Super-Gain Optical Parametric Amplification in Dielectric Micro-Resonators via BFGS Algorithm-Based Non-Linear Programming

Round 1

Reviewer 1 Report

This manuscript theoretically and numerically investigated the high-gain optical parametric amplification in a micro-scale resonator using high-intensity ultrashort pump. The optimum pump frequency was determined using Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization algorithm to maximize the stored electric energy density and polarization density. When the intracavity energy and the polarization density are both high, the input wave can be strongly amplified. In the end, the theoretical results and computational results are compared.

The theoretical derivation and the simulation procedure are complete and clear. However, the reviewer has following questions and suggestion in other aspects need to be addressed by the authors.

1. The reviewer suggests the authors to provide the mathematical definition for gain factor.
2. The review suggests the authors to provide more explanation and references for polarization density in the manuscript.
3. In Section 6, Why is the dielectric constant of the gain medium set to be 12? What kind of material is it corresponding to?
4. In this manuscript, the waves are described using frequency (THz). The reviewer suggests the authors to add the corresponding wavelength to these waves.
5. In this manuscript, the pump frequency ranges from ~100 THz to to ~300 THz corresponding to ~1 µm to ~3 µm. What kind of commercial sources can cover this mid-infrared span?
6. What is the wavelength span for the amplified wave? And what kind of low-loss material can cover this span and be used for the fabrication of the cavity?

Comments for author File: Comments.pdf

Author Response

(1) The mathematical definition for the gain factor is provided at each numerical experiment (pages 12 and 16).

(2) The Polarization Density is further explained with references in section 2 (pages 2 and 3)

(3) The Dielectric constant for numerical experiment 1 is chosen as 12, and the dielectric constant for numerical experiment 2 is chosen as 10. These "background permittivity" values are common for solid dielectric media in the near-infrared and the visible range. Nonetheless these values are just examples, the presented optimization algorithm provides high-gain for any background permittivity.  A very large gain factor is also achievable for smaller values of the background permittivity at the expense of a slight increase in ultrashort pump wave intensity to compensate for the decreased stored energy in the micro-resonator. Similarly, for a gain material with a higher background permittivity, the required pump wave intensity can be lowered in order to achieve the same gain factor.  This is now explained in the manuscript in the discussion section (section 8, page 20). 

(4) The corresponding wavelengths are added for each simulation and for each set of optimal ultrashort pump wave frequencies.

(5) The frequencies of the ultrashort pump wave pulses can be practically tuned from the far IR (Infra-Red) range to the near UV (Ultra-Violet) range. Commercial IR laser sources such as the Neodymium-YAG laser, or the Helium-Neon laser can be used in the near IR range and also in the visible range via frequency doubling. A visible laser light can be used in combination with a near IR laser light to generate a UV laser light via the process of sum frequency generation. A far IR laser light and even a THz laser light can be generated using two near IR laser lights of slightly different frequencies, through difference frequency generation process. Therefore Nd:YAG lasers, Helium-Neon lasers, or Iodine lasers that work in the near infra-red range can be used along with frequency doublers, triplers, and sum & difference frequency generators to cover a wide spectral range (10THz to 1000THz).  This is now explained in the discussion section (page 20)

(6) Wave amplification by nonlinear wave mixing provides wideband amplification unlike the stimulated emission technique (references [1],[4], [21]). The presented BFGS based nonlinear optimization algorithm can be employed at any wavelength ranging from far IR to Near UV (and depending on the practical setting, even beyond). Optical amplification by nonlinear wave mixing is indeed highly sensitive to the resonator loss factor, especially to the polarization damping rate (dielectric absorption loss). Therefore a dielectric material with a relatively lower damping rate should be selected as the interaction material. Preferably a good dielectric with a damping rate of less than 100GHz can effectively cover this spectral span. This is now explained in the discussion section (page 20) and the study that performed such investigation is referenced.

The Authors would like to thank the Reviewer for His/Her valuable comments & suggestions.

Reviewer 2 Report

I recommend accepting this manuscript after a minor revision, summarized below.  It appears that the authors have provided sufficient computational results to support (for publication) their bold claim that: "... super-gain electromagnetic wave amplification can be achieved even in a small microresonator..."  After publication, readers of this paper could be motivated to apply/adapt the authors' design methodology to construct compact, high-gain microphotonic devices.  That would, indeed, be a significant advance, and this paper could become highly cited.

My suggestions for a revision: 

(1) The authors' theoretical model for the time variation of the electric field in a nonlinear dispersive medium is given in Equations (3) and (4).  Although the Laplacian operator is shown in Equation (3), the authors' application of these equations in their manuscript is clearly one-dimensional, and all field quantities are scalars.  I suggest that the authors cite the following paper: 

J. H. Greene and A. Taflove, "General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics," Optics Express, vol. 14, pp. 8305-8310, Sept. 1, 2006.

This paper shows how, using FDTD, Maxwell's full-vector curl equations in 2-D and 3-D can be efficiently time-stepped concurrently with a system of auxiliary differential equations modeling the combined physics of: (a) an instantaneous nonlinearity; (b) a dispersive nonlinearity; and (c) multiple linear dispersions, including Lorentzians.  This is especially useful when modeling real photonic device materials and geometries, the latter including microring and microdisc resonators, where the authors' scalar model is inadequate.  In principle, the Greene-Taflove algorithm could be inserted directly into the authors' Figure 5 flowchart diagram in the yellow block designated "Finite-diference time-domain analysis", thereby expanding the functionality of the authors' technique to arbitrary-shaped 2-D and 3-D microresonators.

(2) The authors' Figures 2, 3, 4, and 6 are very much alike.  Perhaps these could be replaced by a single figure with a more general caption.

Author Response

(1) The mentioned article is cited in the following paragraph (page 21 of the revised manuscript):

" Our main aim in this article is to show that super-gain optical parametric amplification can be achieved in a simple fabry-perot type micro-resonator. For more complicated resonators including microring and microdisc resonators, the one-dimensional model presented in this article must be extended to a two- dimensional model. Our algorithm, which is presented in Figure 4, can be extended to a two-dimensional or a three-dimensional micro-resonator analysis using the technique that is presented in the article referenced in [22]. This article shows how, using FDTD, Maxwell's full-vector curl equations in 2-D and 3-D can be efficiently time-stepped concurrently with a system of auxiliary differential equations modeling the combined physics of: (a) an instantaneous nonlinearity; (b) a dispersive nonlinearity; and (c) multiple linear dispersions, including Lorentzians. In principle, the Greene-Taflove algorithm presented in [22] could be inserted directly into our flowchart diagram given in Figure 4 in the yellow block designated "Finite-diference time-domain analysis", thereby expanding the functionality of our technique to arbitrary-shaped 2-D and 3-D micro-resonators."

(2) The authors believed, although the figures are very much alike, they add visual support to the understanding of the concept and the given numerical experiments, via repetition. Based on the suggestion of the reviewer, figure 4 is removed and the remaining figures (given in sections 6.1 and 6.2) are modified to include the gain medium parameters in order to present a quick visual summary of the simulation parameters, and to reduce the similarity between the stated figures.

The Authors would like to thank the Reviewer for His/Her valuable suggestions.