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Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 1 — Jan. 14, 2002
  • pp: 83–97
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Exact analytical solutions and their applications for interacting waves in quadratic nonlinear medium

Xueming Liu, Hanyi Zhang, and Mingde Zhang  »View Author Affiliations


Optics Express, Vol. 10, Issue 1, pp. 83-97 (2002)
http://dx.doi.org/10.1364/OE.10.000083


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Abstract

The exact analytical solutions for the interacting waves in the quadratic nonlinear medium with the periodic structure are detailedly derived and obtained, and the properties of solutions are analyzed. Three applicable examples employing the exact solutions in the all-optical processing are given and analyzed. The optimized results show that the phase of signal can obviously be increased by proper choosing Δk, and that the intensity of pump can greatly be decreased in the all-optical switching by means of optimizing Δk, increasing medium length, and choosing sum-frequency generation.

© Optical Society of America

1. Introduction

In future all-optical communication systems, optical frequency converters, all-optical switchings, all-optical modulators, optical transistors, etc. will become the primary elements of optical devices [1–3

1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

]. They can be realized via the quadratic nonlinear processes, e.g., the sum-and difference-frequency generation (SFG and DFG) and second-harmonic generation (SHG), of three interacting waves in nonlinear mediums such as semiconductor materials, electrooptic crystals, silica fibers, etc. [4–6

4. G. Assanto and I. Torelli, “Cascading effects in type II second-harmonic generation: application to all-optical processing,” Opt. Commun. 119, 143–148 (1995). [CrossRef]

]. A quantitative description of the quadratic nonlinear effects normally employs a group of coupled-mode equations (CME). And the three lightwaves participated in the three-wave mixing (TWM) process exchange their energies and momentums in the quadratic nonlinear medium, as leads to light frequency conversions [7

7. J. A. Armstrong, N. Bloembergen, and N, J. Ducuing, et al. “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]

, 8

8. A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996). [CrossRef] [PubMed]

]. Wavelength division multiplexing (WDM) networks usually require wavelength conversion techniques that need not only considering the intensities of the waves but also including their phase messages [2

2. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]

]. Since cross-gain/phase modulations in semiconductor optical amplifiers only depend on the intensities of the waves but no their phases, they can not realize full-transparency wavelength conversion [2

2. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]

, 9

9. M. Asghari, I. H. White, and R. V. Penty, “Wavelength conversion using semiconductor optical amplifiers,” J. Lighteave Technol. 15, 1181–1190 (1997). [CrossRef]

]. However, DFG in the TWM process can offer the full-transparency wavelength conversion including the messages of the wave phase and intensity without adding excess noise to the signal [2

2. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]

]. All-optical switchings and modulators, optical transistors, etc. are normally implemented by the nonlinear phase shifts induced by the cascaded SFG (including its degenerated case of SHG) and DFG, which are determined by the phase in the TWM process, and can be obtained from the solutions of CME [1

1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

, 4

4. G. Assanto and I. Torelli, “Cascading effects in type II second-harmonic generation: application to all-optical processing,” Opt. Commun. 119, 143–148 (1995). [CrossRef]

]. Therefore, if an exact solution can be solved from CME, main properties of the all-optical devices can almost be known.

Though TWM-CME has been studied extensively, to our best knowledge, the exact analytical solutions are not completely obtained. Armstrong et al. only obtained a few analytical approaches towards the description of the output modulation via the cascading of the quadratic nonlinearities [7

7. J. A. Armstrong, N. Bloembergen, and N, J. Ducuing, et al. “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]

]. Boyd had detailedly given a solving process for TWM-CME on the assumption that one electrical field amplitude was unaffected by the interaction [10

10. R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992), Chap.2.

]. The evolution of the output intensities was studied using different approximations from zero phase mismatching to only two input waves [11

11. M. J. T. Milton, “General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves,” IEEE. J. Quantum Elecron. 28, 739–749 (1992). [CrossRef]

, 12

12. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap.6-7.

]. Numerical calculations for the phase evolution of the output wave were studied by Sibilia et al. [13

13. A. R. C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effect in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995). [CrossRef]

]. Ironside et al. derived an analytical formula for the scalar case whereas the vectorial case was not covered [14

14. C. N. Ironside, J. S. Aitchison, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE. J. Quantum Elecron. 29, 2650–2654 (1993). [CrossRef]

]. Kobyakov gave an exact analytical description of the output modulation in the vectorial case without SH seed input, and a comprehensive analytical approach to the scalar and vectorial interaction in the cascaded quadratic nonlinearities but not obtaining a concrete solution for TWM-CME [8

8. A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996). [CrossRef] [PubMed]

, 15

15. A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities-an analytical approach,” Opt. Commun. 124, 184–194 (1996). [CrossRef]

]. Three-wave interaction in a quadratic nonlinear medium under perfect phase matching conditions was investigated by D’Aguanno et al. [16

16. G. D’Aguanno, C. Sibilia, E. Fazio, and M. Bertolotti, “Three-wave mixing in a quadratic material under perfect phase-matching,”Opt. Commun. 142, 75–78 (1997). [CrossRef]

]. Obviously, if exact analytical results for TWM-CME can be achieved, they can provide a clear understanding of the underlying physics, disclose the effect of the various parameters on the output modulation, and represent a more flexible and reliable tool for optimization issues than a numerical way. The aims of the present paper are to seek the exact solutions of CME in the TWM process such as SFG, SHG and DFG, and to apply these solutions to optimizing the parameters in the all-optical processing. CME of SFG, SHG and DFG are detailedly derived under the initial condition with two fundamental wave inputs, and their exact solutions are obtained in this paper. Results are analyzed and their physical mechanisms are explained. Moreover, three applicable examples are given and the optimizations are analyzed.

