OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 20428–20438
« Show journal navigation

Adiabatic and diabatic process of sum frequency conversion

Ren Liqing, Li Yongfang, Li Baihong, Wang Lei, and Wang Zhaohua  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 20428-20438 (2010)
http://dx.doi.org/10.1364/OE.18.020428


View Full Text Article

Acrobat PDF (1748 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Based on the dressed state formalism, we obtain the adiabatic criterion of the sum frequency conversion. We show that this constraint restricts the energy conversion between the two dressed fields, which are superpositions of the signal field and the sum frequency field. We also show that the evolution of the populations of the dressed fields, which in turn describes the conversion of light photons from the seed frequency to the sum frequency during propagation through the nonlinear crystal. Take the quasiphased matched (QPM) scheme as an example, we calculate the expected bandwidth of the frequency conversion process, and its dependence on the length of the crystal. We demonstrate that the evolutionary patterns of the sum frequency field’s energy are similar to the Fresnel diffraction of a light field. We finally show that the expected bandwidth can be also deduced from the evolution of the adiabaticity of the dressed fileds.

© 2010 OSA

1. Introduction

In recent years, the crystal’s characteristics can be modulated by periodic or aperiodic electric field. Using this technique in QPM scheme one can solve the problem of the broadband frequencies conversion [2

2. M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30(1), 34–35 (1994). [CrossRef]

5

5. M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004). [CrossRef] [PubMed]

], but with small efficiency. However, Chirped QPM gratings as an efficient technique can achieve the robust efficiency conversion among broad frequencies range. This scheme can be used to manipulate short laser pulse in second harmonic generation [6

6. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22(17), 1341–1343 (1997). [CrossRef] [PubMed]

8

8. D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8(2), 180–198 (2007). [CrossRef]

], difference frequency generation [9

9. G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B 18(4), 534–539 (2001). [CrossRef]

], and in parametric amplification [7

7. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17(2), 304–318 (2000). [CrossRef]

,10

10. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phasematching gratings. I. Practical design formulas,” J. Opt. Soc. Am. B 25(4), 463–480 (2008). [CrossRef]

]. Comparing with the way of describing the interaction between a laser pulse and a two-level atomic system [11

11. L. D. Allen, and J. H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975)

], Haim Suchowski et al. [12

12. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A 78(6), 063821 (2008). [CrossRef]

,13

13. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009). [CrossRef] [PubMed]

] recently proposed the concept of adiabatic frequency conversion, and successfully realized a robust, highly efficient broadband wavelength conversion in the laboratory. Although the proposed physical ideas are very novel and were successfully demonstrated in the lab, they did not illustrate which physical quantities are adiabatic and how the adiabatic restriction affects the sum frequency process.

2. Propagation equation and its adiabatic solution during the sum frequecy

Consider the sum frequency generation process under the first kind phase-matching condition. The two input fields are called as pump field and signal field, respectively. It is assumed that the pump field is so strong that we regard it to be constant during its propagation. Under the undepleted pump approximation, the coupled equation can be simplified as [12

12. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A 78(6), 063821 (2008). [CrossRef]

,13

13. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009). [CrossRef] [PubMed]

]:
{dA1dz=iqA3eiΔkzdA3dz=iq*A1e+iΔkz,
(1)
where A1, A2, and A3 are the amplitudes of the signal field, pump field, and the sum frequency field, respectively, Δk=k1+k2k3 is the phase mismatch, and z is the propagation distance of the light field in the crystal. k1, k2, k3 is the wave number of the signal field, pump field, and the sum frequency field, respectively. q=4πω1ω3k1k3c2χ(2)(ω)A2 is the coupling coefficient, and proportional to the second order susceptibility χ(2)(ω) of the crystal. c is the speed of light in vacuum. ω1 and ω3 are the frequencies of the signal and sum frequency field, respectively. Assume that the energy conservation condition holds, i.e., ω1+ω2=ω3, where ω2 is the frequency of the pump field. If we set q=|q|eiφ(z),A1=A˜1ei[Δkzφ(z)]/2, and A3=A˜3ei[Δkzφ(z)]/2, then Eq. (1) becomes iddzA˜(z)=GA˜(z) with A˜(z)=(A˜1(z)A˜3(z)), G=(ΔK(z)2|q||q|ΔK(z)2) and ΔK(z)=Δk+Δk'zφ'(z). Δk' is the derivative of Δk to z, while φ'(z) is that of φ(z) to z. The eigenvalues or the G are λ1=12ΔK(z)2+4|q|2 and λ3=12ΔK(z)2+4|q|2, respectively. Definingtan(2θ)=2|q|/ΔK(z), one may obtain the rotation matrix R=(cos(θ)sin(θ)sin(θ)cos(θ)) and its inversion R1=(cos(θ)sin(θ)sin(θ)cos(θ)). Do operation about A˜(z) as B˜(z)=(B˜1(z)B˜3(z))=RA˜(z) with R. Borrowing the name of the dressed-pulse fields [17

