ( 1987 ).
[Crossref]

Theimer, J.

Tomas, M. S.

M. S. Tomas , “ Green-function for multilayers - light-scattering in planar cavities ,” Phys. Rev. A 51 , 2545 – 2559 ( 1995 ).
[Crossref] [PubMed]

Trull, J.

Vidakovic, P.

Vilaseca, R.

Viswanathan, R.

Wang, H. T.

Wang, X.

B. Shi , Z. M. Jiang , and X. Wang , “ Defective photonic crystals with greatly enhanced second-harmonic generation ,” Opt. Lett. 26 , 1194 – 1196 ( 2001 ).
[Crossref]

Appl. Phys. Lett. (2)

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

J. E. Sipe , “ New green-function formalism for surface optics ,” J. Opt. Soc. Am. B 4 , 481 – 489 ( 1987 ).
[Crossref]

Opt. Lett. (3)

B. Shi , Z. M. Jiang , and X. Wang , “ Defective photonic crystals with greatly enhanced second-harmonic generation ,” Opt. Lett. 26 , 1194 – 1196 ( 2001 ).
[Crossref]

Phys. Rev. A (3)

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)

Phys. Rev. E (1)

Phys. Rev. Lett. (1)

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 ).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1. A schematic illustration of the system under study. The Bragg stack consists of alternating linear (dielectric function ε_A and thickness d_A ) and nonlinear (dielectric function ε_B , thickness d_B , 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.

Download Full Size | PPT Slide | PDF

Fig. 2. The photonic band structure of the Bragg stack under consideration. The Upper panel shows the real part of Bloch wave number k_z , while the lower panel shows the imaginary part of k_z . 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.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

Fig. 4. The SHG intensity in the Bragg stack having N = 101. The horizontal axis is again the normalized frequency of the FH wave.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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 d_D = 1.2a. The horizontal axis is again the normalized frequency of the FH wave.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

Tables (1)

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

View Table

Equations (119)

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

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) = \sum_{j, k} χ_{ijk}^{(2)} (x) E_{j} (x; ω) E_{k} (x; ω) .

E (x; 2 ω) = {(\frac{2 ω}{c})}^{2} \int d^{3} x' \hat{G} (x, x'; 2 ω) P (x'; 2 ω),

\nabla \times \nabla \times \hat{G} (x, x'; ω) - q^{2} ε (x) \hat{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} = {\begin{matrix} k_{A} (for odd i) \\ k_{B} (otherwise) \end{matrix},

k_{α} = \sqrt{q_{α}^{2} - k_{∥}^{2}}, q_{α} = \frac{ω}{c} \sqrt{ε_{α}} (α = A, B)

E (z; ω) = {\begin{matrix} e^{ikz} + r_{N} e^{- ikz} for z < 0 \\ t_{N} e^{ik (z - D)} for z < D \end{matrix} .

t_{N} = \frac{\det T_{N}}{{(T_{N})}_{22}},

r_{N} = \frac{{- (T_{N})}_{21}}{{(T_{N})}_{22}},

T_{N} = T_{u} T^{\frac{N - 1}{2}} T_{l},

(\begin{matrix} a^{(i + 2)} \\ b^{(i + 2)} \end{matrix}) = T (\begin{matrix} a^{(i)} \\ b^{(i)} \end{matrix}),

(\begin{matrix} a^{(i)} \\ b^{(i)} \end{matrix}) = T^{\frac{i - 1}{2}} T_{l} (\begin{matrix} 1 \\ r_{N} \end{matrix}) .

(\begin{matrix} a^{(i + 1)} \\ b^{(i + 1)} \end{matrix}) = T_{AB} (\begin{matrix} a^{(i)} \\ b^{(i)} \end{matrix}) .

T = {U Λ U}^{- 1}, Λ = (\begin{matrix} λ_{1} & 0 \\ 0 & λ_{2} \end{matrix}) .

s = \frac{1}{∣ k_{∥} ∣} (k_{y}, - k_{x}, 0),

p_{i}^{\pm} = \frac{1}{q_{i} ∣ k_{∥} ∣} ({\pm k}_{x} k_{i}, \pm k_{y} k_{i}, - k_{∥}^{2}) .