2. Exact solutions of CME

Under approximations of the slowly-varying-envelope, plane-waves, and ignoring loss and higher-order nonlinear processes, TWM-CME for the quadratic nonlinear medium with nonmagnetic and nonconducting dielectric can be derived from the Maxwell equations, as [11

11. M. J. T. Milton, “General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves,” IEEE. J. Quantum Elecron. 28, 739–749 (1992). [CrossRef]

, 13

13. A. R. C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effect in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995). [CrossRef]

, 17–19

17. X. -M. Liu, H. - Y. Zhang, and Y. -L. Guo, “Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference-frequency generation,” J. Lightwave Technol. 19, 1785–1792 (2001). [CrossRef]

]

Et1z=iωt1deffnt1cEt2*Et3exp(iΔkz),
(1.a)
Et2z=iωt2deffnt2cEt1*Et3exp(iΔkz),
(1.a)
Et3z=iωt3deffnt3cEt1Et2exp(iΔkz),
(1.c)

where c and deff are the speed of light in the vacuum and the effective nonlinear coefficient, respectively. ktj , ntj and Etj are, respectively, the wave vector, the index of refraction, and the electric field under light-frequencies ωtj (j=1, 2, 3). In a homogeneous quadratic nonlinear medium, the wave vector mismatching Δk=kt3 -k t2-k t1. If the quadratic nonlinearity of the medium is a uniform periodical structure (i.e., periodic domain inversion), CME will have the same formation as Eqs. (1.a)~(1.c) under the approximation of the first-order periodic perturbation effect and the grating phase matching of fundamental diffraction order, but the wave vector mismatching relates to its period ʌ (i.e., Δk=k t3-k t2-k t1±2π/ʌ) [18

18. T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matchedwith uniform and chirped gratings,“ IEEE. J. Quantum Elecron. 26, 1265–1270 (1990). [CrossRef]

].

Provided that the two input waves are at lower frequencies ω 1 and ω 2 it causes SFG (ω 3=ω 1+ω 2), and if they are degenerated (ω 1=ω 2) it is a special case of SFG called SHG. Then the subscripts t1, t2 and t3 in Eqs. (1.a)~(1.c) are labeled 1, 2 and 3, respectively. If ω 3=ω 1-ω 2, it refers to DFG and the subscripts t1, t2 and t3 are named 2, 3 and 1, respectively. ω 1, ω 2 and ω 3 are the light frequencies of the two input fundamental waves and their produced idler wave in the TWM process including SFG, SHG and DFG, and ω 1 and ω 2 present the higher and lower frequency waves in DFG Making use of the relation between the light intensity I and the electric field E [11

11. M. J. T. Milton, “General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves,” IEEE. J. Quantum Elecron. 28, 739–749 (1992). [CrossRef]

]

I=ε0cnE2/2,
(2)

and introducing the photon flux F = I/(⋇ω) and its square root f = √F, then the electric field E is given as

Ej=ejfjexp(iφj),
(3.a)
ej=2ħωj/(njε0c)(j=1,2,3),
(3.b)

where fj (j=1, 2, 3) is the square root of the photon flux at ωj , φj is the phase of the electric field Ej , ε 0 and are the dielectric permittivity in the vacuum and the Planck coefficient, respectively. Assuming C 2=(2⋇ω 1 ω 2 ω 3 deff2)/(n 1 n 2 n 3 ε 0 c 3) and ζ=C·z, and substituting Eq. (3) into Eqs. (1.a)~(1.c), we have a set of coupled equations

ζ(f1f2f3)=(f2f3f1f3f1f2)sinθ,
(4.a)
ζ(φ1φ2φ3)=(f2f3/f1f1f3/f2f1f2/f3)cosθ
(4.b)

for the QPM-SFG process, and

ζ(f1f2f3)=(f2f3f1f3f1f2)sinθ′,
(5.a)
ζ(φ1φ2φ3)=(f2f3/f1f1f3/f2f1f2/f3)cosθ′
(5.b)

for the QPM-DFG process. In the above equations, the relative phase angle θ=φ 3-φ 2-φ 1kz for QPM-SFG and θ′=-φ 3-φ 2+φ 1kz for QPM-DFG. 1)

2.1. Exact solutions for SFG-CME

Doing some manipulations of Eq.(4.a), one can obtain

ζ(ln(f1)ln(f2)ln(f3))=(f2f3/f1f1f3/f2f1f2/f3)sinθ.
(6)

Taking the third expression to resubstitute the first and second expressions of Eq. (4.b), and considering φ 3-φ 2-φ 1=θkz and Eq.(6), we have

=ΔkC+ctan(θ)dln(f1f2f3).
(7)

Considering tan(θ) = -d[ln(cosθ)] and yd[ln(y)]=dy, and changing the third expression in Eq. (4.a) to sin(θ) = d(f32)/(2f 1 f 2 f 3), Eq. (7) can be integrated to give

f1f2f3cosθ+Δkf32/(2C)=Γ1,
(8)

where Γ1 is a constant of integration which is independent of θ and ζ From Eq. (4.a) it is given as

f1df1=f2df2=f3df3=f1f2f3sinθ.
(9)

After a short manipulation, Eq. (9) can be given as

(f12(0)+f32(0)f22(0)+f32(0)f12(0)f22(0))=(f12(ζ)+f32(ζ)f22(ζ)+f32(ζ)f12(ζ)f22(ζ)).
(10)

Eq. (10) is well-known as the Manley-Rowe relations. After a simple manipulation, then, Eqs. (8) and (9) give rise to

12d(f32)=f12f22f32(Γ1Δkf32/(2C))2.
(11)

Assuming that the numbers of photons for the two input fundament-frequency waves in SFG are f12(0)=F 1 and f22(0)=F 2, and considering f32(0)=0 (i.e., without the input idler wave),

then Γ1=0 from Eq. (8). From Eqs. (10) and (11), it can be obtained

0ξ=120F3d(f32)/(F1f32)(F2f32)f32[Δkf32/(2C)]2,
(12)

where ξ=lC, l is the interaction length of the three waves. F 3=f32(ξ) is the number of photons at the normalized distance ξ. Defining the relative phase error over the interaction length δkl/2, and the normalized phase error κ=δ 2/(ξ 2 F 1)=Δk 2/(4C 2 F 1), and introducing the dimensionless variable x 2=f32/F 1 and σ=F 2/F 1, the integral for Eq. (12) can be simplified to