17. J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72(1), 56–59 (1994). [CrossRef] [PubMed]

], we call B˜1(z) and B˜3(z) as dressed fields. The dressed fields B˜(z) obey the new propagation equation
iddzB˜(z)=GaB˜(z),
(2)
where (λ1iθ'iθ'λ3)=Ga=(R1GRiR1R') with two eigenvalues being λB1=12[(λ1λ3)2+4θ'2]1/2 and λB3=12[(λ1λ3)2+4θ'2]1/2. Comparing with the dressed state formalism [14

14. M. Shapiro, and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003)

16

16. A. Massiah, Quantum Mechanics (North Holland, Amsterdam, 1962), Vol. II.

], the adiabatic criterion about the dressed fields B˜1(z) and B˜3(z) is
|θ'|(λ1λ3)/2,
(3)
where θ' is the derivative of θ to z. Equation (3) is a general expression for the sum frequency conversion. When the adiabatic condition (3) holds, Eq. (2) becomes
iddz(B˜1(z)B˜3(z))=(λ100λ3)(B˜1(z)B˜3(z)).
(4)
Equation (4) means that there is no energy conversion between the dressed fields B˜1(z) and B˜3(z), and the dressed fields are entirely determintated by the boundary condition. Evidently, λB1=λ1, λB3=λ3. Then Eq. (3) can be replaced by |dΔK/dz|(ΔK2+4|q|2)3/2/2|q|, which is identical to that presented by the literature [13

13. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009). [CrossRef] [PubMed]

]. It shows that the dressed fields B˜1(z) and B˜3(z) satisfies the adiabatic restriction only when the variation rate of the phase mismatch with the position z is very slowly.

When the second order susceptibility χ(2)(ω) is a constant, then the coupling coefficient q is real and independent of the position z, i.e., q=|q| with φ(z)=0. Given that the phase mismatch ΔK=Δk is also independent of z, one can easily show the adiabatic condition is satisfied well. Under the adiabatic criterion, the solution of Eq. (4) is B˜1(z)=B˜10Exp(iλ(z)dz), B˜3(z)=B˜30Exp(iλ(z)dz) withλ(z)=12Δk2+4|q|2. This solution manifestly shows that there is not any energy exchange between B˜1(z) and B˜3(z), i.e., they are adiabatic. At the input side, the signal field is A1(0)=A10, then the dressed fields are B˜10=A˜10cosθ and B˜30=A˜10sinθ. In this way, the signal field and the sum frequency field are expressed as
{A1(z)=A10eiΔkz2[cos(ϕ(z))iΔk2λsin(ϕ(z))]A3(z)=iA10|q|λeiΔkz2sin(ϕ(z))
(5)
where ϕ(z)=0zλdz'=λz. Equation (5) shows that energy conversion between the signal field and the sum frequency field occurs, i.e., they are diabatic. WhenΔk=0, Eq. (5) is reduced to |A1(z)|2=|A10|2cos2(ϕ(z)), |A3(z)|2=|A10|2sin2(ϕ(z)). These two equations clearly show that the energy of the signal field can be completely transformed into that of the sum frequency field during the propagation. It also show that the energy oscillates with the propagation distance, which is similar to the oscillation of an electron on the two levels of an atomic system driven by an electronic field. Thus, the transformation efficiency between the signal field and the sum frequency field is maximal. In this case, ϕ(z)=0zλ(z)dz'=qz corresponds to the Rabi frequency of the transition. Therefore, the coupling coefficient q completely governs the energy conversion between the signal field and the sum frequency field. ForΔk0, |A3(z)|2=|A10|24|q|2Δk2+4|q|2sin2(ϕ(z)) shows that the energy transformation efficiency varies with Δk, and cannot reach the optimal value. Therefore, one should amplify the value of q or decrease that of Δk to improve the efficiency of frequency conversion.

