Abstract

This paper investigates one-third harmonic generation (OTHG) by analytical methods with a nondepletion approximation and with an exact solution in continuous wave conditions. The nondepletion method shows that OTHG with a small initial power is confined in a very small range except that the overlapping integrals of the pump and signal follow a certain relation. The efficiency depends only on the initial conditions. Increasing pump power only shortens the interaction length to reach the maximum conversion. Furthermore, we exactly explore OTHG by the elliptic functions whose expressions depend on the types of roots derived from the integral and the initial power. We find that the output power level is limited by the initial conditions and the structure of the waveguide, while the pump power only determines the period. There is a constant value Γ that is determined by U0, θ0, and the overlap integrals, where U0 and θ0 are the initial power of the signal and the initial phase difference between pump and signal, respectively. We found that a highly efficient conversion only occurs when Γ is larger than a specific value Γc, called a critical value. A Γc provides a relation between U0 and θ0. So a set of critical conditions of U0 and θ0 is obtained. A highly efficient conversion may be supported if U0 is larger than the power in this set. We investigated some typical structure parameters and found the minimum initial power supporting high conversion efficiency. In OTHG, the variation curve has a sharp peak pattern, which means that a variation of the initial phase difference leads to a great change of the conversion. We established a way to get the smallest initial power with a large phase tolerance. Finally, we find a relation among the overlapping integrals and phase mismatching that can support a high conversion efficiency with a small initial power. This study gives valuable suggestions on the experimental design.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Puell and C. Vidal, “Optimum conditions for nonresonant third harmonic generation,” IEEE J. Quantum Electron. 14, 364–373 (1978).
    [CrossRef]
  2. A. Dot, A. Borne, B. Boulanger, K. Bencheikh, and J. A. Levenson, “Quantum theory analysis of triple photons generated by a (3) process,” Phys. Rev. A 85, 023809 (2012).
    [CrossRef]
  3. M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Experimental proposal for the generation of entangled photon triplets by third-order spontaneous parametric downconversion in optical fibers,” Opt. Lett. 36, 190–192 (2011).
    [CrossRef]
  4. F. Gravier and B. Boulanger, “Triple-photon generation: comparison between theory and experiment,” J. Opt. Soc. Am. B 25, 98–102 (2008).
    [CrossRef]
  5. K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C.R. Physique 8, 206–220 (2007).
    [CrossRef]
  6. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  7. A. Dot, A. Borne, B. Boulanger, P. Segonds, C. Felix, K. Bencheikh, and J. A. Levenson, “Energetic and spectral properties of triple photon downconversion in a phase-matched KTiOPO4 crystal,” Opt. Lett. 37, 2334–2336 (2012).
    [CrossRef]
  8. S. Afshar V., M. A. Lohe, T. Lee, T. M. Monro, and N. G. R. Broderick, “Efficient third and one-third harmonic generation in nonlinear waveguides,” Opt. Lett. 38, 329–331 (2013).
    [CrossRef]
  9. R. Ismaeel, T. Lee, M. Ding, N. G. R. Broderick, and G. Brambilla, “Nonlinear microfiber loop resonators for resonantly enhanced third harmonic generation,” Opt. Lett. 37, 5121–5123 (2012).
    [CrossRef]
  10. Y. Chen, “Four-wave mixing in optical fibers: exact solution,” J. Opt. Soc. Am. B 6, 1986–1993 (1989).
    [CrossRef]
  11. V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13, 6798–6806 (2005).
    [CrossRef]
  12. T. Lee, N. G. R. Broderick, and G. Brambilla, “Resonantly enhanced third harmonic generation in microfiber loop resonators,” J. Opt. Soc. Am. B 30, 505–511 (2013).
    [CrossRef]

2013 (2)

2012 (3)

2011 (1)

2008 (1)

2007 (1)

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C.R. Physique 8, 206–220 (2007).
[CrossRef]

2005 (1)

1989 (1)

1978 (1)

H. Puell and C. Vidal, “Optimum conditions for nonresonant third harmonic generation,” IEEE J. Quantum Electron. 14, 364–373 (1978).
[CrossRef]

1962 (1)

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Afshar V., S.