ξF1=0Xf[(1x2)(σx2)κx2]0.5dx,
(13)

where Xf =F 3/F 1. After the simple manipulation and the resubstitution of the above variables, Eq. (13) can be solved by Jacobian elliptic function sn [7

7. J. A. Armstrong, N. Bloembergen, and N, J. Ducuing, et al. “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]

, 20

20. G. A. Kehen [USA] and T. M. Kehen, Handbook of Mathematics (Worker Press, Beijing, 1987), (in Chinese).

]. The result is

f3(ξ)=F1Bsn(AξF1,u),
(14.a)

where A,B=(1+σ+κ±(1+σ+κ)24σ)/2, and u=B 2/A 2in which A takes the addition and B the subtraction. Eq. (14.a) is the general expression for the magnitude of the generated wave in SFG. The intensity of the other two waves at any place in the medium can be calculated by substituting Eq. (14.a) into the appropriate Manley-Rowe relations given in Eq. (10), and it can easily be obtained

fj(ξ)=FjF1B2sn2(AξF1,u)(j=1,2)
(14.b)

Eqs. (14.a) and (14.b) imply that creating a photon at frequency φ 3 is at the price of annihilating two photons at frequency φ 1 and φ 2. Substituting Eq. (8) into the first expression of Eq. (4.b) can give rise to

φ1ζ=(Γ1Δkf322C)/f12.
(15)

Considering Γ1=0, then Eq. (15) can be solved as

φ1(ξ)=0ξΔkf32(ζ)2Cf12(ζ)+φ1(0).
(16.a)

Along to the same derivation process, it can be obtained

φj(ξ)=0ξΔkf32(ζ)2Cfj2(ζ)+φj(0)(j=2,3),
(16.b)

where φ 1(0), φ 2(0) and φ 3(0) are the initial phases of the two fundament-frequency and sum-frequency wave, respectively. φ 3(0) can be acquired from Eqs. (4.a) and (4.b) and its derivation is done as following.

During the initial stage of the interaction coupling for SFG, the square root of the photon flux f 3 for the sum-frequency wave vanishes (i.e., f 3→0) when z→0. Considering the continuity and finitude of the phase evolution of the sum-frequency wave and the finitude of its derivative (i.e., ∂φ 3/∂ζ, we can obtain from the third expression of Eq. (4.b), as

cosθ(z0)=cos[φ3(z0)φ2(z0)φ1(z0)]0.
(17.a)

Consider that the energy of the two fundament-frequency waves convert into that of the sum-frequency wave in the beginning of the SFG process (i.e., the number of photons of sum-frequency wave increases), then the derivative of f 3 should be positive value (i.e., ∂f 3/∂ξ >0). From the third expression of Eq. (4.a), therefore, we can get

sin[φ3(z0)φ2(z0)φ1(z0)]>0.
(17.b)

From Eqs. (17.a) and (17.b), θ(0) = 2 + π/2 (n = 0,±l,±2,⋯)can be obtained. Since sinθ and cosθ in Eq. (4) are the period functions of 2π we define the range of θ(z) from -πto π, and then θ(0)=π/2 and φ 3(0)=π/2+φ l(0)+φ 2(0). From Eqs. (3), (4), (14) and (16), therefore, the exact solutions for SFG-CME [i.e., Eq. (1)] can be written as

Ej(ξ)=ejFjF1B2sn2(F1,u)·ei(φj(0)0ξΔkf32(ζ)2Cfj2(ζ))(j=1,2),
(18.a)
E3(ξ)=e3F1Bsn(AξF1,u)·ei(π2+φ1(0)+φ2(0)lΔk2).
(18.b)

Equations Eqs. (18.a) and (18.b) imply, in the QPM-SFG process, that

  1. The phase evolution of the sum-frequency wave in the TWM process is linear to the interaction length l in the medium, and is independent of the intensities of the three interacting waves.
  2. Contrarily, the phase evolution of the two fundament-frequency waves relates not only to their intensities but also to that of the sum-frequency wave.
  3. The intensity evolutions of the three waves, which are proportional to the square of the electric field magnitude |E|2, are mutually affected by their intensities while are unaffected by their phases.

2.2. Exact solutions for SHG-CME

When the two incident beams are degenerated (i.e., ω 1=ω 2), the special case of the SFG is SHG (i.e., ω 3=2ω 1). The expressions of SHG-CME is still given by Eqs. (18.a) and (18.b), but σ=1. In the ideal case of perfect phase matching (i.e., Δk=0), σ, A, B, and u become equal to unit and κ is zero, and then sn(t, u)=tanh(t). Therefore the exact solutions of SHG-CME can be simplified to the primary function by

E1(ξ)=E2(ξ)=e1F1F1tanh2(ξF1)exp(iφ1(0)),
(19.a)
E3(ξ)=e2F1tanh(ξF1)exp(iφ3(0)).
(19.c)

2.3. Exact solutions for DFG-CME

After a simple manipulation and the resubstilution of Eq. (5), we can obtain

tan(θ′)dθ′=(Δk/C)tan(θ′)+d[ln(f1f2f3)].
(20)

This can be integrated to give

f1f2f3cosθ′Δkf32/(2C)=Γ1,
(21)

where Γ1′ is a constant of integration which is independent of θ and ζ Considering the initial condition of f32(0)=0 (corresponding to no input idler wave), definingf12(0)= F 1 and f22(0)=F 2, and introducing another Jacobian elliptic function of dn(x, t)=[1-t 2sn2(x, t)]1/2 [20

20. G. A. Kehen [USA] and T. M. Kehen, Handbook of Mathematics (Worker Press, Beijing, 1987), (in Chinese).

], in accordance with the derivation process of SFG we can obtain

fj(ξ)=Fj+(1)jF2B′2u′2sn2(τ,u′)/dn2(τ,u′)(j=1,2),
(22.a)
f3(ξ)=F2Busn(τ,u′)/dn(τ,u′),
(22.b)

where u2 =A2/(A2 + B2), A′,B′=((σ′1κ′)2+4σ′±(σ′1κ′))/2, in which + for A′ and - for B′. τ=ξF2(σ′1κ′)2+4σ′4, , σ′ = F 1 /F 2, κ′ = δ 2/(ξ 2 F 2)=Δk 2/ (4C 2 F 2).