3. Diabatic processes and sum frequency conversion

The quasi-phase matching technique allows us to design almost any desired function of the phase mismatched parameter [18

18. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]

]. In particular, we choose just one specific case of adiabatic frequency conversion, namely, the case that implements the scheme using a linearly chirped grating. Here the second order susceptibility of crystal χ(2)(ω) can be expressed as [7

7. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17(2), 304–318 (2000). [CrossRef]

]
χ(2)(ω)=dmExp[iφ(z)].
(6)
In Eq. (6), dm is a constant, and φ(z)=(K0+Dgz)z with Dg being the spatial chirp coefficient. If the frequency of the pump field ω2 is equal to the central frequency Ω0, then that of the signal field can be written as ω1=Ω0+δΩ with a frequency variation δΩ. Thus, the frequency of the sum frequency field is ω3=ω1+ω2=2Ω0+δΩ. Given that the first dispersion approximation of the light field in the crystal, i.e., k1=k10+(Ω0+δΩ)/u1, k2=k20+Ω0/u2, and k3=k30+(2Ω0+δΩ)/u3, with u1,2,3 being the group velocity of the pump field, signal field, and sum frequency field, respectively. Thus, the phase mismatch can be given as ΔK=ΔK0+δvδΩ2Dgz, with ΔK0=Δk0+δuΩ0K0, Δk0=k10+k20k30, and the group velocity mismatch δu=(1/u1+1/u22/u3) and δv=(1/u11/u3). The first dispersion approximation of the light field in the crystal is called as group velocity mismatch effects. Using the quasi-phase-matching scheme, the coupling coefficient becomes q=4πω1ω3k1k3c2dmA2eiφ(z) with |q|=4πω1ω3k1k3c2dmA2.

3.1. Energy conversion evolving with the propagation distance

Due to ΔK(z)=(ΔK0+δvδΩ2Dgz), the adiabatic criterion (3) becomes

|2θ'(z)/(λ1λ3)|=|4|q|Dg/[4|q|2+ΔK(z)2]3/2|1.
(7)

The left side of the inequality (7) reaches the maximal value |Dg/2|q|2| when ΔK(z)=ΔK0+δvδΩ2Dgz=0, which corresponds to z=zd=(ΔK0+δvδΩ)/2Dg. From the dotted line of Fig. 1
Fig. 1 |2θ'(z)/(λ1λ3)| varies with the propagation distance z. The two dotted lines divide the z axis into two regions, with one being called as diabatic region (I) and another one as adiabatic region (II). The dashed line corresponds to the position zd of the energy conversion between the two dressed fields. The crystal’s length is 9mm, and it’s central point corresponds to the origin z=0. The parameters are chosen as: |q|=4/mm, Dg=8/mm2, ΔK0=2/mm, δνδΩ=2/mm.
one see that the adiabatic condition is not valid around zd, and thus there is energy conversion between the dressed fields B˜1 and B˜3. Because zd is a function of the frequency variation δΩ, for the different frequencies of the signal field, the positions of the energy conversion are different. Thus, for the given Dg, the adiabatic restraint is fully satisfied when |q|, while it is not valid for the small value of |q|. In order to well interprets the adiabaticity and the diabaticity, we divide the z axis into two regions, with one being called as diabatic region (I) and another one as adiabatic region (II). In the diabatic region (I) the adiabatic restriction is not valid, while it holds in the adiabatic region (II). The analytical solution of Eq. (1) in the adiabatic region (II) is equivalent to that of Eq. (4), which is readily reached by the same way as obtaining Eq. (5). However, it is difficult to solve the propagation Eq. (1) in the diabatic region (I). Numerically solving the Eq. (1) with the fourth-order Runge–Kutta method, the energy conversion evolution with the propagation distance can be obtained as shown in Fig. 2
Fig. 2 Numerical results for the energy conversion between he signal field and the sum frequency field. The solid line and dashed line correspond to the energy of the signal field and the sum frequency field, respectively. The two dotted lines devide the z axis into two regions, the diabatic region (I) and the diabatic region (II). The parameters are same as that shown in Fig. 1.
. The dashed line represents the energy variation of the signal field, while the solid line corresponds to that of the sum frequency field. It is shown that the energy of the signal field can be completely transformed into that of the sum frequency field. The energy evolution of the sum frequency field is similar to the Fresnel diffraction of a light by a straight edge, while the diabatic point corresponds to the “straight edge”. The strong oscillation occurs at the two “sides” of the “edge”. From what discussed above, z>zd and |Dg/2|q|2|>0.1 are the two essential restriction for achieving efficient sum frequency conversion.

