1 May 1998
Optics Communications 150 Ž1998. 235–238
A novel quasi-phase-matching frequency doubling technique Xu Guang Huang, Michael R. Wang
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Department of Electrical and Computer Engineering, UniÕersity of Miami, Coral Gables, FL 33124, USA Received 8 December 1997; revised 27 January 1998; accepted 28 January 1998
Abstract A new quasi-phase-matching technique for efficient second-harmonic generation is reported. It is based on the spatial periodic modulation of the light intensity along the propagation direction, rather than the conventional spatial periodic modulation of the nonlinear optical coefficients. It can be realized by using a novel dual-channel waveguide frequency doubler structure for the desired light intensity distribution. This dual-channel waveguide device has major advantages including very small beam size, high light intensity within long nonlinear-waveguide interaction length, highly efficient second-harmonic generation, ease in fabrication of the nonlinear channel waveguides without any spatially periodic poling, and low waveguide propagation losses. The new quasi-phase-matching technique can also be applied to third-harmonic generation and other nonlinear optics processes. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear optics; Second-harmonic generation; Dual-channel waveguide
Frequency doubling of near-infrared or red laser light beams to blue or UV lights are useful for a variety of applications such as high capacity optical data storage, high-resolution laser pattern generation, and industrial and medical spectroscopic systems. Waveguide-type nonlinear devices are particularly suitable for the frequency doubling of low power light beams, from diode and fiber lasers, due to their long interaction length and strong modal confinement. Conventional quasi-phase-matching ŽQPM. w1–3x, modal dispersion phase matching ŽMDPM. w4–6x, anomalous dispersion phase matching ŽPM. w7x, and Cherenkovtype phase matching w8,9x are four phase matching techniques suitable for second-harmonic generation on nonlinear waveguides with little birefringent effect. Conventional QPM is achieved by spatial periodic modulation of the nonlinear optical coefficients that unavoidably results in significant surface deformations during the spatial periodic poling or growing processes. Conventional QPM based waveguide devices have thus high propagation losses and are difficult to fabricate. In MDPM, the exact PM is achieved by using different guided modes and their associ-
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Corresponding author. E-mail:
[email protected]
ated modal dispersion for the fundamental and the harmonic fields. The poor transverse-mode overlapping integral, however, leads to a low energy conversion efficiency that can be improved by sophisticated multilayer structure for sequentially inverted poled-polymer-layers in a waveguide w10x. Recent introduction of anomalous dispersion PM technique is attractive owing to its simplicity and flexibility. It, however, suffers from serious waveguide absorption losses. The Cherenkov-type PM is a well-known technique for waveguide geometry since any wavelength in the Cherenkov regime can automatically satisfy the longitudinal PM condition. Its spread-beam output has very large diverging angle, very poor mode distribution, and poor coherence. The problem could be improved by small radiation-angle restriction through birefringence compensation of the refractive index dispersion or periodic reversal of the second-order nonlinearity w11x. In this paper, we report on a novel QPM technique for efficient SHG. It is based on a spatial periodic modulation of the light intensity rather than the spatial periodic modulation of the nonlinear optical coefficients. The new QPM can thus be called intensity modulation QPM ŽIMQPM.. The IMQPM condition is that the length of the periodic modulation of the intensity is equal to twice the coherent
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 0 6 1 - 3
X.G. Huang, M.R. Wangr Optics Communications 150 (1998) 235–238
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length of SHG determined by the wave vector mismatching between the fundamental and harmonic waves. We also present a dual-channel waveguide frequency doubler structure with nonlinear optical material to realize this new IMQPM for SHG. With proper design the directional-coupling energy exchange results in a spatial periodic intensity distribution in both channels to achieve the required IMQPM. Without periodic poling or periodic growing, this dual-channel waveguide device may offer lower propagation loss. This new QPM method can be extended to other nonlinear optical processes. The second harmonic field at the end of the nonlinear medium of length L is given by the well-known equation E2 v Ž L . s
iv n2 v c
H0
L
A dual-channel nonlinear waveguide is one of the possible structures for achieving the IMQPM. The dualchannel waveguide is in fact nonlinear channel waveguides with a directional coupler structure. The directional coupling results in the spatial periodic modulation of light intensity. When the period of the intensity modulation satisfies the IMQPM condition, efficient SHG can be achieved. Assume that the input fundamental light is coupled to the channel waveguide a1 at the point z s 0 Žsee Fig. 1., the light intensities of the propagating mode in the two channel waveguides can be evaluated by the coupled mode theory w12x and are given as follows, I1Ž z . s I0 cos 2 Ž k z . ,
Ev2 Ž z . d eff exp Ž iD b z . d z ,
Ž1.