Armstrong, J.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bencheikh, K.

A. Dot, A. Borne, B. Boulanger, K. Bencheikh, and J. A. Levenson, “Quantum theory analysis of triple photons generated by a (3) process,” Phys. Rev. A 85, 023809 (2012).
[CrossRef]

A. Dot, A. Borne, B. Boulanger, P. Segonds, C. Felix, K. Bencheikh, and J. A. Levenson, “Energetic and spectral properties of triple photon downconversion in a phase-matched KTiOPO4 crystal,” Opt. Lett. 37, 2334–2336 (2012).
[CrossRef]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C.R. Physique 8, 206–220 (2007).
[CrossRef]

Bloembergen, N.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Borne, A.

A. Dot, A. Borne, B. Boulanger, K. Bencheikh, and J. A. Levenson, “Quantum theory analysis of triple photons generated by a (3) process,” Phys. Rev. A 85, 023809 (2012).
[CrossRef]

A. Dot, A. Borne, B. Boulanger, P. Segonds, C. Felix, K. Bencheikh, and J. A. Levenson, “Energetic and spectral properties of triple photon downconversion in a phase-matched KTiOPO4 crystal,” Opt. Lett. 37, 2334–2336 (2012).
[CrossRef]

Boulanger, B.

A. Dot, A. Borne, B. Boulanger, P. Segonds, C. Felix, K. Bencheikh, and J. A. Levenson, “Energetic and spectral properties of triple photon downconversion in a phase-matched KTiOPO4 crystal,” Opt. Lett. 37, 2334–2336 (2012).
[CrossRef]

A. Dot, A. Borne, B. Boulanger, K. Bencheikh, and J. A. Levenson, “Quantum theory analysis of triple photons generated by a (3) process,” Phys. Rev. A 85, 023809 (2012).
[CrossRef]

F. Gravier and B. Boulanger, “Triple-photon generation: comparison between theory and experiment,” J. Opt. Soc. Am. B 25, 98–102 (2008).
[CrossRef]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C.R. Physique 8, 206–220 (2007).
[CrossRef]

Brambilla, G.

Broderick, N. G. R.

Chen, Y.

Corona, M.

Ding, M.

Dot, A.

A. Dot, A. Borne, B. Boulanger, K. Bencheikh, and J. A. Levenson, “Quantum theory analysis of triple photons generated by a (3) process,” Phys. Rev. A 85, 023809 (2012).
[CrossRef]

A. Dot, A. Borne, B. Boulanger, P. Segonds, C. Felix, K. Bencheikh, and J. A. Levenson, “Energetic and spectral properties of triple photon downconversion in a phase-matched KTiOPO4 crystal,” Opt. Lett. 37, 2334–2336 (2012).
[CrossRef]

Douady, J.

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C.R. Physique 8, 206–220 (2007).
[CrossRef]

Ducuing, J.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Felix, C.

Garay-Palmett, K.

Gravier, F.

F. Gravier and B. Boulanger, “Triple-photon generation: comparison between theory and experiment,” J. Opt. Soc. Am. B 25, 98–102 (2008).
[CrossRef]

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C.R. Physique 8, 206–220 (2007).
[CrossRef]

Grubsky, V.

Ismaeel, R.

Lee, T.

Levenson, A.

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C.R. Physique 8, 206–220 (2007).
[CrossRef]

Levenson, J. A.

A. Dot, A. Borne, B. Boulanger, P. Segonds, C. Felix, K. Bencheikh, and J. A. Levenson, “Energetic and spectral properties of triple photon downconversion in a phase-matched KTiOPO4 crystal,” Opt. Lett. 37, 2334–2336 (2012).
[CrossRef]

A. Dot, A. Borne, B. Boulanger, K. Bencheikh, and J. A. Levenson, “Quantum theory analysis of triple photons generated by a (3) process,” Phys. Rev. A 85, 023809 (2012).
[CrossRef]

Lohe, M. A.

Monro, T. M.

Pershan, P.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Puell, H.

H. Puell and C. Vidal, “Optimum conditions for nonresonant third harmonic generation,” IEEE J. Quantum Electron. 14, 364–373 (1978).
[CrossRef]

Savchenko, A.

Segonds, P.

U’Ren, A. B.

Vidal, C.

H. Puell and C. Vidal, “Optimum conditions for nonresonant third harmonic generation,” IEEE J. Quantum Electron. 14, 364–373 (1978).
[CrossRef]

C.R. Physique (1)

K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” C.R. Physique 8, 206–220 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. Puell and C. Vidal, “Optimum conditions for nonresonant third harmonic generation,” IEEE J. Quantum Electron. 14, 364–373 (1978).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. (1)

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A (1)

A. Dot, A. Borne, B. Boulanger, K. Bencheikh, and J. A. Levenson, “Quantum theory analysis of triple photons generated by a (3) process,” Phys. Rev. A 85, 023809 (2012).
[CrossRef]

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.


Figures (8)

Fig. 1.
Fig. 1.

Two real roots of f(u)=0, η3<η4, are plotted depending on Γ. There is an interruption of η3 in the dashed area that is plotted in the inset graph. There are four real roots when γ is between zero and the critical value, highlighted in the inset graph. The distribution of corresponding initial power, shown as the shaded area, is in the interval [η1,η2], which means only a low conversion is supported in this interval.

Fig. 2.
Fig. 2.

Critical value, Γc, determines a set of U0 and θ0, shown in (a), from which we obtain the range of the corresponding U0. The first three roots depending on initial power U0 are plotted with different θ0 as π, π/2, and 0, shown in (b), (c), and (d), respectively. When U0 is larger than the value determined on the line in (a), a highly efficient conversion is supported. Critical U0 increases with the increase of θ0 from 0 to π.

Fig. 3.
Fig. 3.

(a) Initial power is put at 0.012 for a tolerance of phase. The variations of U with different θ0 are plotted. It is shown that power of the signal decreases when θ0=π, which is a THG process. Other values of θ0 support high efficiency. But the variation curves are shifted by the change of θ0. So the initial phase must be controlled strictly in order to get a stable high signal output with a specific interaction length. (b) A standard ODE numerical method is used to evaluate the exact coupled wave equations [Eqs. (1) and (2)] at the corresponding conditions in (a). The results in (a) and (b) are consistent.

Fig. 4.
Fig. 4.

(a) Extremely small initial power is put as 104. θ0 is chosen as π, π/2, and 0. θ0=0 leads to high efficiency, whereas θ0=π leads to reduction of power. (b) A standard ODE numerical method is used to evaluate the exact coupled wave equations [Eqs. (1) and (2)] at the corresponding conditions in (a). The results in (a) and (b) are consistent.

Fig. 5.
Fig. 5.

Value of two real roots, η3 and η4, depending on a and Γ, is shown in (a) and (b), respectively. It is found that the interval in which U varies has a large upper limit and a small lower limit at the circle region, which means a possible high conversion with small initial power. In the rest region, the difference between the upper and lower limits is small and does not lead to highly efficient conversion.

Fig. 6.
Fig. 6.

(a) η4 is the upper limit of the conversion. A large a causes a small upper limit so that the maximum of conversion is small. (b) Variations with a as 0.1 and 1, simulated by the exact coupled wave equations [Eqs. (1) and (2)]. A large a leads to a long length with a higher efficiency and a smaller maximum value. So a small value of a is desirable.

Fig. 7.
Fig. 7.

Highly efficient conversion of OTHG is supported when b equals U0, evaluated by the ODE numerical method by the exact coupled wave equations [Eqs. (1) and (2)].

Fig. 8.
Fig. 8.

OTHG process discussed on the condition of a=1, U0=104, and θ0=0.5. (a) When b=ΔS is satisfied, a high conversion can be obtained. A larger value of b leads to a longer length to get the maximum conversion. (b) When b=0.1 is supposed, there are only two real roots at this condition. The smaller one, η3, is plotted depending on ΔS. It is found that η3 is larger than U0 when ΔS is in the interval [0.095, 0.13], which means that a high conversion is supported in this interval. (c) Variations of U with different ΔS, as 0.095, 0.105, and 0.115, respectively, are plotted. The lengths to get the maximum conversion are different according to the values of ΔS. (a) and (c) are simulated by the exact coupled wave equations [Eqs. (1) and (2)].

Tables (1)

Tables Icon

Table 1. U0 and Roots

Equations (42)

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

A1z=iγ0[(J1|A1|2+2J2|A3|2)A1+J3(A1*)2A3eiδβz],
A3z=iγ0[(6J2|A1|2+3J5|A3|2)A3+J3A13eiδβz],
A1z=iγ0[2J2|A3|2A1+J3(A1*)2A3eiδβz],
A3z=i3γ0J5|A3|2A3.
dρ1dzγ0J3ρ12P3sinθ,
dθdz=P3(δβP3+3γ0J56γ0J23γ0J3ρ1P3cosθ)P3(δβP3+3γ0(J52J2))=P3δβ,
ρ1(z)=P3δβρ1(0)P3δβγ0J3ρ1(0)P3[cos(P3δβz+θ0)cosθ0]=ρ1(0)1γ0J3ρ1(0)P3δβ[cos(P3δβz+θ0)+cosθ0],
ρ1(πδβ)=ρ1(0)12γ0J3ρ1(0)P3δβ,
ρ1(πδβ)ρ1(0)+2γ0J3P3δβρ1(0)2.
J52J2J343111μρP3.
J52J2J343ρP3.
δβP3J3+γ03(J52J2)J3=111μρ1(0)P3.
dUdζ=2U3(1U)sinθ,
dθdζ=(ΔS+b)+cosθdU3(1U)dζsinθU3(1U)+(ab)U,
ΔS=δβγ0J3P,
a=3(2J2J1)J3,
b=3(J52J2)J3.
Γ=U3(1U)cosθab4U2b+ΔS2U,
ζ=U0U(ζ)1f(U)dU,
f(U)=4(1U)U34(14(ab)U2+Γ+12U(b+ΔS))2.
U=η4N+η3M(η4Mη3N)cn(ζ±ζ0ζc,k)M+N+(MN)cn(ζ±ζ0ζc,k),
m=Re((η1η2)22),
n=Re(η1+η22),
M2=(η4n)22+m2,
N2=(η3n)2+m2,
g=1MN,
k2=(η4η3)2(MN)24MN,
c02=4+a24ab2+b24,
ζc=gc0,
ζ0=ζcF(cos1((η4U0)N(U0η3)M(η4U0)N+(U0η3)M),k).
U(ζ)=η1η4η2η1+(η4η2η1η4)sn2(ζ±ζ0ζc,k)η4η2+(η2η1)sn2(ζ±ζ0ζc,k),
ζ0=ζcF(sin1(η4η2)(U0η1)(η2η1)(η4U0),k),
g=2(η4η2)(η3η1),
k=(η4η3)(η2η1)(η4η2)(η3η1).
U=η4η3η3η2+(η4η2η4η3)sn2(ζ±ζ0ζc,k)η4η3+(η2η3)sn2(ζ±ζ0ζc,k),
k=(η3η2)(η4η1)(η4η2)(η3η1),
ζ0=ζcF(sin1(η4η2)(η3U0)(η3η2)(η4U0),k).
U=η4η1η4η3+(η3η1η4η1)sn2(ζ±ζ0ζc,k)η1η3+(η3η4)sn2(ζ±ζ0ζc,k),
k=(η4η3)(η2η1)(η4η2)(η3η1),
ζ0=ζcF(sin1(η3η1)(η4U0)(η4η1)(U0η1),k).
f(U)=4(1U)U3(a4U2+Γ)2.
U(ζ)=44+(ζ0ζ)2,

Metrics