The effects of the diabatic process on the sum frequency field can be further illustrated by using the two dressed fields. Using B˜(z)=RA˜(z), A1=A˜1ei[Δkz2φ(z)], and A3=A˜3ei[Δkz2φ(z)], then the dressed fields can be expressed as
{|B˜1(ω1,ω3,z)|2=cos2θ|A1(ω1,z)|2+sin2θ|A3(ω3,z)|212sin2θ[A1(ω1,z)A3*(ω3,z)eiΔkz+A1*(ω1,z)A3(ω3,z)eiΔkz]|B˜3(ω1,ω3,z)|2=sin2θ|A1(ω1,z)|2+cos2θ|A3(ω3,z)|2+12sin2θ[A1(ω1,z)A3*(ω3,z)eiΔkz+A1*(ω1,z)A3(ω3,z)eiΔkz],
(8)
Both the two equations consist of three parts, the first two parts of each equation shows the influence of sin2θ and cos2θ on the intensity of the signal field and the sum frequency field, while their last part is the same interaction term between A1 and A3. Figure 3(a)
Fig. 3 The relation between energy conversion of the two dressed fields and the diabatic process. (a) Numerical results for the energy evolution of the dressed fields. The solid line shows the energy evolution |B˜1|2, while the dashed line corresponds to |B˜3|2. (b) cos2[θ], sin2[θ] and sin[2θ] are denoted with the solid line attached with △, ○and □ respectively. The two dotted lines divide the z axis into two regions, the diabatic region (I) and the diabatic region (II). The parameters are same as that shown in Fig. 1.
shows the energy transfer between the dressed fields |B˜1|2 and |B˜3|2. The solid line shows the energy evolution of |B˜1|2, while the dashed line corresponds to that of |B˜3|2. Figure 3(b) gives the evolution of cos2θ, sin2θ and sin(2θ), which are denoted with the solid line attached with △, ○and □ respectively. In the adiabatic region (II), the physical process is similar to the total population transfer principle. In this region, there is not energy conversion between the dressed fields. From Fig. 3(b) one see that cos2θ1 and sin2θ0 in the adiabatic region (II), but sin(2θ) plays a role only in the diabatic region (I). Consider the boundary condition A1(z=0)=A1 and A3(z=0)=0, then one may obtain |B˜1(ω1,ω3,z=0)|2=|A1(ω1,0)|2 and |B˜3(ω1,ω3,z=0)|2=0. After the complete transformation from the energy of the signal field into that of the sum frequency field, A1(z)=0 and A3(z) becomes maximal, and thus |B˜1(ω1,ω3,z=l)|2=0, |B˜3(ω1,ω3,z=l)|2=|A3(ω3,l)|2, with l being the length of the crystal. The intersection in Fig. 3(a) shows that the dressed fields are in the diabatic region (I), i.e., they are not adiabatic. At the diabatic poin z=zd, in terms of the definition of tan(2θ)=2|q|/ΔK(z) and ΔK(z)=0, then θ=π4(2n+1) with n being the integer, thus cos2θ=sin2θ=1/2, sin(2θ)=±1. By properly choosing the parameters used in Fig. 1-3, the equation Δkzdφ(zd)=2mπ can be satisfied, where m is also a integer. Thus, Eq. (8) becomes
|A1(ω1,zd)|2+|A3(ω3,zd)|2=A1(ω1,zd)A3*(ω3,zd)+A1*(ω1,zd)A3(ω3,zd).
(9)
Equation (9) indicates that the energy of the dressed fields meet the maximal coherence at the diabatic point, and under this condition the energy is transferred from the signal field to the smu frequency field completely. For different frequency variation δΩ, there will be different diabatic point z=zd. In this way, the restriction Δkzdφ(zd)=2mπ is critical to the total energy conversion between the signal field and the sum frequency field. The eigenvalues of the matrix Ga in Eq. (2) varies nonlinearly, and meet |λB1(zd)|=|λB3(zd)|. This shows that both conservation law of energy and that of momentum hold at the diabatic point during sum frequency. Furthermore, the momentum is minimum at this point.