E^{dip} (x) = \int \frac{d^{2} k_{∥}}{{(2 π)}^{2}} e^{{i k}_{∥} \cdot {(x - x_{d})}_{∥}} E^{dip} (z),

E^{dip} (x) = ε_{\pm} e^{{ik}_{j} ∣ z - z_{d} ∣} - \hat{z} p_{0 z} δ (z - z_{d}),

ε_{\pm} = {(ε_{\pm})}_{s} s + {(ε_{\pm})}_{p} p_{j}^{\pm},

{(ε_{\pm})}_{s} = \frac{{iq}^{2}}{{2 k}_{j}} s \cdot p_{0}, {(ε_{\pm})}_{p} = \frac{{iq}^{2}}{{2 k}_{j}} p_{j}^{\pm} \cdot p_{0},

E (z) = {\begin{matrix} t_{+}^{(j)} e^{{ik}_{j} (z - D)} for z > D \\ t_{-}^{(j)} e^{- {ik}_{j} z} for z < 0 \end{matrix},

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)} = \frac{\det (T_{u} T^{\frac{N - j - 1}{2}} T_{BA})}{{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22}} {(T_{AB} T^{\frac{j - 2}{2}} T_{l})}_{22},

t_{+ -}^{(j)} = \frac{\det (T_{u} T^{\frac{N - j - 1}{2}} T_{BA})}{{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22}} {(T_{AB} T^{\frac{j - 2}{2}} T_{l})}_{12},

t_{- +}^{(j)} = - \frac{1}{{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22}} {(T_{u} T^{\frac{N - j - 1}{2}} T_{BA})}_{21},

t_{- -}^{(j)} = - \frac{1}{{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22}} {(T_{u} T^{\frac{N - j - 1}{2}} T_{BA})}_{22},

c_{+ +}^{(j)} = \frac{{(T_{AB} T^{\frac{j - 2}{2}} T_{l})}_{12}}{{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22}} {(T_{u} T^{\frac{N - j - 1}{2}} T_{BA})}_{21},

c_{+ -}^{(j)} = \frac{{(T_{AB} T^{\frac{j - 2}{2}} T_{l})}_{12}}{{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22}} {(T_{u} T^{\frac{N - j - 1}{2}} T_{BA})}_{22},

c_{- +}^{(j)} = \frac{{(T_{u} T^{\frac{N - j - 1}{2}} T_{BA})}_{21}}{{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22}} {(T_{AB} T^{\frac{j - 2}{2}} T_{l})}_{22},

c_{- -}^{(j)} = - \frac{{(T_{u} T^{\frac{N - j - 1}{2}} T_{BA})}_{21}}{{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22}} {(T_{AB} T^{\frac{j - 2}{2}} T_{l})}_{12} .

\hat{G} (x, x'; ω) = \int \frac{d k_{∥}}{{(2 π)}^{2}} e^{i k_{∥} \cdot (x_{∥} - x_{∥}^{'})} \hat{G} (z, z'; ω),

\hat{G} (z, z'; ω) = e^{i k (z - D)} \frac{j}{{2 k}_{j}} ({(t_{+ +}^{(j)})}_{s} s \otimes s e^{- {ik}_{j} (z' - z_{j})} + {(t_{+ -}^{(j)})}_{s} s \otimes s e^{{ik}_{j} (z' - z_{j})}

+ {(t_{+ +}^{(j)})}_{p} p^{+} \otimes p_{j}^{+} e^{- {ik}_{j} (z' - z_{j})} + {(t_{+ -}^{(j)})}_{p} p^{+} \otimes p_{j}^{-} e^{- {ik}_{j} (z' - z_{j})}),

E (z; 2 ω) = {\begin{matrix} e^{2 ik (z - D)} T_{+} for z > D \\ e^{- 2 ikz} T_{-} for z < 0 \end{matrix},

T_{\pm} = T_{1}^{\pm +} Z_{1} + T_{2}^{\pm +} Z_{2} + T_{3}^{\pm +} Z_{3} + T_{1}^{\pm -} Z_{3} + T_{2}^{\pm -} Z_{2} + T_{3}^{\pm -} Z_{1},

T_{1}^{αβ} = \sum_{j : even} t_{αβ}^{(j)} ε_{β} χ^{(2)} {(a^{(j)})}^{2},

T_{2}^{αβ} = \sum_{j : even} t_{αβ}^{(j)} ε_{β} χ^{(2)} (a^{(j)} b^{(j)} + b^{(j)} a^{(j)}),

T_{3}^{αβ} = \sum_{j : even} t_{αβ}^{(j)} ε_{β} χ^{(2)} {(b^{(j)})}^{2} (α, β = \pm),

Z_{1} = \int_{\frac{- d_{B}}{2}}^{\frac{d_{B}}{2}} dz e^{{i (k}_{B} (2 ω) - {2 k}_{B} (ω)) z},

Z_{2} = \int_{\frac{- d_{B}}{2}}^{\frac{d_{B}}{2}} dz e^{{ik}_{B} (2 ω) z},

Z_{3} = \int_{\frac{- d_{B}}{2}}^{\frac{d_{B}}{2}} dz e^{{i (k}_{B} (2 ω) + {2 k}_{B} (ω)) z} .