In the beginning stage of DFG, considering the increase of the photon number of the difference-frequency wave, the continuity and finitude of the evolution of phase φ 3, and the finitude of ∂φ 3/∂ζ we can obtain that sin θ′(z→0)<0 and cosθ′(z→0)→0 from the third expressions of Eqs. (5.a) and (5.b), and then φ 3(0)=π/2+φ 1(0)-φ 2(0) [the range of θ′(z) has been limited between -π and π]. Considering f 3(0)=0 and Γ1′=0, from Eqs. (5) and (21) we can obtain

φj(ξ)=0ξΔkf32(ζ)2Cfj2(ζ)+φj(0)(j=2,3),
(23)

where φ 1(0) and φ 2(0) are the initial phases of the two fundament-frequency waves, respectively. Therefore, the explicit solutions of CME [i.e., Eq. (1)] for QPM-DFG can be written, from Eqs. (3), (5), (22) and (23), as

Ej(ξ)=ejFj+(1)jF2B′2u′2sn2(τ,u′)dn2(τ,u′)·ei(φj(0)+0ξΔkf32(ζ)2Cfj2(ζ))(j=1,2),
(24.a)
E3(ξ)=e3F2B′u′[sn(τ,u′)/dn(τ,u′)]·ei(π2+φ1(0)φ2(0)+lΔk2).
(24.b)

Comparing Eqs. (18) with (24) exhibits that the phase evolution of the three waves in QPM-DFG is contrary to that in QPM-SFG. For example the former increases monotonously but the latter decreases monotonously when Δk>0. The magnitude |E| of electric field of the three waves in QPM-DFG is independent of their phase, which is similar to |E| in QPM-SFG.

2.4. Small-signal approximation

Sections 2.1 and 2.3 have dealt with the general expressions of SFG- and DFG-CME. They can be simplified under the small-signal approximation, where the intensity of the pump wave ω 1 is much stronger than intensities of the signal wave ω 2 and the idler wave ω 3. Then σ<<1, and A, B can be expanded by Taylor function as A≈(κ+1)1/2 and Bσ/(κ+1), and u<<1 and sn(t, u)≈sin(t). Thus the electric field of the idler wave in the SFG process can be reduced to

E3(ξ)=e3F2/(κ+1)sin(κ+1ξF1)·ei(π2+φ1(0)+φ2(0)lΔk2).
(25)

Substituting the definition of the variables of e 3, F 1, σ, κ and ξ into Eq. (25), it can be further simplified to

E3(ξ)=[ME1(0)E2(0)/g]sin(gl)·eilΔk2,
(26)

where M = -iCe 3/(e 1 e 2), g=Δk2/4+C2F1.

With similar deriving process of the SFG, the solution of the idler wave under small-signal approximation for DFG-CME is given by

E3(ξ)=i[M2E2*(0)/Q]sinh(Ql)·eilΔk2,
(27)

where Q=M2M3Δk2/4, M j = ω j deff E 1(0)/(nj c) (j=2, 3). In the manipulation, for A′>> B′ and u′≈1, we use the relationships of sn(τ, u′)≈tanh(τ) and sinh(τ)≈sn(τ, u′)/dn(τ u′).

Results of Eqs. (26) and (27) can also be obtained on the assumption that the electric field E 1 of the pump wave is unaffected by the interaction and is taken as constant in CME [10

10. R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992), Chap.2.

].

3. Calculation and analysis

In Section 2, without the incident idler field, the general expressions for the photon fluxes fj2, phase φ j and electric filed Ej (j=1, 2, 3) of the three interacting waves in any SFG or DFG process have been obtained. In this part we go on to draw a number of conclusions from these results and verify them.

3.1. Comparing the results from Eq. (1) with those from Eqs. (18) and (24)

Eqs. (18) and (24) exhibit the complete and exact information about the evolution of the phase and intensity of the two fundamental waves and their creating wave provided that no idler field is incident. It can be easily found from Eqs. (18) and (24) that the magnitude of the electric field |Ej | (j=1, 2, 3) is independent of the phase φj . Then the evolution of exchanging energy for the two fundament-frequency waves and their creating wave in the quadratic nonlinear medium is only dependent on the number of their photons whereas independent of their phases. Since sn(K·z, u) and dn(K·z, u) are period functions with T=(02π1u2sin2ϕ)K [20

20. G. A. Kehen [USA] and T. M. Kehen, Handbook of Mathematics (Worker Press, Beijing, 1987), (in Chinese).

, 12

12. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap.6-7.

], the energy evolutions of the three waves in the nonlinear medium are periodic, whereas their phase evolutions monotonously step-like increase or decrease (the evolution for the tendency of increasing or decreasing is determined by the sign of the phase mismatching Δk). They are shown in Fig. 1 and 2, which describe the evolution of the intensities and phases of the three waves with the interaction length z during the QPM-DFG and SFG process, respectively. Dot lines in the those figures are obtained after numeral calculating Eq. (1) by 4/5-order Runge-Kutta method (the precision of 10-6), and solid lines are directly calculated from Eqs. (18) and (24) (the precision of sn and dn is 10-15)[21

21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Dover, 1965), Chap. 16–17.

