Abstract

We introduce a rigorous theory of third-harmonic generation in optical waveguides and apply it to design a micro-fiber waveguide for efficient generation of third-harmonic radiation from infrared lasers. Phase-matching with efficient mode overlap is achieved in micro-fibers having a diameter roughly equal to half of the fundamental wavelength. Using a typical solid-state or fiber laser for pumping, high conversion efficiency is possible in only a few centimeters of a micro-fiber.

© 2005 Optical Society of America

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References

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  1. A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall Ltd, 1983).
  2. C. Vassallo, Optical Waveguide Concepts, (Elsevier, 1991).
  3. J. D. Jackson, Classical Electrodynamics, Third Edition (John Wiley & Sons, Inc.,1998).
  4. R. Boyd, Nonlinear Optics, Second Edition (Academic Press, 2002).
  5. A. Efimov, A. J. Taylor, F. G. Omenetto, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Phase-matched third harmonic generation in microstructured fibers,” Opt. Express 11, 2567–2576 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2567.
    [Crossref] [PubMed]
  6. F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,” Opt. Lett. 26, 1158–1160 (2001).
    [Crossref]
  7. S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2864.
    [Crossref] [PubMed]
  8. A. M. Vengsarkar, P. J. Lemaire, and J. B. Judkins, “Long-period gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
    [Crossref]
  9. J. J. Zayhowski, “Ultraviolet generation with passively Q-switched microchip lasers,” Opt. Lett. 21, 588–590 (1996).
    [Crossref] [PubMed]

2004 (1)

2003 (1)

2001 (1)

1996 (2)

A. M. Vengsarkar, P. J. Lemaire, and J. B. Judkins, “Long-period gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

J. J. Zayhowski, “Ultraviolet generation with passively Q-switched microchip lasers,” Opt. Lett. 21, 588–590 (1996).
[Crossref] [PubMed]

Arriaga, J.

Birks, T. A.

Boyd, R.

R. Boyd, Nonlinear Optics, Second Edition (Academic Press, 2002).

Efimov, A.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, Third Edition (John Wiley & Sons, Inc.,1998).

Judkins, J. B.

A. M. Vengsarkar, P. J. Lemaire, and J. B. Judkins, “Long-period gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

Knight, J. C.

Lemaire, P. J.

A. M. Vengsarkar, P. J. Lemaire, and J. B. Judkins, “Long-period gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

Leon-Saval, S. G.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall Ltd, 1983).

Moores, M. D.

Omenetto, F. G.

Russell, P. St. J.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall Ltd, 1983).

Taylor, A. J.

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts, (Elsevier, 1991).

Vengsarkar, A. M.

A. M. Vengsarkar, P. J. Lemaire, and J. B. Judkins, “Long-period gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

Wadsworth, W. J.

Zayhowski, J. J.

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Figures (2)

Fig. 1.
Fig. 1.

Phase-matching of the fundamental (HE11) mode of the fundamental wave (λ1=1.064 μm, orange) and the modes of the third-harmonic wave (λ3=0.355 μm, blue). Thick blue line represents the fundamental mode of the third-harmonic wave. The amplitude of J 3 integral (in μm-2) is shown for each mode-matching combination. The most efficient interaction is with the HE12 mode of the third harmonic (marked with a red circle). Modes of high angular orders are not considered due to their negligible overlap with the fundamental mode of the fundamental wave.

Fig. 2.
Fig. 2.

Calculated conversion of a 1.064-μm fundamental wave (input power P0=1 kW) into its third harmonic in a micro-fiber with diameter 0.52 μm. Fiber parameters from Eq. (24): ν1=2.07, ν2=0.39, ν3=1.25, detuning δβ * L NL = -5.08. Dotted lines represent the ideal conversion (Eqs. (17,21,22)) when ν13= -δβ * L NL /3. Competing nonlinear effects are ignored.