How to interpret the concepts of the adiabatic and diabatic process? A thorogh understanding will be reached by the analogy of these processes with the atomic transition in a two-level system driven by a laser light field. If we consider that the signal field A1 is similar to the groud state |g of the two-level system, while the sum frequency field A3 resembles the excited state |e, then the pump field in the sum frequency process corresponds to the driving field. In this way, the frequency conversion between the signal field and the sum frequency field has strong similarity to the transition from the groud state |g to the excited state |e. Thus, the dressed states |ϕ+ and |ϕ constructed by |g and |e are similar to the dressed fields B˜1 and B˜3 in this paper. The adiabatic and diabatic process in the two-level atomic system are to the dressed states |ϕ+ and |ϕ, thus that of sum frequcny conversion ocuurs only between the dressed fields B˜1 and B˜3. One can achieve the total population transfer with the theory of the rapid adiabatic passage in the two-level atomic system [15

15. L. P. Yatsenko, N. V. Vitanov, B. W. Shore, T. Rickes, and K. Bergmann, “Creation of coherent superpositions using Stark-chirped rapid adiabatic passage,” Opt. Commun. 204(1-6), 413–423 (2002). [CrossRef]

]. Similarly, one also can achieve the complete frequency conversion during the sum frequency process, as shown in Fig. 2. The energy conversion is closely associated with the diabatic process, as shown in Fig. 2 and Fig. 3. Because the diabatic point is a function of the signal field’s frequency, thus one can achieve the broadband frequency conversion through adjusting the frequency of the signal field. This will be discussed in the following part.

3.2. Energy transfer evolving with the frequency of the signal field

In terms of Eqs. (1) and (8) one may reach the calculated results described in Fig. 5
Fig. 5 The adiabatic and diabatic process during the light propagation accompanied by the energy transfer. (a) The solid line corresponds to the energy evolution of the sum frequency field |A3|2 with δνδΩ, while the dashed line denotes that of the signal field |A1|2 with δνδΩ. (b) The variation for |2θ'/(λ1λ3)| with δνδΩ. (c) The energy transfer between the two dressed fields with δνδΩ. The parameters are chosen as: |q|=2 .5/mm, Dg=8/mm2, ΔK0=2/mm, l=8mm.
. Here, the central position of the crystal is set as the origin of coordinate, and the crystal’s length modulated by the external field is l. Figures 5(a) and 5(b) shows the energy evolution of the sum frequency and the left side of the inequality (7) |2θ'(z)/(λ1λ3)| with δνδΩ, respectively. The dashed- and solid- line denotes the energy evolution of |A˜1(z)|2 and |A˜3(z)|2 respectively. Figure 5(c) dictates the energy conversion between the two dressed fields with δνδΩ. It is shown from Figs. 5(a) and 5(c) respectively that the traces describing |A3|2 and |B˜3(ω1,ω3)|2 are similar. The two intersections of |B˜1(ω1,ω3)|2 and |B˜3(ω1,ω3)|2 coincides with the two diabatic points, as shown in Figs. 5(b) and 5(c). Therefore, the energy transfer between |B˜1(ω1,ω3)|2 and |B˜3(ω1,ω3)|2 determines the energy conversion of |A˜1(z)|2 (dashed line) and |A˜3(z)|2 (solid line) as shown in Fig. 5(a).

Using the adiabatic condition (7) one may calculate the positions of the two diabatic points are δΩ1=[2Dg(l/2)ΔK0]/δv and δΩ2=[2Dg(l/2)ΔK0]/δv respectively. The distance between these two points is obtained as δΩd=|δΩ1δΩ2|=2Dgl/δv, which corresponds to the range of the frequency response for the generated field. This result is same as the outcome determined by the gating period [6

6. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22(17), 1341–1343 (1997). [CrossRef] [PubMed]

8

8. D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8(2), 180–198 (2007). [CrossRef]

,12

12. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A 78(6), 063821 (2008). [CrossRef]

]. However, our result may manifestly shows the energy conversion during the sum frequency. The range of the frequency response δΩd is proportional to the crystal’s length and thus confines the bandwidth of the frequency. Therefore, using the adiabatic and diabatic process may well explain the sum frequency.

4. Conclusion

In this paper we analyzed the adiabatic and diabatic processes of sum frequency conversion. We showed that the adiabatic and diabatic processes are to the dressed fields, which are superpositions of the signal field and the sum frequency field. A rigorous adiabatic criterion was obtained by using the dressed state formalism, rather than by the simply analogy with the two-level sysytem. We also demonstrated that the diabatic process occurs between the dressed fields when the adiabatic criterion is not satisfied, and that the positions of the energy conversion are entirely determined by the diabatic points. We demonstrated that the evolutionary patterns of the sum frequency field’s energy are similar to the Fresnel diffractions of a light field. Based on the intensity expression of the dressed fields, we showed that the dressed fields meet the maximal coherence at the diabatic points when the energy of the signal field is completely transfered to that of the sum frequency field. We calculated the expected bandwidth of the frequency conversion process, and its dependence on the length of the crystal. We finally showed that the expected bandwidth can be also deduced from the evolution of the adiabaticity of the dressed fileds.

