Ultrabroadband midinfrared generation by using group-velocity-dispersion tailoring in a Bragg reflection waveguide for a differencefrequency-generation process Ritwick Das1,* and K. Thyagarajan2 1
Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
2
Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India *Corresponding authors email:
[email protected] Received 19 August 2009; revised 21 September 2009; accepted 25 September 2009; posted 25 September 2009 (Doc. ID 115912); published 12 October 2009
We propose a novel scheme for ultrabroadband midinfrared (mid-IR) generation using quasi-phasematched difference-frequency generation (DFG) in a GaN=Alx Ga1−x N based Bragg reflection waveguide (BRW). By optimally tailoring the phase- and group-velocity dispersion properties of symmetric BRWs, we show that the phase-matching condition for a DFG process could be maintained over a broad range of signal wavelengths. This could lead to generation of an ∼700 nm broad idler close to 3:26 μm wavelength. Since the idea is based on dispersion compensation using photonic bandgap geometry, we can shift the broadband features to any desired spectral region and for any material system within the constraints imposed by the transparency of nonlinear materials. We also investigate the possibility of broadband mid-IR generation using pump sources with broad spectral width. © 2009 Optical Society of America OCIS codes: 230.7390, 230.1480, 230.7405, 190.4975.
1. Introduction
The generation of broadband optical radiation in the midinfrared (mid-IR) is of substantial importance in a variety of applications such as single-shot spectroscopy of large molecules [1], spectral-comb generation for spectroscopy and metrology [2], fundamental studies on the electronic transitions in highly correlated materials [3], high-harmonic generation by optical ion ionization for attosecond pulse generation [2], and as potential seeds for ultrafast optical parametric amplifiers [4]. The most attractive approach for mid-IR generation is based on utilizing the extended mid-IR transparency range for various periodically poled nonlinear crystals, such as LiNbO3 (PPLN), LiTaO3 (PPLT), and KTiOPO4 (PPKTP), in order to realize difference-frequency generation 0003-6935/09/305678-05$15.00/0 © 2009 Optical Society of America 5678
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(DFG) between near-IR emitting lasers in a singlepass configuration [5]. However, due to the narrow phase-matching bandwidth (BW) accommodated by the quasi-phase-matching (QPM) process, most of the schemes exploit the material dispersion properties of nonlinear crystals in order to achieve extended phase matching [6]. For example, Brida et al. [7] have shown that the group-velocity-matching (GVM) properties of MgO:PPLN and MgO:PPLT around 1 μm wavelength could be favorably exploited to generate transform-limited ultrashort (∼30 fs) mid-IR pulses in the 3–4 μm wavelength region. Another technique that is widely being used for broadband optical parametric generation (OPG) is to choose the pump wavelength to be half of the wavelength at which the group-velocity dispersion (GVD) vanishes for a nonlinear crystal, resulting in a few hundreds of nanometer BW around the degenerate wavelength [8]. The latter technique is a highly promising approach for generating ultrabroad radiation in the mid-IR,
