We describe a theoretical analysis of the nonlinear dynamics of third-harmonic generation ͑ → 3͒ via Kerr ͑ ͑3͒ ͒ nonlinearities in a resonant cavity with resonances at both and 3. Such a doubly resonant cavity greatly reduces the required power for efficient harmonic generation, by a factor of ϳV / Q 2 , where V is the modal volume and Q is the lifetime, and can even exhibit 100% harmonic conversion efficiency at a critical input power. However, we show that it also exhibits a rich variety of nonlinear dynamics, such as multistable solutions and long-period limit cycles. We describe how to compensate for self-and cross-phase modulation ͑which otherwise shifts the cavity frequencies out of resonance͒, and how to excite the different stable solutions ͑and especially the high-efficiency solutions͒ by specially modulated input pulses.
We show that the difficulty of cloaking is fundamentally limited by delay-loss and delaybandwidth/size limitations that worsen as the size of the object to be cloaked increases relative to the wavelength, using a simple model of ground-plane cloaking. These limitations must be considered when scaling experimental cloaking demonstrations up from wavelength-scale objects.PACS numbers: 78.67. Pt, 42.81.Dp We will argue that the problem of cloaking becomes intrinsically more difficult as the size of the object to be cloaked increases compared to the wavelength, and is ultimately limited by fundamental considerations involving the delay-bandwidth and delay-loss products, even for ground-plane cloaks [1][2][3] where bandwidth is not limited by causality constraints. The difficulty is greatest for cloaking objects many wavelengths in diameter (unlike experiments cloaking wavelength-scale objects [3][4][5][6][7][8][9][10][11]), but unfortunately this is the most useful regime for resolving an object of interest. We illustrate these limitations with an idealized one-dimensional (1d) system in which cloaking is much simpler than in three dimensions (3d)-only one incident wave need be consideredbut in which the same limitations appear. We argue that the results and conclusions from this simplified model apply even more strongly to 2d and 3d, and are consistent with recent numerical calculations for 3d cloaks [12]. We conclude that cloaking of human-scale objects is challenging at radio frequencies (RF), while cloaking such objects at much shorter (e.g. visible) wavelengths is rendered impractical by the delay-loss product. Despite the simplicity of this analysis, we arrive at fundamental criteria that may help guide future research on the frontiers of cloaking phenomena.There has been intensive interest in cloaking, both theoretically and experimentally, since the inspiring original papers describing how coordinate transformations, mapped into inhomogeneous materials ("transformation optics") [13] could theoretically render an object invisible [14,15]. Since then, many authors have proposed variations on the original cloaking designs [1,2,5,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], and there have also been attempts at experimental realization [3][4][5][6][7][8][9][10][11]33]. Most theoretical work, however, has considered only lossless materials. In experiments, significant reductions in the scattering cross-section (partial cloaking) have been demonstrated mainly for objects on the scale of the wavelength, with one recent exception [33] discussed below. Two practical concerns about cloaking have been bandwidth limitations and the impact of losses/imperfections, and we argue that these two difficulties become fundamentally more challenging as the size of the object to be cloaked increases.Pendry pointed out that perfect cloaking in air/vacuum is impossible over nonzero bandwidth, because rays traveling around the object must have velocity > c to mimic empty space [14]; this can be interpreted...
By directly simulating Maxwell's equations via the finite-difference time-domain (FDTD) method, we numerically demonstrate the possibility of achieving high-efficiency second harmonic generation (SHG) in a structure consisting of a microscale doubly-resonant ring resonator side-coupled to two adjacent waveguides. We find that ≳ 94% conversion efficiency can be attained at telecom wavelengths, for incident powers in the milliwatts, and for reasonably large bandwidths (Q ∼ 1000s). We demonstrate that in this high efficiency regime, the system also exhibits limit-cycle or bistable behavior for light incident above a threshold power. Our numerical results agree to within a few percent with the predictions of a simple but rigorous coupled-mode theory framework.
We show that cloaking of isolated objects is subject to a diameter-bandwidth product limitation: as the size of the object increases, the bandwidth of good (small cross-section) cloaking decreases inversely with the diameter, as a consequence of causality constraints even for perfect fabrication and materials with negligible absorption. This generalizes a previous result that perfect cloaking of isolated objects over a nonzero bandwidth violates causality. Furthermore, we demonstrate broader causality-based scaling limitations on any bandwidth-averaged cloaking cross-section, using complex analysis and the optical theorem to transform the frequency-averaged problem into a single scattering problem with transformed materials.
We prove that, for arbitrary three-dimensional transformation-based invisibility cloaking of an object above a ground plane or of isolated objects, there are practical constraints that increase with the object size. In particular, we show that the cloak thickness must scale proportional to the thickness of the object being cloaked, assuming bounded refractive indices, and that absorption discrepancies and other imperfections must scale inversely with the object thickness. For isolated objects, we also show that bounded refractive indices imply a lower bound on the effective crosssection.
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