I_{ab} = \sum_{j : even} {(\frac{λ_{b, ω}^{2}}{λ_{a, 2 ω}})}^{\frac{j - 2}{2}} (a, b = 1,2),

J_{a} = \sum_{j : even} {(\frac{λ_{1, ω} λ_{2, ω}}{λ_{a, 2 ω}})}^{\frac{j - 2}{2}} (a = 1,2) .

λ_{1} = λ_{2}^{*} = e^{{ik}_{z} (ω) a},

λ_{1} = \frac{1}{λ_{2}} = e^{- Im (k_{z} (ω)) a} .

I_{11} = I_{22}^{*} = \sum_{j = 2,4, \dots, N - 1} e^{i ({2 k}_{z} (ω) - k_{z} (2 ω)) a \frac{j - 2}{2}},

I_{12} = I_{21}^{*} = \sum_{j = 2,4, \dots, N - 1} e^{- i ({2 k}_{z} (ω) + k_{z} (2 ω)) a \frac{j - 2}{2}} .

{2 k}_{z} (ω) - k_{z} (2 ω) = \frac{2 π}{a} Z .

{2 k}_{z} (ω) + k_{z} (2 ω) = \frac{2 π}{a} Z .

J_{1} = J_{2}^{*} = \sum_{j = 2,4, \dots N - 1} e^{- {ik}_{z} (2 ω) a \frac{j - 2}{2}},

k_{z} (2 ω) = \frac{2 π}{a} Z .

ρ_{N} (ω) \propto \frac{d \arg (t_{N})}{dω} .

e^{{ik}_{z} (ω) a (N - 1)} = \frac{{(T_{u} U)}_{22} {(U^{- 1} T_{l})}_{21}}{{(T_{u} U)}_{21} {(U^{- 1} T_{l})}_{11}} (\equiv e^{iϕ (ω)}) .

ω = ω_{c} + {αk}_{z} {(ω)}^{2} .

ω_{FP} = ω_{c} + α {(\frac{ϕ (ω_{c}) + 2 πM}{a (N - 1)})}^{2} .

\det U = \frac{- {iT}_{+ -} \sqrt{1 - {(Re T_{- -})}^{2}}}{∣ Im T_{- -} ∣ \sqrt{{∣ T_{- -} ∣}^{2} - 1}},

Re T_{- -} = \pm 1 + (ω - ω_{c}) \frac{d Re T_{- -}}{dω} ∣ ω_{c} + \dots .

a^{(i)}, b^{(i)} \propto U^{- 1} \propto {(ω_{FP} - ω_{c})}^{- γ} \propto N .

{(T_{u} T^{\frac{N - 1}{2}} T_{l})}_{22} = - e^{{ik}_{z} (2 ω) a \frac{N - 1}{2}} \frac{\det T_{l} {(T_{u} U)}_{21}}{\det U {(U^{- 1} T_{l})}_{21}} .

\frac{k}{q} ({∣ t_{+}^{(j)} ∣}^{2} + {∣ t_{-}^{(j)} ∣}^{2}) = q Im [p_{0}^{*} \cdot (E^{dip} (z_{d}) + E^{ind} (z_{d}))]

= q^{3} Im [p_{0}^{*} \hat{G} (z_{d}, z_{d}) p_{0}],

(\begin{matrix} a^{(D)} \\ b^{(D)} \end{matrix}) = T_{AD} T^{\frac{N - 3}{4}} T_{l} (\begin{matrix} 1 \\ r_{N} \end{matrix}),

T_{N} = T_{u} T^{\frac{N - 3}{4}} T_{DA} T_{AD} T^{\frac{N - 3}{4}} T_{l} .