Equations (37)

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E r t = j = 1,3 A j ( z ) 2 Z 0 1 2 F j ( r ) exp [ i ( β j z ω j t ) ] + c . c .
H r t = j = 1,3 A j ( z ) 2 Z 0 1 2 G j ( r ) exp [ i ( β j z ω j t ) ] + c . c .
r = x y z , r = x y
β j = ω j c n j eff ( ω j ) , Z 0 = μ 0 ε 0
1 4 A ( F j × G j * + F j * × G j ) · z ̂ dS = 1 ,
P r t = A E r t × H r t t · z ̂ dS = j = 1,3 A j 2 4 A ( F j × G j * + F j * × G j ) · z ̂ dS = j = 1,3 A j ( z ) 2
d A j dz = Z 0 1 2 2 A exp [ i ( β j z ω j t ) ] F j * · P NL t t dS ,
P NL r t = ε 0 χ ( 3 ) ( r ) ( E r t · E r t ) E r t
χ ( 3 ) ( r ) = { χ ( 3 ) = const , within the glass ( A NL part of the cross section ) 0 , outside of the glass
A 1 z = i n ( 2 ) k 1 [ ( J 1 A 1 2 + 2 J 2 A 3 2 ) A 1 + J 3 A 1 * 2 A 3 e iδβz ]
A 3 z = i n ( 2 ) k 1 [ ( 6 J 2 A 1 2 + 3 J 5 A 3 2 ) A 3 + J 3 * A 1 3 e iδβz ] ,
J 1 = 1 3 A NL ( 2 F 1 4 + F 1 2 2 ) dS = A NL F 1 4 ( 2 + e ̂ 1 · e ̂ 1 2 3 ) dS
J 2 = 1 3 A NL ( F 1 2 + F 3 2 + F 1 · F 3 2 + F 1 · F 3 * 2 ) dS = A NL F 1 2 F 3 2 ( 1 + e ̂ 1 · e ̂ 3 2 + e ̂ 1 · e ̂ 3 2 3 ) dS
J 3 = A NL ( F 1 * · F 3 ) ( F 1 * · F 1 * ) dS = A NL F 1 3 F 3 ( e ̂ 1 * · e ̂ 3 ) ( e ̂ 1 * · e ̂ 1 * ) dS
J 5 = 1 3 A NL ( 2 F 3 4 + F 3 2 2 ) dS = A NL F 3 4 ( 2 + e ̂ 3 · e ̂ 3 2 3 ) dS
z ( A 1 2 + A 3 2 ) = 0
P 3 ( z ) P 0 3 ( k 1 n ( 2 ) J 3 ) 2 ( 2 δ β ˜ ) 2 sin 2 ( δ β ˜ 2 z )
P 3 z P 0 = 2 ( n ( 2 ) J 3 P 0 ) 2 ( k 1 δ β ˜ ) 2
A 1 = P 0 u 1 exp ( i φ 1 ) , A 3 = P 0 u 0 exp ( i φ 3 )
ν 1 = 2 J 2 J 1 1 , ν 2 = J 3 J 1 , ν 3 = J 5 J 1 2 J 2 J 1
Φ = δβ z + φ 3 3 φ 1
L NL = 1 ( n ( 2 ) k 1 J 1 P 0 ) , ξ = z L NL
u 1 2 + u 3 2 = 1
d u 1 = ν 2 u 1 2 u 3 sin Φ
d u 3 = ν 2 u 1 3 sin Φ
d Φ = δβ L NL + 3 ( ν 1 u 1 2 + ν 3 u 3 2 ) + ν 2 u 1 u 3 ( u 1 2 3 u 3 2 ) cos Φ
u 3 ( z ) = ν 2 sin Φ z z L NL 2 + ( ν 2 sin Φ z z ) 2 ,
sin Φ z = 1 z 0 z sin Φ ( z ) dz
Φ ( ξ ) = π 2 = const
A 3 ( z ) 2 = P 0 ( ν 2 z ) 2 L NL 2 + ( ν 2 z ) 2
L THG = L NL ν 2 = 1 k 1 n ( 2 ) J 3 P 0
φ 3 ( 0 ) = 3 φ 1 ( 0 ) + π 2
Φ ( 0 ) = π 2
ν 1 = ν 3 J 2 = J 1 + J 5 4
δβ = 3 ν 1 L NL = 3 ( 2 J 2 J 1 ) k 1 n ( 2 ) P 0
δβ 3 ν 1 + ν 3 2 L NL = 3 2 ( J 5 J 1 ) k 1 n ( 2 ) P 0
J 1 = 1.85 μ m 2 , J 2 = J 4 = 2.84 μ m 2 , J 3 = J 6 = 0.70 μ m 2 , J 5 = 7.99 μ m 2

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