] where the calculation of the phases evolution employs 7-order Richardson extrapolate method 2) [the precision of 1/(47+1)]. In the simulation calculations we employ the presently representative data of periodic poling LiNbO3 (PPLN) [6

6. G. S. Kanter and P. Kumar, “Optical devices based on internally seeded cascaded nonlinearities,” IEEE. J. Quantum Elecron. 35, 891–896 (1999). [CrossRef]

, 22

22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and I. Yokohama, “All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide,” IEEE. Photon. Technol. Lett. 11, 328–330 (1999). [CrossRef]

, 23

23. C. Q. Xu, H. Okayama, and M. Kawahara, “Optical frequency conversions in nonlinear medium with periodically modulated linear and nonlinear optical parameters,” IEEE. J. Quantum Elecron. 31, 981–987 (1995). [CrossRef]

], i.e., the length l of PPLN is 10mm, the intensities I 1 and I 2 of the two fundament-frequency waves are 104 and 102 kW/cm2 respectively, the wavelengths λ 1 and λ 2 of the two fundament-frequency waves are 1555nm and 1545nm in QPM-SFG, respectively, and 775nm and 1545nm in QPM-DFG, and deff =20.5pm/V, φ 1(0)=0.5rad, φ 2(0)=0, ʌ=16.21 μm, Δk=0.0419 mm-1 for QPM-SFG and Δk=-0.0507 mm-1 for QPM-DFG. Sellmeier equation comes from the reference [22

22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and I. Yokohama, “All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide,” IEEE. Photon. Technol. Lett. 11, 328–330 (1999). [CrossRef]

].

Fig.1. Evolutions of the intensities and phases in QPM-SFG with the interacting length z, where Δk=0.0419 mm-1.
Fig.2. Evolutions of the intensities and phases in QPM-DFG with the interacting length z, where Δk=-0.0507 mm-1.

Fig. 1 and 2 show that the results of numeral calculating Eq. (1) and direct calculating Eqs. (18) & (24) are extremely consistent under the applied precision. The intensity evolutions of the three interacting waves in the QPM-SFG process, see Fig.1, show that increasing or decreasing for the intensity of the sum-frequency wave simultaneously accompanies decreasing or increasing for intensities of the two fundament-frequency waves. Fig.2 exhibits that, during the QPM-DFG process, the intensity of the higher frequency wave increases or decreases accompanied by decreasing or increasing intensities of the lower and idler frequency wave. This mechanism is that creating a photon of the lower and idler frequency wave comes from destroying a photon of the higher frequency wave, or generating a photon of the higher frequency wave is at the price of annihilating a photon of the lower and idler frequency wave.

3.2. Phase evolutions of three interacting waves

Fig.1 and 2 show that the phase evolutions of the two fundament-frequency waves step-like change, which are dependent on the sign of Δk. The phase evolutions of the idler wave, however, increases (or decreases) linearly. In addition, the phase evolutions for QPM-SFG and QPM-DFG are contrary to each other under the same sign of Δk. If the TWM process is content with perfect phase matching (i.e., Δk =0), it can easily be found from Eqs. (16) and (23) that the phases of the three interacting waves do not change in the whole interaction process which are decided by the initial phases, and that they have the relationship like

φ3(ξ)φ2(ξ)φ1(ξ)=π/2.
(28)

In the SFG process, the left-hand side in Eq. (28) takes the plus sign while it takes the minus plus in the DFG process. Eq. (28) shows that, when Δk =0, the phase relation of the three waves in the TWM process is determined at an arbitrary distance z in the medium. Provided Δk≠0, their relation are plotted in Fig.3, which is obtained by numerical calculating Eq. (1), and the used parameters are the same as in the Fig.1 and 2, but ʌ (corresponding to Δk) is suitably changed. Fig.3 show that the numerical results from Eq. (1) are the same as the derivational results of Eq. (28), and that the derived expressions [i.e., Eqs. (17) and (23)] can accurately predict the phase evolution in the QPM-TWM process.

Fig.3. The phase evolutions of three interacting waves with z under the difference phase mismatching Δk. (a) for SFG, and (b) for DFG

3.3. The effects under the imperfect phase matching

Fig.4. The influence of phase mismatching Δk in the QPM-SFG process, (a) for the normalized electrical field |E 3N| of the sum-frequency wave, (b) for one-forth period (i.e., T/4) which corresponds to the interacting length of |E 3N|, and (c) for the phase evolution of the signal wave.
Fig.5. The influence of phase mismatching Δk in the QPM-DFG process, (a) for the normalized electrical field |E 3N| of the difference-frequency wave, (b) for one-forth period (i.e., T/4) which corresponds to the interacting length of |E 3N|, and (c) for the phase evolution of the signal wave.

4. Optimizations and applications

Although Eq.(1) can straightforward numerically be calculated by the ordinary differential equations (ODE) algorithm (e.g., Runge-Kutta method, Euler method, Gill method, and so on), it does not intuitionisticly provide the influences of the parameters, and is difficult to obtain some optimal results [17

17. X. -M. Liu, H. - Y. Zhang, and Y. -L. Guo, “Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference-frequency generation,” J. Lightwave Technol. 19, 1785–1792 (2001). [CrossRef]

, 24

24. X. -M. Liu, H. - Y. Zhang, and Y. -H Li, “Optimal design for the quasi-phase-matching three-wave mixing,” Opt. Express 9, 631–636 (2001), http://www.opticsexpress.org/oearchive/source/37804.htm. [CrossRef] [PubMed]

]. The analytical solutions [i.e., Eqs. (18) and (24)], however, can offer a clear understanding in the physical property and show the effects of the various parameters. What is more, it can be easy to obtain some optimal solutions. In the following sections, we employ Eqs. (18) and (24) to optimize the wavelength conversion, all-optical switching and the nonlinear phase shift. These optimal results have important help in the all-optical processing.

