Abstract

The possibility of second-harmonic generation based on surface dipole and bulk multipole nonlinearities in silica nanowires is investigated numerically. Both circular and microstructured nanowires are considered. Phase matching is provided by propagating the pump field in the fundamental mode, while generating the second harmonic in one of the modes of the LP11 multiplet. This is shown to work in both circular and microstructured nanowires, although only one of the LP11 modes can be phase-matched in the microstructure. The prospect of obtaining large conversion efficiencies in silica-based nanowires is critically discussed, based on simulations of second-harmonic generation in nanowires with a fluctuating phase-matching wavelength. It is concluded that efficient wavelength conversion will require strong improvements in the nanowire uniformity, peak powers well in excess of 10KW, increase of the second-order nonlinearity by an order of magnitude by use of a different base material, or highly polarizable surface coatings.

© 2010 Optical Society of America

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References

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2010

2009

2008

F. Rodriguez, F. X. Wang, and M. Kauranen, “Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass,” Opt. Express 16, 8704–8710 (2008).
[CrossRef] [PubMed]

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[CrossRef]

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[CrossRef]

2007

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[CrossRef]

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[CrossRef]

F. Rodriguez, F. X. Wang, B. Canfield, S. Cattaneo, and M. Kauranen, “Multipolar tensor analysis of second-order nonlinear optical response of surface and bulk of glass,” Opt. Express 15, 8695–8701 (2007).
[CrossRef] [PubMed]

2006

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

L. Shi, X. Chen, H. Liu, Y. Chen, Z. Ye, W. Liao, and Y. Xia, “Fabrication of submicron-diameter silica fibers using electric strip heater,” Opt. Express 14, 5055–5060 (2006).
[CrossRef] [PubMed]

2004

1991

1987

1986

Alt, W.

Anderson, D.

Birks, T.

Brambilla, G.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. Sessions, E. Koukharenko, X. Feng, G. Murugan, J. Wilkinson, and D. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1, 107–161 (2009).
[CrossRef]

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[CrossRef]

Broderick, N. G. R.

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[CrossRef]

Canfield, B.

Cattaneo, S.

Chen, X.

Chen, Y.

Dan, C.

Dominguez Juarez, J.

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[CrossRef]

Dominguez-Juarez, J.

J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

Ebendorff-Heidepriem, H.

Feinberg, J.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[CrossRef]

Feng, X.

Grubsky, V.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[CrossRef]

Heflin, J.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[CrossRef]

Horak, P.

Irsen, S.

Jung, Y.

Karapetyan, K.

Kauranen, M.

Koizumi, F.

Koukharenko, E.

Kozyreff, G.

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[CrossRef]

J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

Leon-Saval, S.

Liao, W.

Liu, H.

Liu, Z.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[CrossRef]

Margulis, W.

Martorell, J.

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[CrossRef]

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[CrossRef]

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

Mason, M.

Meschede, D.

Mizrahi, V.

Monro, T. M.

Murugan, G.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

Osterberg, U.

Petrovich, M.

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[CrossRef]

Pritzkau, D.

Richardson, D.

Rodriguez, F.

Russell, P. S. J.

Sessions, N.

Shi, L.

Sipe, J.

Stolen, R.

Terhune, R.

Tom, H.

Vidal, X.

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

Vukovic, N.

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[CrossRef]

Wadsworth, W.

Wang, A.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[CrossRef]

Wang, F. X.

Warren-Smith, S. C.

Weinberger, D.

Wiedemann, U.

Wilkinson, J.

Xia, Y.

Xu, F.

Xu, Y.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[CrossRef]

Ye, Z.

Adv. Opt. Photon.

Appl. Phys. Lett.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[CrossRef]

IEEE Photon. Technol. Lett.

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[CrossRef]

Phys. Rev. Lett.

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

Other

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

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Figures (8)

Fig. 1
Fig. 1

Schematic of the threefold symmetric microstructure investigated. The fiber core is characterized by the surface curvature radius r c and the bridge width W b . The outer radius of the microstructured region was set to 6 μ m in the calculations.

Fig. 2
Fig. 2

Relation between diameter and phase matched SHG wavelength for a circular silica nanowire for the case where the SHG mode is either TM 01 or HE 21 .

Fig. 3
Fig. 3

Nonlinear coupling parameter ρ versus SHG wavelength for silica nanowires with the phase matched SHG radiation being in either the TM 01 mode (top panel) or the HE 21 mode (bottom panel).

Fig. 4
Fig. 4

SHG wavelength λ SHG versus r c (top panel) and sum of surface nonlinear coefficients versus λ SHG (bottom panel) for microstructured nanowires with different relative bridge widths.

Fig. 5
Fig. 5

Product of Δ λ FWHM as found from Eq. (30) and the fiber length L for a circular nanowire with phase matching to either the TM 01 or HE 21 mode.

Fig. 6
Fig. 6

Typical variation of SHG phase matching wavelength for a particular realization of the random-coefficient Fourier expansion discussed in the text.

Fig. 7
Fig. 7

SHG conversion efficiency versus propagation distance in a 10 cm microstructured nanowire. Results averaged over 200 realizations of the random structure are compared with results from the individual realizations giving either maximal or minimal conversion efficiency at the output end.

Fig. 8
Fig. 8

SHG conversion efficiency versus nanowire length for various levels of fluctuations in the SHG phase matching wavelength. The fiber lengths are 10 cm (top) and 50 cm (bottom). The curves and Δ λ L values are averaged over 200 realizations of the random structure.