but the operation is restricted close to degeneracy and strongly dependent on the availability of suitable pump sources. Therefore the main challenge is to design QPM based geometry for which the GVD of the signal and idler mutually cancel each other, at wavelengths significantly away from degeneracy, for any choice of pump wavelength. This would be particularly advantageous for designing configurations pumped by well-established high-power solidstate laser sources such as Ti:sapphire, Nd:YAG, or Yb fiber lasers. In order to have a better appreciation of the point, we expand the phase-mismatch factor [Δβ ¼ βp − βs − βi − 2π=ΛQPM , where βp , βs , and βi are modal propagation constants at pump (λp ), signal (λs ), and idler (λi ) wavelengths, respectively] for a QPM based DFG process around a central signal wavelength λs ¼ λs0 (corresponding to λi ¼ λi0 ), given by ∂ðΔβÞ Δλ Δβ ¼ ½Δβλ¼λs0 þ ∂λ λ¼λs0 1 ∂2 ðΔβÞ 1 ∂3 ðΔβÞ 2 ðΔλÞ þ ðΔλÞ3 þ 2 6 ∂λ2 λ¼λs0 ∂λ3 λ¼λs0 1 ∂4 ðΔβÞ þ ðΔλÞ4 þ :::: ð1Þ 24 ∂λ4 λ¼λs0
The first term in the expansion of Δβ could be made zero by choosing an appropriate QPM poling period (ΛQPM ) for the nonlinear crystal at λs0 . For a fixed pump, the second term, also known as the group-velocity-mismatch term, can be simplified to ∂ðΔβÞ ∂ 2π ðβ ¼ − þ β Þ ¼ − 2 ðN s − N i Þ; ð2Þ s i ∂λ λ¼λs0 ∂λ λ λ¼λs0 s0 where N s and N i are the group indices at the signal and idler wavelengths, respectively. It is well known that for a parametric downconversion process involving waves of identical polarization, at the degenerate wavelength (λs0 ¼ λi0 ), all the odd-order derivative terms vanish as a consequence of GVM (N s − N i ¼ 0). Therefore the phase-matching BW is primarily determined by the third term in Eq. (1), ∂2 ðΔβÞ ∂λ2 λ¼λs0 2λ2i0 ∂βi ∂2 βs ∂2 βi −2 2 4 ¼ −λs0 λs0 2 þλi0 2 þ : ∂λ λ¼λs0 ∂λ λ¼λi0 λs0 ∂λ2 λ¼λi0 ð3Þ The most common recipe for ultrabroadband parametric generation is to choose a pump wavelength that is close to half the zero GVD wavelength for a nonlinear crystal, leading to simultaneous vanishing of the first four terms in the expansion of Δβ in Eq. (1). Using this technique, Prakash et al. [8] have achieved OPG (signal þ idler) BW of ∼1200 nm at de-
generate wavelength of 1:87 μm in 8 mm long PPLN crystal. For a similar configuration, Kuo et al. [9] generated ultrabroadband OPG in the mid-IR from 4:5 μm to 10:7 μm using a 17:5 mm long orientationpatterned GaAs crystal that was pumped in the 3:1–3:3 μm wavelength range. However, the operation close to degeneracy involving ultrashort pulses may lead to mutual cross talk between the broadband signal and idler if the signal spectrum crosses the degeneracy point. The interference effects arising due to the spectral overlap of different ultrashort pulses in a parametric downconversion process can be minimized by employing a noncollinear geometry, and therefore it could be employed for carrierenvelope-phase stabilization under suitable conditions [10]. This would, however, lead to small conversion efficiency due to angular dispersion of the idler beam and hence, smaller spatial overlap of the interacting modes. Interestingly, it has been demonstrated that broadband mid-IR generation using DFG could also be realized by exploiting the dispersion properties of a few nonlinear crystals that satisfy the GVM condition (N s − N i ¼ 0) for wavelengths spectrally located significantly away from the degeneracy [6,7]. Employing this technique, Cao et al. [11] have generated a 3:7 THz (Δλ ≈ 220 nm) broad midIR idler at 4:2 μm using a 40 mm long PPLN crystal where the BW is essentially limited by the factor in Eq. (3). In another work, Brida et al. [12]. reported the generation of 30 THz (Δλ ≈ 1000 nm) broadband mid-IR idlers at 3:6 μm in a 1:2 mm long PPLT crystal using pump and broadband near-IR signals. The prime objective of the present work is to employ a suitable Bragg reflection waveguide (BRW) based design for simultaneously achieving ∂ðΔβÞ=∂λ λ¼λs0 2 2 and ∂ ðΔβÞ=∂λ at wavelengths significantly λ¼λs0
far from degeneracy. 2.