where Ev Ž z . and E2 v Ž z . are the amplitudes of the fundamental and second-harmonic fields, z is the distance along the propagation direction, d eff is the effective nonlinear coefficient for SHG. D b s b 2 v y 2 bv is the propagation constant mismatch between the fundamental and the second harmonic fields caused by dispersion in the nonlinear material. n 2 v is the refractive index of the material at the harmonic wavelength. v and c are respectively the fundamental frequency and the speed of light. It can be seen that the SHG amplitude E2 v Ž L. will increase linearly with the increasing L as: Ži. D b s 0, this is the so-called exact PM condition; or Žii. d eff f 0 or d eff - 0, when D b z s Ž2 m y 1.p ; 2 mp , m s 1,2,3, . . . . This is the conventional QPM condition of the spatial periodic modulation of the nonlinear optical coefficients. The above exact PM Žincluding MDPM, anomalous dispersion PM, and Cherenkov-type PM. and conventional QPM conditions are well-known conditions for waveguide SHG. Beside these conditions, we recognize that the SHG signal will also increase with the propagation distance, if Ev2 Ž z . f 0 when D b z s Ž2 m y 1.p ; 2 mp , or generally, if Ev2 Ž z . is a special periodic distribution of z. Suppose the fundamental light intensity Ev2 Ž z . has the form Ev2 Ž z . s E02 cos 2 Ž p zrL . ,
I2 Ž z . s I0 sin2 Ž k z . ,
Ž5.
where I0 s E02. k is the coupling coefficient between the modes in the two waveguides which depends strongly on the shape of the mode tails in the channel waveguides and the channel separation. k can be determined by
ks
2 h 2 peyp s
b Ž w q 2rp . Ž p 2 q h2 .
,
Ž6.
where p 2 s b 2 y n2s k 02 , tan Ž hw . s
2 ph h2 y p 2
.
h2 s n2g k 02 y b 2 ,
Ž7.
Here, k 0 s vrc, w is the channel width, s is the channel separation, b is the waveguide mode propagation constant in the z-direction, h and p are respectively the propagation constant and the extinction coefficient in the y-direction, n g and n s are the refractive indices of channel waveguides and the substrate, respectively. From Eq. Ž5., it can be seen that the fundamental guided beam intensity does indeed transfer back and forth between the two parallel waveguides as a function of
Ž2.
where L is the spatial period of the light intensity and E0 is the field amplitude. Substituting Eq. Ž2. into Eq. Ž1. and performing the integration, the harmonic amplitude is found to increase with L only when L s 2prD b . In the case of L being an integer number of L, we have < E2 v Ž L . < s
v d eff E02 L 4 n2 v c
.
Ž3.
If we define the coherence length l c of SHG as l c s prD b , the QPM condition requires that the fundamental light intensity is spatially modulated with a period of
L s 2 lc . This is the new IMQPM condition for SHG.
Ž4.
Fig. 1. Schematic of the dual-channel waveguide frequency doubler.
X.G. Huang, M.R. Wangr Optics Communications 150 (1998) 235–238
propagation length. The period L of the light intensity within one of the waveguides is given by L s prk . The periodic coupling can be utilized for IMQPM to enhance the SHG in the dual-channel waveguide structure, provided that L s 2 l c , or Db
ks
eff 2p Ž neff 2 v y nv .
, Ž8. 2 l where nveff and neff 2 v are the effective refractive indices of the fundamental and harmonic guided modes, respectively, and l is the fundamental light wavelength. It is known that the light beam with shorter wavelength has stronger modal confinement. Thus, the inter-channel energy coupling of the harmonic waves can be made much weaker than that of the fundamental waves. With proper waveguide dimensions such as the channel width w, the separation s, and the length L, the lowest harmonic mode in the two waveguides can be made nearly uncoupled within the device length while the fundamental waves are strongly coupled. The harmonic signals, therefore, grow and propagate independently in their waveguides. With a nonlinear waveguide interaction length L being a positive integer number of L, the second-harmonic amplitude at the end of each channel waveguide can be evaluated by Eq. Ž3.. The two second-harmonic waves have identical propagation phases. If a suitable passive Y-junction waveguide is used to bring the two channel waveguide outputs into one, the two second-harmonic waves can be added constructively in the single Y-junction waveguide output to yield its amplitude of s
< E2 v Ž L . < s
v d eff E02 L 2 n2 v c
.