4.1. Applications in the wavelength conversion

In all-optical wavelength converter based on QPM-DFG, the length l of the nonlinear medium needs to be designed suitably in order that the conversion efficiency (corresponding to the power of the difference-frequency wave) reaches maximum [17

17. X. -M. Liu, H. - Y. Zhang, and Y. -L. Guo, “Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference-frequency generation,” J. Lightwave Technol. 19, 1785–1792 (2001). [CrossRef]

]. If the designed wavelength l is not proper, the conversion efficiency η of the creating wave can not be optimized. Fig.5 (a) & (b) show that when Δk=0 and l=T/4, the conversion efficiency η reaches maximum. With the data of Fig.5, T/4 is about 15.43 mm [see Fig.5(b)], the value of which is dependent on deff , I 1, 2 λ 1, 2 and etc., provided that Δk=0. In the WDM all-optical wavelength converter, not only has the conversion efficiency to be obtained but also the phase of creating wave should be known [2

2. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]

]. Using the “ellipj” and “quad” functions in MATLAB and calculating Eq. (24), we can easily obtain the conversion efficiency η and the phase φ 3 of creating wave, which are shown in Fig.6, for the different signal waves λ 2 In the calculation of Fig.6, all parameters are the same as those in Fig.5, except l=T/4 (=15.43 mm) and ʌ=16.21 μm. Fig.6 shows that the signal λ 2 can be uniformly converted with an extremely wide bandwidth (3-dB bandwidth >90 nm: from ~1510 to ~1600 nm), whereas the phases φ 3 of its converted waves are nonuniform. The nonuniformity of phase can quite easily be interpreted from Eq. (24.b): a different Δk for the various signals under the condition of the fixed grating period ʌ leads to the changes of the phases.

Fig.6. Relationships of the conversion efficiency η and the phase φ 3 of the converted wave with the signal wavelength λ 2.

4.2. Optimization of the nonlinear phase shift

To effectively design all-optical processing devices, the nonlinear phase shift Δφ of signal wave needs to be optimized. From Eqs. (14), (16), (22) and (23), as well as Fig.4(c) and 5(c), it can easily be seen that if Δk is suitably designed, the phase φ 2 of the signal beam will reach the maximum. To obtain the maximum of φ 2, we let the derivative of φ 2 be zero, namely ∂φ 2/∂Δk=0, as

(Δk)0ξΔk2Cf32(ζ,Δk)2Cf22(ζ,Δk)=0.
(29)

After a simple manipulation, it becomes

0ξf32(ζ,Δk)f22(ζ,Δk)+Δk(Δk)0ξf32(ζ,Δk)f22(ζ,Δk)=0.
(30)

Provided that Δk in Eq. (30) can be solved, the corresponded φ 2 will be maximum. Under the ordinary condition, Eq. (30) must be numerically calculated. However, under the small-signal approximation, Eq. (30) can be simplified. From Eqs. (14), (22), (25) and (27), we can obtain

f3=(CF1F2/g)sin(gl),
(31.a)
f2=F21(C2F1/g2)sin2(gl),
(31.b)

for QPM-SFG, and

f3=(F2N/Q)sinh(Ql),
(32.a)
f2=F21+(N2/Q2)sinh2(Ql),
(32.b)

for QPM-DFG, where N= [M 2/E l (0)]·e l e 2F 1 . Substituting Eqs. (31) and (32) into Eq. (30) respectively, we can obtain

(Δk){Δk2g[(1N0N02N0)1N1atan(tan(gl/2)N1)+(1+N0N02N0)1N2atan(tan(gl/2)N2)2atan(tan(gl2))]}=0
(33)

for QPM-SFG, and

(Δk){Δk2Q[(1+N0N′02N0)1N1atan(tanh(Ql/2)N1)+(1N0N′02N0)1N2atan(tanh(Ql/2)N2)ln(tanh(Ql/2)1tanh(Ql/2)+1)]}=0
(34)

for QPM-DFG, where N 0=C 2 F 1/g 2, N1=2N0+1+2N02N0, N0=N 2/Q 2, N2=2N0+12N02N0, N1=2N01+2N′02+N0, N2=2N012N′02+N0.

Eqs. (33) and (34), which escape the integral operation, are the simplification of Eq. (30) under small-signal approximation, and can easily be solved by the numerical calculation. Employing Eqs. (16), (23), (33) and (34), we obtain the maximum phase φ 2_max(l) of signal (see Fig.7), which corresponds to the optimized Δk, under the different medium length l. The corresponded I 2_out /I 2 (I 2_out and I 2 denote the output and input intensity of signal, respectively, in the TWM) comes from Eqs. (14) and (22). All parameters in Fig.7, where figures (a) correspond to Fig.1 and figures (b) correspond to Fig.2, are the same as those in Fig.1 and 2 but the optimized Δk. Fig.7 (a) and (b) show that with the increase of l, ①the maximum phaseφ 2_max of signal for QPM-DFG approximately linearly increases, but |φ 2_max| for QPM-SFG step-like increases; ②the optimized Δk for DFG gradually approaches a certain value after it rapidly decreases at first, but the evolution of Δk for SFG step-like decreases; ③ I 2_out /I 2 >1 for DFG, but I 2_out /I 2 <1 for SFG. These results can be well explained from Eqs. (18) and (24): due to f 3(ζ)/f 2(ζ) >0, the phase of signal always increases with l (see Fig.1 and 2), and then | φ2_max| also increases; because f 2(ζ) >f 2(0) (ζ ≠0) for DFG and f 2( ζ) ≤f 2(0) [even f 2(ζ) →0] for SFG, and f32( ζ)/f22( ζ) of SFG is always more than that of DFG under the conditions of having the same f 2(0) and f 3(ζ), |φ 2_max| of SFG is also always over that of DFG and I 2_out /I 2 of DFG or SFG is more or less the unit respectively. Comparing Fig.7(a) and (b), it is easily found that although the optimized phase | φ 2_max| of SFG is over that of DFG, but I 2_out /I 2 of SFG may be far less than the unit (i.e., I 2_out/I 2 < <1). Thus, to design effectively all-optical switching under the same conditions (e.g., deff , l, P 1 and P 2), we should choose the SFG process rather than the DFG process. This result can interpret that almost all experiments for all-optical switching are based on SFG (or its degenerated case of SHG) 5), not DFG [1

1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

, 22

22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and I. Yokohama, “All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide,” IEEE. Photon. Technol. Lett. 11, 328–330 (1999). [CrossRef]

].