Bragg Reflection Waveguide Design
The BRWs, originally proposed by Yeh and Yariv [13], are a class of interferometric waveguides with high or low index core and periodically stratified cladding that act as Bragg reflectors for confining energy in the core, as shown in Fig. 1. The choice of BRW for addressing the present issue is essentially motivated by the fact that an additional degree of freedom is afforded by the periodic cladding geometry to appreciably tailor the GVD properties at any desired wavelength of operation [14]. Also, it is to be appreciated that the BRWs exhibit strong phasevelocity dispersion characteristics that can be favorably exploited to achieve GVM [14]. We analyze a DFG process in a planar high-index core symmetric BRW consisting of a periodically poled GaN (PPGaN) core (nx ) to facilitate QPM and periodic layers of Al0:01 Ga0:99 N (n1 ) and Al0:04 Ga0:60 N (n2 ) as Bragg cladding (Fig. 1). The thicknesses of layers with refractive indices nc , n1 , 20 October 2009 / Vol. 48, No. 30 / APPLIED OPTICS
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is relatively weak for having a significant impact on the ∂2 ðΔβÞ=∂λ2 term. Alternatively, a slight shift λ¼λs0
Fig. 1. (Color online) Schematic of the proposed symmetric BRW for a DFG process that has periodically poled core (PPGaN) and periodic cladding of Al0:01 Ga0:99 N (n1 ) and Al0:40 Ga0:60 N (n2 ) layers. The thicknesses of the layers corresponding to refractive indices nc , n1 , and n2 are dc , d1 , and d2 , respectively. The propagation direction is chosen to be the z direction.
and n2 are dc , d1 , and d2 , respectively. The choice of GaN for the present study is inspired by the investigations carried out to establish GaN as a potential nonlinear material for visible and IR operation owing to its wide transparency range extending deep into mid-IR (365 nm–13:6 μm), a high nonlinear coefficient comparable to LiNbO3 and LiTaO3 [15], and the possibility of periodic poling in GaN structures for achieving QPM [16]. Moreover, GaN has an appreciably high optical damage threshold (34 TW=cm2 at 400 nm) and exhibits no photorefractive effect, which implies that GaN based devices could be pumped with high-power visible radiation at room temperature [17,18]. Exploiting the high secondorder nonlinearity of GaN, Chowdhury et al. [17] have experimentally demonstrated efficient secondharmonic generation in PPGaN. It is also to be noted that the use of epitaxially grown III-V semiconductors for the design of optoelectronic devices is particularly advantageous because of the wide possibility of exploiting the mature fabrication technology for designing monolithically integrated components as compared to bulk ionic crystals.
of the idler toward the band edges entails stronger waveguide dispersion to the idler (BRW) mode without any substantial fall in modal confinement. So the prime task is to obtain an optimum λd and core thickness (dc ) that leads to significantly strong phasevelocity dispersion ð∝ ∂βi =∂λi Þ and GVD ð∝ ∂2 βi =∂λ2i Þ for the idler mode so as to efficiently counter the modal dispersion of the corresponding signal. Exploiting this design feature, we have found that, for λd ¼ 3:58 μm and dc ¼ 1:428 μm with ΛQPM ¼ 7:09 μm, the variation of Δβ as a function of signal wavelength exhibits a negligible variation close to phase matching (Δβ ¼ 0) over a signal wavelength range of ∼40 nm (1040–1080 nm), as can be seen in Fig. 2. This is essentially a consequence of the fact that dðΔβÞ=dλs ≃ 0 and d2 ðΔβÞ=dλ2s ≃ 0 simultaneously close to λs0 ¼ 1060 nm, shown in Figs. 3(a) and 3(b). The simultaneous vanishing of these two terms at a desired wavelength of operation is essentially brought about by (i) strong dispersion exhibited by the idler (BRW) mode away from the bandgap center and (ii) an additional degree of freedom afforded by the Bragg cladding geometry to achieve an optimum waveguide dispersion of the idler (BRW) mode. The simulations also reveal that the contribution made by the idler dispersion on the sum of the last two terms on the right-hand side of Eq. (3) enables exact cancellation of the first term (signal) at λs0 ¼ 1060 nm. For the analysis, we have taken the thicknesses of the periodic cladding layers to be d1 ¼ 0:716 μm and d2 ¼ 1:317 μm. We have also found that 12 unit cells of the periodic cladding are sufficient to achieve negligible leakage loss. In order to illustrate the idea, we have calculated the DFG idler power (Pi ) for a given signal power (Ps ) by solving the time-independent coupled-wave equations, under the low pump-depletion approximation and considering material dispersion. It is given by [20]
3. Results and Analysis
We assume a single-frequency pump (λp ¼ 800 nm) and signal (λs0 ¼ 1060 nm) to be transversemagnetic (TM) polarized, fundamental total-internal-reflection (TIR) guided modes in the high index core BRW. The analysis of TIR-guided modes is carried out using a standard transfer-matrix method for one-dimensional multilayer structures [19]. We choose the idler to be a TM-polarized BRW mode. The BRW design is such that the quarter-wave stack (QWS) condition [14] is satisfied at a design wavelength (λd ) longer than the idler wavelength (λi0 ¼ 3:26 μm). This choice is governed by the fact that when the idler is at the center of the bandgap (satisfying the QWS condition), the idler mode dispersion 5680
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Fig. 2. Plot for the variation phase-mismatch factor (Δβ) as a function of signal wavelength (fixed pump wavelength) with dc ¼ 1:428 μm, d1 ¼ 0:716 μm, and d2 ¼ 1:317 μm.