Ž9.
Comparing Eq. Ž9. with the second harmonic amplitudes in the cases of the exact PM and the conventional QPM, the amplitude and effective nonlinear coefficient in our dualchannel structure are half of those of the exact PM and similar to those of the conventional QPM cases, respectively. The SHG efficiency of the new structure is thus high. Without the periodic domain inversion and associated surface deformation, the channel waveguides are expected to have lower propagation loss than conventional QPM based devices. From Eq. Ž6., it can be seen that k increases exponentially with decreasing s. k also depends on b , a nonmonotonic functional dependence, through Eqs. Ž6. and Ž7. in the region n s k 0 - b - n g k 0 . Numerical analysis shows that k is of the order of 10y3 –10y2 mmy1 for the lowest fundamental mode given the waveguide parameters such as D n s n g y n s s 10y2 , l s 1 mm, w s 3 mm, and s s 0.5 mm. For typical index dispersion Ž nv y n 2 v . of the order of 10y2 the corresponding D b is of the order of 10y2 mmy1. Therefore, the IMQPM condition for SHG, i.e. Eq. Ž8., is possible to realize. In addition, our calculation shows that the coupling constant k of the lowest harmonic mode is one order of magnitude smaller than that
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of the fundamental mode. The reverse contribution of the SHG has much lower conversion efficiency than the forward SHG contribution. The forward SHG is thus the dominating effect. In general, decreasing waveguide separation s increases the coupling coefficient k . In the case of very small s, the modes of the directionally coupled waveguides can no longer be described as the modes of each individual waveguide. Instead, the modes must be expressed in terms of the supermodes of the compound structure consisting of both channel waveguides. There is still, however, a similar exchange of power between the two waveguides with the period of L w13x. Owing to the difference of the propagation constants of the two supermodes, the energy couples from one waveguide to the other with the same periodic distribution along the propagation direction as in Eq. Ž5.. In this case the coupling coefficient k is a lot larger than that of the weak directional coupling case described by Eq. Ž6. w14x. Due to the dependence of the coupling strength k on the channel separation s, the fluctuation of the channel separation s caused by device fabrication errors will have an influence on the IMOPM condition, and thus on the SHG intensity. Given the maximum relative deviation of s to be "20%, the calculation from Eq. Ž6. shows the corresponding relative deviation of L of about "14%. Suppose that the fluctuation of L is random, or the distribution of its relative deviation d is a Gaussian function, the ratio P between the SHG amplitudes with and without the fluctuation can be roughly estimated by Ps
0.14
Hy0 .14 H0 L
L
0.14
cos Ž 2pd zrL . d z exp Ž yd 2 . d d
Hy0 .14exp Žyd
2
. d d f 0.96.
There is no significant reduction of the SHG signal within "20% deviation on the channel separation s. The new IMQPM technique based on the spatial modulation of light intensity or field is also suitable to other nonlinear optics processes. For example, for third-harmonic generation ŽTHG. the coherent length l c is given by prŽ b 3 v y 3 bv .. The IMQPM condition for THG is to set the period L of the intensity modulation of the fundamental light to be L s l c . The dual-channel device design for efficient THG is to realize the intensity modulation period L through suitable choices of channel waveguide parameters. To our knowledge, waveguide THG has not yet been demonstrated owing to the difficulty of achieving the required PM or QPM conditions. Our IMQPM technique provides the possibility of high efficiency waveguide THG without any periodic modulation of the third order nonlinearity along the waveguides. In conclusion, a new intensity modulation quasi-phasematching technique for efficient SHG and THG is reported. It is based on the spatial periodic modulation of the light intensity along the propagation direction without
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X.G. Huang, M.R. Wangr Optics Communications 150 (1998) 235–238
artificial modulation of the nonlinear optical property. A novel dual-channel waveguide frequency doubler can achieve the desired light intensity distribution for IMQPM with lower propagation loss than conventional QPM based devices. Because the period of the intensity modulation depends on the waveguide thickness, the channel separation distance, the channel width, and the refractive indices of the two waveguides, one can easily match the period of the intensity modulation to the IMQPM condition. This could make the device implementation for IMQPM based SHG or THG easier. The IMQPM technique should prove valuable to optical data storage, sensing, and spectroscopic applications.
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