a^{(D)} ≃ {(T_{AD} U)}_{12} e^{[Im k_{z} (ω_{d})] a \frac{N - 3}{4}} {(U^{- 1} T_{l})}_{21},

b^{(D)} ≃ {(T_{AD} U)}_{22} e^{[Im k_{z} (ω_{d})] a \frac{N - 3}{4}} {(U^{- 1} T_{l})}_{21},

t_{+ +}^{(D)} = \frac{\det (T_{u} T^{\frac{N - 3}{4}} T_{DA})}{{(T_{N})}_{22}} {(T_{AD} T^{\frac{N - 3}{4}} T_{l})}_{22},

t_{+ -}^{(D)} = \frac{\det (T_{u} T^{\frac{N - 3}{4}} T_{BA})}{{(T_{N})}_{22}} {(T_{AD} T^{\frac{N - 3}{4}} T_{l})}_{12},

t_{- +}^{(D)} = - \frac{1}{{(T_{N})}_{22}} {(T_{u} T^{\frac{N - 3}{4}} T_{DA})}_{21},

t_{- -}^{(D)} = \frac{1}{{(T_{N})}_{22}} {(T_{u} T^{\frac{N - 3}{4}} T_{DA})}_{22}

{(T_{N})}_{22} ≃ - \frac{\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) = \frac{4 ω}{π c^{2}} ImTr \hat{G} (z, z; 2 ω),

ε_{B} = n_{o}^{2} (λ) = 4.9048 + \frac{0.11768}{λ^{2} - 0.04750} - {0.027169 λ}^{2},

χ_{zzz}^{(2)} = 34.4 [\frac{pm}{V}],

χ_{zxx}^{(2)} = χ_{zyy}^{(2)} = 5.95 [\frac{pm}{V}],

χ_{xxz}^{(2)} = χ_{yyz}^{(2)} = 5.95 [\frac{pm}{V}],

χ_{yyy}^{(2)} = {- χ}_{yxx}^{(2)} = - χ_{xxy}^{(2)} = 3.07 [\frac{pm}{V}] .

E (x; 3 ω) = \int d^{3} x' \hat{G} (x, x'; 3 ω) P (x'; 3 ω),

P_{i} (x; 3 ω) = \sum_{j, k, l} χ_{ijkl}^{(3)} E_{j} (x; ω) E_{k} (x; ω) E_{l} (x; ω) .

k_{z} (3 ω) \pm {3 k}_{z} (ω) = \frac{2 π}{a} Z,

k_{z} (3 ω) \pm k_{z} (ω) = \frac{2 π}{a} Z .

T_{1}^{+ +} = {C \tilde{χ}}_{1 + +}^{(2)} (G_{1} F_{2} + G_{2} F_{1}),

T_{2}^{+ +} = {2 C \tilde{χ}}_{2 + +}^{(2)} (G_{1} F_{4} + G_{2} F_{3}),

T_{3}^{+ +} = {C \tilde{χ}}_{3 + +}^{(2)} (G_{1} F_{6} + G_{2} F_{5}),

T_{1}^{+ -} = C {\tilde{χ}}_{1 + -}^{(2)} (G_{3} F_{2} + G_{4} F_{1}),

T_{2}^{+ -} = 2 C {\tilde{χ}}_{2 + -}^{(2)} (G_{3} F_{4} + G_{4} F_{3}),

T_{3}^{+ -} = C {\tilde{χ}}_{3 + -}^{(2)} (G_{3} F_{6} + G_{4} F_{5}),

T_{1}^{- +} = - C' {\tilde{χ}}_{1 - +}^{(2)} (G_{5} F_{1} + G_{6} F_{2}),

T_{2}^{- +} = - 2 C' {\tilde{χ}}_{2 - +}^{(2)} (G_{5} F_{3} + G_{6} F_{4}),

T_{3}^{- +} = - C' {\tilde{χ}}_{3 - +}^{(2)} (G_{5} F_{5} + G_{6} F_{6}),

T_{1}^{- -} = C' {\tilde{χ}}_{1 - -}^{(2)} (G_{7} F_{1} + G_{8} F_{2}),

T_{2}^{- -} = 2 C' {\tilde{χ}}_{2 - -}^{(2)} (G_{7} F_{3} + G_{8} F_{4}),

T_{3}^{- -} = C' {\tilde{χ}}_{3 - -}^{(2)} (G_{7} F_{5} + G_{8} F_{6}) .