Fig. 4. Plot of idler power as a function of idler wavelength for a 10 mm long BRW.
Fig. 3. (a) Plot for the variation of dðΔβÞ=dλs as a function of signal wavelength, (b) plot for the variation of d2 ðΔβÞ=dλ2s as a function of signal wavelength.
Pi ðz ¼ lÞ ¼
2 4Ps ½γðβÞAp 2 ωs ωi gl sinh ; 2 2 g
ð4Þ
where g2 ¼ 4½γðβÞAp 2 ωs ωi − ðΔβÞ2 , Ap is the incident pump amplitude calculated assuming pump power of 1 mW=μm launched in the TIR guided mode, l is the length of the BRW, and γðβÞ, known as the modal overlap integral, is defined as [20] γðβÞ ¼
ϵ0 2
Z
deff Ep Es Ei dx;
ð5Þ
Ep , W s , and Ei being the power-normalized pump, signal, and idler transverse field profiles, respectively, and deff ¼ ð2=πÞd33 is the effective nonlinear coefficient for GaN. A lower limit of parametric idler power is estimated by taking into account the nonlinearity of only the GaN core. In order to obtain the idler power variation, we kept the pump (λp ¼ 800 nm) fixed and varied the signal wavelength from 1:0 μm to 1:11 μm. For every signal wavelength, we assumed the signal power to be 1 mW in Eq. (4). From the spatial extent of the TIR-guided pump and signal modes in the x direction at λp ¼ 800 nm and λs0 ¼ 1060 nm, we have calculated the pump and signal intensities to be 44:8 kW=cm2 and 42:9 kW=cm2 , respectively. The modal field is assumed to extend up to 1 μm in the y direction. In Fig. 4, we plot the variation of idler power as a function of corresponding idler wavelength for a 10 mm long BRW, calculated using Eq. (4). The curve exhib-
its a 3 dB idler BW of Δvi ¼ 20 THz (Δλi ≈ 700 nm) at the central mid-IR wavelength of λi0 ¼ 3:26 μm, as shown in Fig. 4, which is close to the fundamental absorption bands for trace gases such as methane and formaldehyde [1]. This essentially implies that we can utilize the entire spectral BW of 16 fs transform limited ultrashort sech2 shaped pulse of a suitable power level in the near-IR (close to 1 μm) so as to generate mid-IR radiation above 3 μm. We would like to mention that, for situations involving a broadband pump source, the group-velocity mismatch between the pump and signal is 235 fs=mm and that between pump and idler is 574 fs=mm for the present BRW design. It is also to be noted that the conventional configurations facilitate broadband mid-IR generation only in certain regions of the mid-IR spectrum and are not tunable. The primary advantage offered by the BRW design is that we can shift the broadband features exhibited by the BRW configuration to any region of the mid-IR spectrum with appropriate choice of interacting modes, for any choice of pump wavelength, and for any material system within the constraints of material transparency and possibility of periodic poling. As an example, for the same configuration, when we choose λd ¼ 4:35 μm and dc ¼ 1:712 μm (ΛQPM 8:33 μm), the broadband features are shifted to the mid-IR idler at λi0 ¼ 3:74 μm with Δvi ¼ 18 THz (Δλi ≈ 829 nm) for a central signal wavelength of λs0 ¼ 1:018 μm. Also, the semiconductor (GaN/AlGaN) based design facilitates the fine tuning of the waveguide parameters using current injection. The unavoidable trade-off for the broad phasematching BW is the fall in maximum idler power generated as compared to the conventional TIRbased waveguiding geometry. This is mainly because of the reduction in the modal overlap integral defined in Eq. (5). The modal overlap integral in this case is about 0.60 times that for a case where all the interacting waves are fundamental TIR-guided modes of a conventional TIR waveguide designed for realizing a QPM-DFG process. The fall in modal overlap would essentially be attributed to the presence of a significant fraction of the idler (BRW) modal field in the periodically stratified cladding [13,14]. Nevertheless, 20 October 2009 / Vol. 48, No. 30 / APPLIED OPTICS