Tables (1)

Tables Icon

Table 1 Surface and Bulk χ ( 2 ) Components (in Units of pm 2 V ; from [7])

Equations (37)

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d A 2 d z i ρ 2 A 1 2 exp ( i Δ β z ) = 0 ,
ρ 2 = ω 2 2 A 1 2 d r e 2 * P ( 2 ) Re d r [ e 2 * × h 2 ] z ,
E ( r , ω j ) = A j ( z ) e j ( r , ω j ) exp ( i ( β j z ω j t ) ) ,
H ( r , ω j ) = A j ( z ) h j ( r , ω j ) exp ( i ( β j z ω j t ) ) ,
P b ( 2 ) ( r ) = ε 0 γ ( E 1 E 1 ) + ε 0 δ ( E 1 ) E 1 .
P s ( 2 ) ( r ) = δ ( r S ) [ P ( 2 s ) + P ( 2 s ) + P ( 2 s ) ] ,
P ( 2 s ) = ε 0 χ ( 2 s ) E 1 2 r ̂ ,
P ( 2 s ) = ε 0 χ ( 2 s ) | E 1 | 2 r ̂ ,
P ( 2 s ) = 2 ε 0 χ ( 2 s ) E 1 E 1 ,
E n r ( r , θ ) = A n β n a u n R n r ( r ) cos ( n θ + φ ) ,
E n θ ( r , θ ) = A n β n a u n R n θ ( r ) sin ( n θ + φ ) ,
E n z ( r , θ ) = i A n J n ( u n a r ) cos ( n θ + φ ) ,
R n r ( r ) = 1 s n 2 J n 1 ( u n a r ) 1 + s n 2 J n + 1 ( u n a r ) ,
R n θ ( r ) = 1 s n 2 J n 1 ( u n a r ) + 1 + s n 2 J n + 1 ( u n a r ) ,
u n = a k n n s 2 n n 2 , w n = a k n n n 2 1 , s n = n ( u n 2 + w n 2 ) J n ( u n ) u n J n ( u n ) + K n ( w n ) w n K n ( w n ) ,
[ J n ( u n ) u n J n ( u n ) + K n ( w n ) w n K n ( w n ) ] [ J n ( u n ) u n J n ( u n ) + 1 n s 2 K n ( w n ) w n K n ( w n ) ] = n 2 ( 1 u n 2 + 1 w n 2 ) [ 1 u n 2 + 1 ( n s w n ) 2 ] .
E r ( r ) = A 2 β 2 a u 2 J 1 ( u 2 a r ) ,
E θ = 0 ; E z ( r ) = i A 2 J 0 ( u 2 a r ) ,
J 1 ( u 2 ) u 2 J 0 ( u 2 ) = 1 n s 2 K 1 ( w 2 ) w 2 K 0 ( w 2 ) .
TM 01 mode :
ρ s = π a 1 2 a 2 β 1 2 β 2 a 4 u 1 2 u 2 R 1 r 2 ( a ) J 1 ( u 2 ) χ ( 2 s ) ,
ρ s = π a 1 2 a 2 β 2 a 2 u 2 J 1 ( u 2 ) χ ( 2 s ) [ ( β 1 a u 1 ) 2 R 1 θ 2 ( a ) J 1 2 ( u 1 ) ] ,
ρ s = 2 π a 1 2 a 2 β 1 β 2 a 3 u 1 u 2 J 0 ( u 2 ) J 1 ( u 1 ) R 1 r ( a ) χ ( 2 s ) ;
HE 21 mode :
ρ s = π 2 a 1 2 a 2 β 1 2 β 2 a 4 u 1 2 u 2 R 1 r 2 ( a ) R 2 r ( a ) χ ( 2 s ) ,
ρ s = π 2 a 1 2 a 2 β 2 a 2 u 2 R 2 r ( a ) χ ( 2 s ) [ ( β 1 a u 1 ) 2 R 1 θ 2 ( a ) + J 1 2 ( u 1 ) ] ,
ρ s = π a 1 2 a 2 β 1 a 2 u 1 R 1 r ( a ) χ ( 2 s ) [ a 2 β 1 β 2 u 1 u 2 R 2 θ ( a ) R 1 θ ( a ) J 1 ( u 1 ) J 2 ( u 2 ) ] .
P 2 ( z ) = [ ρ 2 P 1 z ] 2 P 2 P 1 = ( ρ 2 z ) 2 P 1 .
A 2 ( λ , L ) = 2 i ρ 2 sin Δ β ( λ ) L 2 Δ β ( λ ) .
Δ β ( λ ) d Δ β d λ ( λ λ 2 c ) = 4 π λ 2 c 2 ( n g 1 n g 2 ) ( λ λ 2 c ) ,
Δ λ FWHM 1.39156 λ 2 c 2 2 π L ( n g 1 n g 2 ) .
d A 2 ( ω j , z ) d z = i ρ 2 k A 1 ( ω k ) A 1 ( ω j ω k ) exp ( i 0 z d z Δ β j ( z ) ) .
Δ β = β ( ω k ) + β ( ω j ω k ) β ( ω j ) 2 β ( ω j 2 ) β ( ω j ) Δ β j
λ 2 c ( z ) = W λ m Φ m exp ( 1 2 k m 2 ( L c 2 ) 2 ) exp ( i k m z ) ,
d z λ 2 c ( z ) λ 2 c ( z + z ) exp [ ( z L c ) 2 ] ;
A 2 ( ω j , z m + 1 ) = A 2 ( ω j , z m ) + i ρ 2 Δ z k A 1 ( ω k ) A 1 ( ω j ω k ) sin ( Δ β j m Δ z 2 ) Δ β j m × exp ( i 0 z m d z Δ β j ( z ) ) exp ( i Δ β j m Δ z 2 ) ,
0 z d z Δ β j ( z ) 1 ,

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