Fig.7. The optimized maximum phase φ 2_max of signal, and the corresponded Δk and I 2_out/I 2 (I 2_out and I 2 denote the output and input intensity of signal, respectively) with the different medium length l. (a) for QPM-SFG, and (b) for QPM-DFG

4.3. Applications in all-optical switching

From Fig.7 and 8, we conclude that by means of increasing l, optimizing Δk and choosing SFG, the intensity I 1 of pump can greatly be decreased when designing the all-optical switching. These results have great benefit to the practical applications.

Fig.8. the minimum intensity I 1 of pump and its corresponding Δk for the “push-pull” all-optical switching (i.e., Δφ=π/2) under I 2_out /I 2>0.8 with the different medium length l. (a) for QPM-DFG, and (b) QPM-SFG.

5. Conclusion

Under the slowly-varying-envelope and the first-order diffraction approximation, the exact analytical expressions for the mutual coupling of three interacting waves in the nonmagnetic quadratic nonlinear medium with the uniform periodic structure are obtained in the initial condition of two input waves in the paper. The properties of these solutions are analyzed, and the physical mechanism of the properties is explained. Both of the results from numerical calculating and analytical solutions show good agreements between them. The intensity evolutions of the interacting waves periodically change while the evolutions of their phases step-like increase (or decrease). In the TWM process with the perfect phase matching, though the conversion efficiency can reach the maximum value, the phase evolution of the three waves are unaffected. To effectively obtain the maximum of the nonlinear phase shift Δφ, quasi-phase matching Δk must be appropriately chosen, which can be obtained from the numerical calculation. The sign of Δk decides the tendency of the phase evolution, as is quite valuable in designing the “push-pull” all-optical switching.

Employing the obtained Eqs. (18), (24) and (30), three examples (i.e., applications in the wavelength conversion and all-optical switching, and optimization of the nonlinear phase shift) are given. The numerical results show that the functions in the all-optical process can obviously be improved by suitable choosing such parameters as Δk and l, and that, by means of increasing l, optimizing Δk and choosing SFG, the intensity I 1 of pump can greatly be decreased when designing the all-optical switching. The results obtained have great significance in designing all-optical wavelength converters and all-optical switchings for the all-optical communication networks.

Acknowledgement:

The authors thank the supports by the National Natural Science Foundation of China (69990540) and the 863 High Technology Research Development Program.

Footnotes

1) Eqs.(4) and (5) have the generality for the TWM processing. Provided ʌ→∞, Δk=k t3-k t2-k t1, as belongs to TWM for the homogeneous medium without the spatial periodical structure. In this situation, the most common procedure for achieving phase-matching is to make use of the birefringence displayed by many crystals [7

7. J. A. Armstrong, N. Bloembergen, and N, J. Ducuing, et al. “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]

, 10

10. R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992), Chap.2.

, 12

12. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap.6-7.

]. Provided Δk=0, θ=φ 3-φ 2-φ 1 and θ′=-φ 3-φ 2+φ 1, as is the exact phase-matching of the TWM processing. Therefore, Eqs.(4) and (5) are equally valid for any method of phase-matching, and the theoretical method in this paper is suitable not only to QPM but also the other phase-matching technique.
2) sn and dn in Eqs. (18) and (24) can be directly utilized by function of “ellipj” in the MATLAB, and the integral operation for the calculations of phase can also be obtained from the function of “quad” in MATLAB after only a few changes.
3) This periodical process is also named as the “cascading” of SFG-DFG in [1

1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

, 22

22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and I. Yokohama, “All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide,” IEEE. Photon. Technol. Lett. 11, 328–330 (1999). [CrossRef]

].
4) The periodical change of the DFG and its reverse process is called as the “cascading” of DFG-SFG.
5) This SFG process is consistent with the “cascading” of SFG-DFG in [1

1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

, 22

22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and I. Yokohama, “All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide,” IEEE. Photon. Technol. Lett. 11, 328–330 (1999). [CrossRef]

].

References and links

1.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

2.

S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]

3.

G. P. Banfi, P. K. Datta, V. Degiorgio, and D. Fortusini, “Wavelength shifting and amplification of optical pulses through cascaded second-order processes in periodically poled lithium niobate,” Appl. Phys. Lett. 73, 136–138 (1998). [CrossRef]

4.

G. Assanto and I. Torelli, “Cascading effects in type II second-harmonic generation: application to all-optical processing,” Opt. Commun. 119, 143–148 (1995). [CrossRef]

5.

J. Leuthold, P. A. Besse, E. Gamper, M. Dulk, S. Fischer, G. Guekos, and H. Melchior,, “All-optical Mach-Zehnder interferometer wavelength converters and switches with integrated data- and control-signal separation scheme,” J. lightwave Technol. 17, 1056–1065 (1999). [CrossRef]

6.

G. S. Kanter and P. Kumar, “Optical devices based on internally seeded cascaded nonlinearities,” IEEE. J. Quantum Elecron. 35, 891–896 (1999). [CrossRef]

7.

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

8.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996). [CrossRef] [PubMed]

9.

M. Asghari, I. H. White, and R. V. Penty, “Wavelength conversion using semiconductor optical amplifiers,” J. Lighteave Technol. 15, 1181–1190 (1997). [CrossRef]

10.

R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992), Chap.2.

11.

M. J. T. Milton, “General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves,” IEEE. J. Quantum Elecron. 28, 739–749 (1992). [CrossRef]

12.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap.6-7.

13.

A. R. C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effect in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995). [CrossRef]

14.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE. J. Quantum Elecron. 29, 2650–2654 (1993). [CrossRef]

15.

A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities-an analytical approach,” Opt. Commun. 124, 184–194 (1996). [CrossRef]

16.

G. D’Aguanno, C. Sibilia, E. Fazio, and M. Bertolotti, “Three-wave mixing in a quadratic material under perfect phase-matching,”Opt. Commun. 142, 75–78 (1997). [CrossRef]

17.

X. -M. Liu, H. - Y. Zhang, and Y. -L. Guo, “Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference-frequency generation,” J. Lightwave Technol. 19, 1785–1792 (2001). [CrossRef]

18.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matchedwith uniform and chirped gratings,“ IEEE. J. Quantum Elecron. 26, 1265–1270 (1990). [CrossRef]

19.