In conclusion, we believe that understanding the sum frequency conversion (or other nonlinear optical processes) with the adiabatic and diabatic process is a new angle of view. From this profile, we can get more physical informations, and bring new physical insights into the process of the frequency conversion. This scheme is greatly propitious to the practical applications.

Acknowledgments

This research was supported by Doctoral Dissertations Foundation of Shaanxi Normal University (Grants No.X2009YB09).

References and links

1.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100(18), 183601 (2008). [CrossRef] [PubMed]

2.

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30(1), 34–35 (1994). [CrossRef]

3.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phasematched second-harmonic generation,” IEEE J. Quantum Electron. 30(7), 1596–1604 (1994). [CrossRef]

4.

H. Guo, S. H. Tang, Y. Qin, and Y. Y. Zhu, “Nonlinear frequency conversion with quasi-phase-mismatch effect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 066615 (2005). [CrossRef] [PubMed]

5.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004). [CrossRef] [PubMed]

6.

M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22(17), 1341–1343 (1997). [CrossRef] [PubMed]

7.

G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17(2), 304–318 (2000). [CrossRef]

8.

D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8(2), 180–198 (2007). [CrossRef]

9.

G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B 18(4), 534–539 (2001). [CrossRef]

10.

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phasematching gratings. I. Practical design formulas,” J. Opt. Soc. Am. B 25(4), 463–480 (2008). [CrossRef]

11.

L. D. Allen, and J. H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975)

12.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A 78(6), 063821 (2008). [CrossRef]

13.

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009). [CrossRef] [PubMed]

14.

M. Shapiro, and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003)

15.

L. P. Yatsenko, N. V. Vitanov, B. W. Shore, T. Rickes, and K. Bergmann, “Creation of coherent superpositions using Stark-chirped rapid adiabatic passage,” Opt. Commun. 204(1-6), 413–423 (2002). [CrossRef]

16.

A. Massiah, Quantum Mechanics (North Holland, Amsterdam, 1962), Vol. II.

17.

J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72(1), 56–59 (1994). [CrossRef] [PubMed]

18.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]

OCIS Codes
(190.4360) Nonlinear optics : Nonlinear optics, devices
(230.4320) Optical devices : Nonlinear optical devices
(140.3613) Lasers and laser optics : Lasers, upconversion

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 24, 2010
Revised Manuscript: July 22, 2010
Manuscript Accepted: July 25, 2010
Published: September 10, 2010

Citation
Ren Liqing, Li Yongfang, Li Baihong, Wang Lei, and Wang Zhaohua, "Adiabatic and diabatic process of sum frequency conversion," Opt. Express 18, 20428-20438 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20428


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100(18), 183601 (2008). [CrossRef] [PubMed]
  2. M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30(1), 34–35 (1994). [CrossRef]
  3. K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phasematched second-harmonic generation,” IEEE J. Quantum Electron. 30(7), 1596–1604 (1994). [CrossRef]
  4. H. Guo, S. H. Tang, Y. Qin, and Y. Y. Zhu, “Nonlinear frequency conversion with quasi-phase-mismatch effect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 066615 (2005). [CrossRef] [PubMed]
  5. M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004). [CrossRef] [PubMed]
  6. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22(17), 1341–1343 (1997). [CrossRef] [PubMed]
  7. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17(2), 304–318 (2000). [CrossRef]
  8. D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8(2), 180–198 (2007). [CrossRef]
  9. G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B 18(4), 534–539 (2001). [CrossRef]
  10. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phasematching gratings. I. Practical design formulas,” J. Opt. Soc. Am. B 25(4), 463–480 (2008). [CrossRef]
  11. L. D. Allen, and J. H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975)
  12. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A 78(6), 063821 (2008). [CrossRef]
  13. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009). [CrossRef] [PubMed]
  14. M. Shapiro, and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003)
  15. L. P. Yatsenko, N. V. Vitanov, B. W. Shore, T. Rickes, and K. Bergmann, “Creation of coherent superpositions using Stark-chirped rapid adiabatic passage,” Opt. Commun. 204(1-6), 413–423 (2002). [CrossRef]
  16. A. Massiah, Quantum Mechanics (North Holland, Amsterdam, 1962), Vol. II.
  17. J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72(1), 56–59 (1994). [CrossRef] [PubMed]
  18. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited