Optical parametric amplification via non-Hermitian ... - OSA Publishing

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Letter

Vol. 40, No. 21 / November 1 2015 / Optics Letters

Optical parametric amplification via non-Hermitian phase matching R. EL-GANAINY,1,* J. I. DADAP,2

AND

R. M. OSGOOD, JR.2

1

Department of Physics, Michigan Technological University, Houghton, Michigan 49931, USA Microelectronics Sciences Laboratories, Columbia University, New York, New York 10027, USA *Corresponding author: [email protected]

2

Received 1 September 2015; revised 30 September 2015; accepted 1 October 2015; posted 1 October 2015 (Doc. ID 249234); published 29 October 2015

We introduce the notion of dissipative optical parametric amplifiers (DOPA) and demonstrate that, even in the absence of the Hermitian phase-matching condition in these structures, the signal beam can be amplified when the idler mode suffers optical attenuation. We discuss the optical implementation of this concept in waveguide platforms, and we propose different methods to control the optical loss of these configurations only at the wavelength of the idler component. Surprisingly, this spectrally selective dissipation process allows the signal beam to draw more energy from the pump and, as a result, attains net amplification. Similar results also apply if the losses are introduced only to the signal component. This intriguing feature can open new avenues for building long wavelength light sources and parametric amplifiers by using semiconductor planar structures, where Hermitian phase-matching requirements can be difficult to satisfy without adding stringent geometric constraints or relatively complex fabrication steps. © 2015 Optical Society of America OCIS codes: (190.4410) Nonlinear optics, parametric processes; (230.4480) Optical amplifiers; (130.2790) Guided waves. http://dx.doi.org/10.1364/OL.40.005086

Nonlinear optics (NLO) is one of the main pillars of optical science [1]. From laser engineering and biosensing to optical communication and quantum optics, these nonlinear interactions have been at the forefront of many research disciplines. One of the most important applications of NLO is building new light sources having emission frequencies that are not easily accessible using traditional methods. In particular, optical parametric amplifiers (OPAs) and optical parametric oscillators (OPOs) are indispensable tools that have a wide range of applications [1,2]. These devices usually enjoy a wide bandwidth of operation, but also require stringent phase-matching conditions between the different frequency components. This latter constraint can be achieved, for example, by using periodically poled structures or by using birefringent crystals [1–3]. Nonetheless, none of these techniques lends itself to easy integration with semiconductor chip-scale planar technology. 0146-9592/15/215086-04$15/0$15.00 © 2015 Optical Society of America

To overcome this obstacle, new techniques that rely on dispersion engineering in photonic structures have been proposed and employed [4,5]. These approaches, however, often pose tight geometric constraints on the optical waveguide dimensions. Here we propose an altogether new solution to this problem. Our proposal is based on the novel idea of engineering the non-Hermitian properties of the parametric amplifiers. Non-Hermitian optics has gained much attention recently. While any optical system that includes gain or loss is nonHermitian by nature, the introduction of the parity-time (PT) symmetric concept [6] has led to new insights into nonHermitian systems. In particular, PT symmetric Hamiltonians were shown to exhibit real spectra, spontaneous symmetry breaking, and power oscillations among several other intriguing effects [6–8]. This concept was later extended to optical systems [9–11], where the first experimental demonstrations of PT symmetric structures were performed [12,13]. Proposed applications of optical PT symmetric photonics include optical isolators [14] and single mode lasers [15,16]. In addition, several recent works have investigated PT symmetry in different optical setups, such as whispering gallery microcavities and microresonators [17,18]. It is worth noting that most of the characteristics associated with PT symmetric systems can also be observed in general non-Hermitian configurations that exhibit a special type of degeneracy called exceptional points [19,20]. In this Letter, we show that the interplay between nonlinearity and non-Hermiticity can be utilized to build a new generation of parametric amplifiers and oscillators that do not necessarily require the Hermitian phase-matching condition between the real propagation constants of the different frequency components. Downconversion is a nonlinear effect that can be mediated by second-order or higher-order optical nonlinear processes [1]. More specifically, in quadratic nonlinear crystals, this process involves the conversion of an optical pump beam having frequency ωP into two beams called the signal and idler, and having frequencies ωS and ωI . If the initial power of the signal beam is nonzero, the nonlinear interactions lead to difference frequency generation, where the signal beam is amplified and an idler is generated. This process is shown schematically in Fig. 1(a), where the devices that utilize these effects, for

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Fig. 1. (a) Schematic of an OPA process where a quadratic nonlinear interaction can lead to energy transfer from a strong pump beam to a weak signal beam along with the generation of an idler component. Phase-matching between the real propagation constants of the three different beams must be satisfied. (b) Adding dissipation to the idler beam can relax the phase-matching requirement.

example, to amplify weak signals, are the well-known parametric amplifiers. To achieve efficient energy transfer between the different beams, a phase-matching condition must be satisfied. However, this is not always an easy task for most material systems, including the important case of semiconductor planar platforms, since it often leads to more complicated optical designs and fabrication steps. Here we show that, by introducing optical loss for the idler beam only, one can obtain optical gain, even when the propagation constants of the different optical modes do not satisfy the aforementioned phase-matching requirement. To this end, we consider the nonlinear interaction between the pump, signal, and idler beams under slowly varying envelope approximations. Within the coupled mode analysis, this process can be described by [1] dES iκS0 E P E I expiΔβz − γ S E S ; dz dEI iκI0 E P E S expiΔβz − γ I E I ; dz dEP iκP0 E I E S exp−iΔβz; (1) dz where, in Eq. (1), E S;I ;P are the signal, idler, and pump field 0 amplitudes, respectively; κS;I ;P are the coupling coefficients, which are functions of the second-order nonlinearity, as well as the optical frequencies ωS;I ;P ; while Δβ βP − βS βI is the propagation constant mismatch between the three different beams [1]. Finally, γ S;I are the signal and idler linear loss coefficients, z is the propagation distance, and the asterisks denote complex conjugation. We first review the well-known behavior of the optical fields described by Eq. (1) in the absence of any optical loss, i.e., γ S;I . Under these conditions and within the undepleted pump approximation (E P is approximately constant), the solution of Eq. (1) can be expressed as linear superposition of exponentialpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi functions of ffi the form expiΔβz∕2 iλz with λ Δβ∕22 − κ S κI [1], where pffiffiffiffiffiffiffiffiffi 0 E P . If Δβ∕2 is larger than the product κS κI , κ S;I κ S;I the power of both the signal and idler beams remains bounded and oscillates between two extreme values. However, when the pffiffiffiffiffiffiffiffi ffi phase mismatch drops below κ S κI , the value of λ becomes complex, and parametric amplification takes place. Physically, this net gain is a result of power transfer from the pump to the signal and idler. Mathematically, the total power is not conserved, since Eq. (1) is non-Hermitian. Before we proceed, we would like to p remark between the two ffiffiffiffiffiffiffiffiffi that the transition pffiffiffiffiffiffiffiffi ffi regimes Δβ∕2 > κS κI and Δβ∕2 < κ S κ I resembles PT


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spontaneous symmetry breaking. In particular, when Δβ > ffiffiffiffiffiffiffiffiffi p κ S κ I , the system is in the PT-like phase, where the eigenvalues are real and the total ffiffiffiffiffiffiffiffiffi remains bounded. On the other ppower hand, when Δβ∕2 < κ S κ I , the eigenvalues cease to be real and form a complex conjugate pair. Similar to PT systems, under this condition, the intensity profiles associated with the normalized eigenmodes of Eq. (1) become asymmetric. Aside from this interesting analogy, it is clear that a certain level of phase-matching has to be achieved before parametric amplification can take place. Satisfying this condition in semiconductor platforms can be achieved via techniques, such as quasiphase matching or artificial birefringence, which might not be compatible with planar technology and require several additional complex fabrication steps [5]. We now investigate the case where γ S 0, while γ I is finite, and we focus first on the scenario in which the undepleted pump approximation still applies. In the next section, we will discuss how this optical loss can be selectively introduced only for the idler beam without appreciably affecting the pump and signal beams. While the advantage of this additional term might not be obvious at first sight, straightforward analysis shows that the solution of Eq. (1) in this case can be expressed as a superposition of the two exponential functions expiΔβz∕2 iλ z, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where λ iγ I ∕2 Δβ∕2 iγ I ∕22 − κ S κI . The first term of λ accounts for the average linear dissipation. The second term, on the other hand, represents the interplay between the linear loss, phase mismatch, and nonlinear gain, and can contribute extra dissipation or gain depending on its sign. An interesting question now arises: can p these ffiffiffiffiffiffiffiffifficombined effects lead to a net gain, even when Δβ∕2 > κ S κI ? In other words, can these different factors work synergistically and cause the pump to provide energy to either the signal or idler beams at a faster rate than the linear optical dissipation, even when the Hermitian phase-matching condition is violated? To answer − thisp question, ffiffiffiffiffiffiffiffiffi we first consider the normalized eigenvalue λN − part asffi a function of the norλ ∕ κS κI and plot its imaginary p ffiffiffiffiffiffiffiffi Δβ∕ κ κ malized mismatch Δβ N S I and loss coefficient pffiffiffiffiffiffiffiffiffi γ N γ I ∕ κ S κI in Fig. 2(a). Interestingly, we findffi that the imaginary part of λ− in this pffiffiffiffiffiffiffiffi regime (Δβ∕2 > κ S κ I ) is negative. This, in turn, gives rise to parametric amplification. Before we proceed, it is instructive to summarize this straightforward, yet important, finding. For the range of mismatch parameters chosen in Fig. 2(a), and when all linear loss coefficients γ S;I vanish, both the signal and idler beams experience power oscillations between bounded

Fig. 2. (a) Two-dimensional plot of the nonlinear gain parameters of the signal and idler beams represented by Imfλ−N g as a function of the normalized phase mismatch ΔβN and idler loss γ N , as described in the text. The yellow line indicates the locations of the local minima (corresponding to maximum gain) at each vertical cross section. (b) Imfλ−N g as a function of γ N when ΔβN 3 and 5.

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values without any average amplification. On the other hand, and contrary to what one might expect, when the idler mode experiences dissipation γ I ≠ 0 while fixing all other parameters, there is a net energy transfer from the pump to the signal beam, and amplification takes place. This counterintuitive effect may find application in building on-chip planar parametric amplifiers using material systems where the Hermitian phasematching condition is difficult to achieve. The yellow line in Fig. 2(a) indicates the location of the maximum amplification in the parameter space of ΔβN ; γ N . Figure 2(b) depicts two different cross sections of the eigenvalue landscape of Fig. 2(a) [indicated by the dashed white line in Fig. 2(a)] when ΔβN 3 and 5, respectively. We note that, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi when jΔβ∕2j > κS κI and Δβ < 0, λ N λ ∕ κS κI becomes the amplifying term instead of λ−N . To verify our results and gain more insight into the physics of the proposed DOPA, we investigate its dynamical evolution under the relevant initial conditions. To do so, we first scale Eq. (1) according to ξ z∕z 0 and aS;I ;P η−1 S;I ;P E S;I ;P ∕E 0 , where z 0 and E 0 are arbitrary length scale and reference electric field parameters, respectively, while the dimensionless quantity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ηS;I ;P κS;I ;P z 0 E 0 . The normalized equations, expressed in terms of scaled quantities aS;I ;P , take a form similar to that of Eq. (1) with the normalized couplings, phase mismatch, and optical losses given by κ˜S;I ;P ≡ κ˜ ηS ηI ηP, Δβ˜ Δβz 0 , and γ˜S;I γ S;I z 0 , respectively. We first start by considering the undepleted pump approximation, where the normalized pump field aP is treated as a constant throughout the evolution. Figure 3 depicts the normalized powers P S jaS j2 (blue line) and P I jaI j2 (red line) as a function of the normalized propagation distance ξ for four different values of the normalized idler loss γ˜I 0; 0.06; 0.12; 0.24. In all simulations, the Hermitian phase-matching condition is strongly violated with Δβ˜ 3, and the initial conditions were taken to be aS ξ 0 1 and aI ξ 0 0. ˜ P 1 and γ˜S 0. When Additionally, we assumed κa γ˜I 0, the signal and idler powers oscillate around their mean values without any net average amplification as shown in Fig. 3(a). For small values of γ˜I , the systems demonstrate both oscillatory and amplification behavior [see Fig. 3(b)]. Surprisingly, as the value of idler loss coefficient γ˜I increases,

Fig. 3. Evolution of the normalized power of the signal (blue curve starting at one) and the idler (red curve starting at zero) beams as a function of the normalized distance when Δβ˜ 3 under the following different idler loss coefficients: (a) γ˜I 0, (b) γ˜I 0.06, (c) γ˜I 0.12, and (d) γ˜I 0.24. In all cases, the initial conditions were taken to be aS ξ 0 1 and aI ξ 0 0.

Letter amplification starts to dominate the dynamics of the signal beam after a certain propagation distance, as illustrated in Fig. 3(c). Finally, for even larger values of γ˜I , the oscillatory behavior becomes very weak, and the amplification of both signal and idler components completely dominates the dynamics, as observed in Fig. 3(d). Thus, these observations are consistent with our discussion in the previous section. These results are consistent with our discussion about the eigenvalues of λ of Eq. (1). We now treat the general case where the undepleted pump approximation no longer holds. This case can be relevant when the device length allows for the amplified signal power to be comparable to that of the pump beam. Figure 4 depicts the numerical results for this scenario for different design parameters. In our simulations, we assumed that κ˜ 0.1 and aP t 0 10. Oscillatory power transfer between the pump and both signal and idler beams is observed in the absence of loss, as shown in Fig. 4(a). When losses are added to the idler components, the dynamical quantities asymptotically approach the steady state solution: P ssp P ssI 0 and P ssS P P ξ 0 100, as shown in Figs. 4(b)–4(d), for different phase mismatch conditions. We note however that this last result does not imply a complete power transfer between the pump and the signal beams. In fact, the steady state output power of the signal beam, measured in real units, scales with the coefficient κ˜S . In free space, κ˜S varies linearly with signal frequency ωS. In waveguides, κ˜S is still a function of the ωS , though the scaling is no longer linear. This means that, roughly speaking, the total output power will scale with the output beam frequency. Our numerical simulations also indicate that the propagation distance required for achieving that complete power transfer decreases as the idler attenuation factor increases. Finally, we briefly discuss some possible realizations for the dissipative parametric amplifiers proposed in this Letter. This

Fig. 4. Dynamical evolution of normalized power of pump (black curve starting at 100), signal (blue curve starting at one and ending at 100), and idler (red curve starting at zero) beams as a function of the normalized distance when κ˜ 0.1 and under the different conditions indicated on the plots. In all cases, the initial conditions were taken to be aP z 0 10, aS z 0 1, and aI z 0 0. Evidently, adding losses to the idler component alters the nonlinear time evolution dynamics from the oscillatory limiting cycle to a complete power transfer with the system asymptomatically reaching fixed point steady state output. This feature allows for signal amplification, even when the phasematching condition is violated. Note that the normalized amplitude of the idler beam in (a) becomes nearly identical to that of the signal after a short transit time and thus its curve is not visible in the figure.

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Fig. 5. Two different possible realizations for dissipative parametric amplifiers: (a) deep Bragg gratings that efficiently couple the idler frequency to the electromagnetic continuum, resulting in large idler radiation loss without appreciably affecting the pump and signal components; (b) an asymmetric coupler in which the effective index of the auxiliary waveguide (on the right) is equal to that of the idler mode of the main waveguide. Optical loss in the idler beam can be introduced and tuned via a periodic metal film structure, as has been previously demonstrated [12].

can be achieved by using different systems. For instance, material absorption generally varies with wavelength, and it may happen that the required loss condition can be satisfied naturally in certain material systems without any additional effort. The loss factor can also be introduced to one of the frequency components by using nanofabricated structures, such as one of the two schemes shown in Fig. 5. In Fig. 5(a), the pump and signal beams are launched into a waveguide that exhibits a deep Bragg grating on the top [21]. The grating structure is tailored to introduce an efficient coupling between the idler component and the electromagnetic continuum, thus leading to radiation loss. In that case, the pump and signal do not suffer any appreciable loss since their wavelengths do not satisfy the Bragg condition. Another possible implementation that relies on an entirely different concept is shown in Fig. 5(b). Here, the system consists of an asymmetric directional coupler, where the first waveguide serves as the main channel, in which the pump and signal are launched. In this channel, the nonlinear interaction leading to parametric gain actually takes place. The two waveguides are engineered to exhibit the same effective index only at the idler frequency. Optical absorption is then introduced to the idler beam by depositing thin metallic films on the auxiliary waveguide, as shown in Fig. 4(b). The idler extinction factor can then be controlled by choosing the duty cycle of the periodic metal films. We note that this strategy has been implemented previously in studying linear parity-time (PT) symmetric couplers [12]. Additionally, the optical loss in this system also depends on the separation between the two waveguides. Ideally, the PT phase would be preferred, since it will lead to larger losses for the idler beam [12]. Alternatively, one can use other wavelength dependent mechanisms for introducing the optical loss, such as ion implantation, as has been previously demonstrated in silicon waveguide [22]. We note that these same strategies can be used to build long wavelength light sources, an important direction for silicon photonics [23,24], by introducing the optical losses to the signal instead of the idler. It is worth mentioning that similar effects are also expected to persist during other nonlinear interaction processes, such as Raman amplification, four-wave mixing OPA, and nondegenerate second-harmonic generation. For example, the latter case can be implemented in optical nanowires with optimized aspect ratio, where electromagnetic TE and TM modes at the same frequency have completely


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different field profiles and, thus, interact differently with the cladding and surface roughness at the interface between the core and clad layer. We explore all these possibilities elsewhere. It is important to note that in contrast to most previous OPA proposals that focus on achieving the Hermitian phasematching condition [5], the additional design and fabrication steps required to build the structures shown in Fig. 5 are compatible with chip-scale planar semiconductor platforms. In conclusion, we have proposed a new mechanism for achieving parametric gain in waveguide systems. Our scheme is based on introducing spectrally selectively losses only into one of the lower frequency components. We have tested our scheme using nonlinear coupled mode theory, and we have also discussed possible implementations based on single waveguides having Bragg gratings, as well as asymmetric directional couplers with engineered spectral absorption coefficients. This work can open new avenues for new applications, such as low pumping threshold and chip-scale nonlinear light sources, nonlinear wavelength converters, parametric amplifiers, and Raman lasers, to mention a few.

Funding.

Directorate for Engineering (ENG) (1545804).

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