C = \frac{i {(\frac{2 ω}{c})}^{2}}{2 k_{B} (2 ω)} \frac{\det (T_{u, 2 ω} T_{2 ω}^{\frac{N - 3}{2}} T_{BA, 2 ω})}{{(T_{u, 2 ω} T_{2 ω}^{\frac{N - 1}{2}} T_{l, 2 ω})}_{22}},

C' = \frac{i {(\frac{2 ω}{c})}^{2}}{2 k_{B} (2 ω)} \frac{1}{{(T_{u, 2 ω} T_{2 ω}^{\frac{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}^{\frac{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}^{\frac{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}^{\frac{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}^{\frac{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}},

	S(FH)-S(SH)a	S(FH)-P(SH)	P(FH)-S(SH)
${\tilde{χ}}_{1 \pm +}^{(2)}$	$χ_{ijk}^{(2)} s_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {(p_{2 ω}^{+})}_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {s_{i} (p_{2 ω}^{+})}_{j} {(p_{ω}^{+})}_{k}$
${\tilde{χ}}_{2 \pm +}^{(2)}$	$χ_{ijk}^{(2)} s_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {(p_{2 ω}^{+})}_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} s_{i} \frac{1}{2} {((p_{2 ω}^{+})}_{j} {(p_{ω}^{-})}_{k} + {(p_{ω}^{-})}_{j} {(p_{ω}^{+})}_{k})$
${\tilde{χ}}_{3 \pm +}^{(2)}$	$χ_{ijk}^{(2)} s_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {(p_{2 ω}^{+})}_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {s_{i} (p_{2 ω}^{-})}_{j} {(p_{ω}^{-})}_{k}$
${\tilde{χ}}_{1 \pm -}^{(2)}$	$χ_{ijk}^{(2)} s_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {(p_{2 ω}^{-})}_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {s_{i} (p_{2 ω}^{+})}_{j} {(p_{ω}^{+})}_{k}$
${\tilde{χ}}_{2 \pm -}^{(2)}$	$χ_{ijk}^{(2)} s_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {(p_{2 ω}^{-})}_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} s_{i} \frac{1}{2} {((p_{2 ω}^{-})}_{j} {(p_{ω}^{-})}_{k} + {(p_{ω}^{-})}_{j} {(p_{ω}^{-})}_{k})$
${\tilde{χ}}_{3 \pm -}^{(2)}$	$χ_{ijk}^{(2)} s_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {(p_{2 ω}^{-})}_{i} s_{j} s_{k}$	$χ_{ijk}^{(2)} {s_{i} (p_{2 ω}^{-})}_{j} {(p_{ω}^{-})}_{k}$

Abstract

References

2004 (1)

2002 (1)

2001 (3)

1999 (3)

1998 (1)

1997 (1)

1995 (3)

1987 (1)

Abram, I.

Aitchison, J. S.

Akhmanov, S. A.

Angelis, C. D.

Bertolotti, M.

Bloemer, M. J.

Bowden, C. M.

Centini, M.

Cheng, C.

Cole, J. D.

Corbalan, R.

D’Aguanno, G.

Dowling, J. P.

Dumeige, Y.

Gringoli, F.

Hashizume, N.

Haus, J. W.

Hirao, K.

Ito, R.

Jiang, Z. M.

Kalocsai, A. G.

Ketterson, J. B.

Khohlov, R. V.

Kiehne, G. T.

Kondo, T.

Kryukov, A. E.

Levenson, A.

Levenson, J. A.

Li, R.

Manka, A. S.

Martorell, J.

Meriadec, C.

Midrio, M.

Modotto, D.

Monnier, P.

Nalesso, G. F.

Nefe-dov, I.

Ohashi, M.

Qiu, J. R.

Ren, F. F.

Sagnes, I.

Sakoda, K.

Sauvage, S.

Scalora, M.

Shi, B.

Si, J. H.

Sibilia, C.

Sipe, J. E.

Theimer, J.

Tomas, M. S.

Trull, J.

Vidakovic, P.

Vilaseca, R.

Viswanathan, R.

Wang, H. T.

Wang, X.

Appl. Phys. Lett. (2)

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

Opt. Lett. (3)

Phys. Rev. A (3)

Phys. Rev. B (1)

Phys. Rev. E (1)

Phys. Rev. Lett. (1)

Other (3)

Cited By

Figures (8)

Tables (1)

Equations (119)

Metrics

Login or Create Account