X. -M. Liu and M. -D. Zhang, “Theoretical studies for the special states of the cascaded quadratic nonlinear effects”, J. Opt. Soc. Am. B 18, 1659–1666 (2001). [CrossRef]

20.

G. A. Kehen [USA] and T. M. Kehen, Handbook of Mathematics (Worker Press, Beijing, 1987), (in Chinese).

21.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Dover, 1965), Chap. 16–17.

22.

H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and I. Yokohama, “All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide,” IEEE. Photon. Technol. Lett. 11, 328–330 (1999). [CrossRef]

23.

C. Q. Xu, H. Okayama, and M. Kawahara, “Optical frequency conversions in nonlinear medium with periodically modulated linear and nonlinear optical parameters,” IEEE. J. Quantum Elecron. 31, 981–987 (1995). [CrossRef]

24.

X. -M. Liu, H. - Y. Zhang, and Y. -H Li, “Optimal design for the quasi-phase-matching three-wave mixing,” Opt. Express 9, 631–636 (2001), http://www.opticsexpress.org/oearchive/source/37804.htm. [CrossRef] [PubMed]

OCIS Codes
(060.2630) Fiber optics and optical communications : Frequency modulation
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing
(230.1150) Optical devices : All-optical devices

ToC Category:
Research Papers

History
Original Manuscript: December 21, 2001
Published: January 14, 2002

Citation
Xueming Liu, Hanyi Zhang, and Mingde Zhang, "Exact analytical solutions and their applications for interacting waves in quadratic nonlinear medium," Opt. Express 10, 83-97 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-83


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References

  1. G. I. Stegeman, D. J. Hagan, L. Torner, "?(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions," Opt. Quantum Electron. 28, 1691-1740 (1996). [CrossRef]
  2. S. J. B. Yoo, "Wavelength conversion technologies for WDM network applications," J. Lightwave Technol. 14, 955-966 (1996). [CrossRef]
  3. G. P. Banfi, P. K. Datta, V. Degiorgio, D. Fortusini, "Wavelength shifting and amplification of optical pulses through cascaded second-order processes in periodically poled lithium niobate," Appl. Phys. Lett. 73, 136-138 (1998). [CrossRef]
  4. G. Assanto and I. Torelli, "Cascading effects in type II second-harmonic generation: application to all-optical processing," Opt. Commun. 119, 143-148 (1995). [CrossRef]
  5. J. Leuthold, P. A. Besse, E. Gamper, M. Dulk, S. Fischer, G. Guekos, H. Melchior,, "All-optical Mach-Zehnder interferometer wavelength converters and switches with integrated data- and control-signal separation scheme," J. lightwave Technol. 17, 1056-1065 (1999). [CrossRef]
  6. G. S. Kanter and P. Kumar, "Optical devices based on internally seeded cascaded nonlinearities," IEEE. J. Quantum Elecron. 35, 891-896 (1999). [CrossRef]
  7. J. A. Armstrong, N. Bloembergen N, J. Ducuing, et al. "Interaction between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962). [CrossRef]
  8. A. Kobyakov and F. Lederer, "Cascading of quadratic nonlinearities: an analytical study," Phys. Rev. A 54, 3455-3471 (1996). [CrossRef] [PubMed]
  9. M. Asghari, I. H. White, R. V. Penty, "Wavelength conversion using semiconductor optical amplifiers," J. Lighteave Technol. 15, 1181-1190 (1997). [CrossRef]
  10. R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992), Chap.2.
  11. M. J. T. Milton, "General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves," IEEE. J. Quantum Elecron. 28, 739-749 (1992). [CrossRef]
  12. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap.6-7.
  13. A. R. C. Sibilia, E. Fazio, M. Bertolotti, "Field dependent effect in a quadratic nonlinear medium," J. Mod. Opt. 42, 823-839 (1995). [CrossRef]
  14. C. N. Ironside, J. S. Aitchison, J. M. Arnold, "An all-optical switch employing the cascaded second-order nonlinear effect," IEEE. J. Quantum Elecron. 29, 2650-2654 (1993). [CrossRef]
  15. A. Kobyakov, U. Peschel, F. Lederer, "Vectorial type-II interaction in cascaded quadratic nonlinearities-an analytical approach," Opt. Commun. 124, 184-194 (1996). [CrossRef]
  16. G. D'Aguanno, C. Sibilia, E. Fazio, M. Bertolotti, "Three-wave mixing in a quadratic material under perfect phase-matching,"Opt. Commun. 142, 75-78 (1997). [CrossRef]
  17. X. -M. Liu, H. -Y. Zhang, Y. -L. Guo, "Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference-frequency generation," J. Lightwave Technol. 19, 1785-1792 (2001). [CrossRef]
  18. T. Suhara, H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings," IEEE. J. Quantum Elecron. 26, 1265-1270 (1990). [CrossRef]
  19. X. -M. Liu and M. -D. Zhang, "Theoretical studies for the special states of the cascaded quadratic nonlinear effects", J. Opt. Soc. Am. B 18, 1659-1666 (2001). [CrossRef]
  20. [USA] G. A. Kehen, T. M. Kehen, Handbook of Mathematics (Worker Press, Beijing, 1987), (in Chinese).
  21. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Dover, 1965), Chap. 16-17.
  22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, I. Yokohama, "All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide," IEEE. Photon. Technol. Lett. 11, 328-330 (1999). [CrossRef]
  23. C. Q. Xu, H. Okayama, M. Kawahara, "Optical frequency conversions in nonlinear medium with periodically modulated linear and nonlinear optical parameters," IEEE. J. Quantum Elecron. 31, 981-987 (1995). [CrossRef]
  24. X. -M. Liu, H. -Y. Zhang, Y. -H Li, "Optimal design for the quasi-phase-matching three-wave mixing," Opt. Express 9, 631-636 (2001), <a href="http://www.opticsexpress.org/oearchive/source/37804.htm">http://www.opticsexpress.org/oearchive/source/37804.htm</a> [CrossRef] [PubMed]

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