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the vanishing of the d2 ðΔβÞ=dλ2s term leads to significantly weaker dependence of the parametric gain BW on the interaction length (l−1=3 ), which could be judiciously exploited by using a long BRW length. Normally, the interaction BW exhibits an inverse square dependence on length (l−1=2 ) for schemes facilitating GVM. Further enhancement in parametric gain could be achieved by suitably controlling the nonlinearity of AlGaN layers during the growth process. We have also investigated the viability of simultaneously achieving dðΔβÞ=dλp ≃ 0 and d2 ðΔβÞ=dλ2p ≃ 0 at a certain pump wavelength (fixed signal), thereby realizing ultrabroad pump acceptance BW features for broadband mid-IR generation. Assuming again that the idler is a BRW mode, it could be observed that spectral variation of the phase-matching condition is essentially dependent on the factor βp − βi instead of −ðβs þ βi Þ, as could be observed in Eqs. (2) and (3). From the calculations, it could be seen that this requires a positive d2 βi =dλ2i so as to counter the role played by the pump dispersion. However, we have found that this condition is difficult to realize in a conventional BRW geometry, leading to nonvanishing d2 ðΔβÞ=dλ2p so far. Nevertheless, by suitable choice of the design wavelength (λd ), we have been able to obtain dðΔβÞ=dλp ¼ 0 and minimum d2 ðΔβÞ=dλ2p ≃ 24:50 μm−3 . This leads to a pump acceptance BW of Δvp ≈ 4:7 THz (Δλp ≈ 9:7 nm), which eventually results in idler BW of Δvi ≈ 4:7 THz (Δλi ≈ 207 nm) at λi ≈ 3:58 μm. We expect that by employing chirped cladding geometries [21] or appropriately chirped QPM gratings [22], the pump acceptance BW can be further enhanced. 4. Conclusion
In conclusion, we have presented a novel BRW design for generating broadband mid-IR radiation via QPM-based DFG process. By optimally tailoring the dispersion properties of the idler, which is a BRW mode, the phase-matching condition could be maintained over a broad spectrum (>20 THz) around 1 μm signal wavelength. This manifests into mid-IR idler BW, Δvi ¼ 20 THz (Δλi ≈ 700 nm) around 3:26 μm wavelength for a 10:0 mm long BRW. We have also investigated the possibility of broadband mid-IR generation using a broadband pump source. The unique and most advantageous features of the design are (i) simultaneous realization of dðΔβÞ=dλs ≃ 0 and d2 ðΔβÞ=dλ2s ≃ 0 at wavelengths substantially away from degeneracy (for a fixed pump) and (ii) for any choice of pump wavelength the possibility to shift the broadband feature to a desired spectral region in the mid-IR. References 1. N. Gayraud, Ł. W. Kornaszewski, J. M. Stone, J. C. Knight, D. T. Reid, D. P. Hand, and W. N. MacPherson, “Mid-infrared gas sensing using a photonic bandgap fiber,” Appl. Opt. 47, 1269–1277 (2008). 2. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000). 5682
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