Unidirectional Coherent Phonon Emission in an Optomechanic Nanobeam Containing Coupled Cavities
Round 1
Reviewer 1 Report
The manuscript “Unidirectional coherent phonon emission in an optomechanic nanobeam containing coupled cavities” by Alexander V. Korovin and coworkers provides a simulation study of nonreciprocal phonon emission in an optomechanic nanobeam system. The paper systematically investigates unidirectional phonon propagation for this two-resonator system via finite-element method, which is supported with analyses of port output, dispersion relation and Poynting vector distribution. Overall, the whole manuscript is well structured with concise and straightforward sentences to introduce the proposed two-phonon-resonator device to readers.
This manuscript can be considered for publication provided that the authors can address my comments below:
1) I suggest the authors to add labels (a), (b), (c) …. for each panel in Figure 1, Figure 7, Figure 8, Figure 10, Figure 11, and Figure 13.
2) I recommend the authors to improve image quality in Figure 7, Figure 8, Figure 9, and Figure 10. Now, they are not clear to see details.
3) Please also describe the technical details for the numerical simulation. For examples, parameters for port, mesh, etc.
4) The authors should also add units for color bar in Figure 11 and Figure 13.
5) I also recommend the authors to cite resources used in section 2. ‘Fundamentals for two levels switching’.
Author Response
Thank you very much for your useful comments that improved our manuscript.
The answer to the reviewer comments:
1) I suggest the authors to add labels (a), (b), (c) …. for each panel in Figure 1, Figure 7, Figure 8, Figure 10, Figure 11, and Figure 13.
The labels have been added.
2) I recommend the authors to improve image quality in Figure 7, Figure 8, Figure 9, and Figure 10. Now, they are not clear to see details.
We corrected quality of Figure 8, Figure 9. And all other images are prepared in the vector format in the word and unfortunately the low quality is the result of pdf generation in the mdpi submission engine.
3) Please also describe the technical details for the numerical simulation. For examples, parameters for port, mesh, etc.
Simulation details have been added (page 5, line 132 in the old version of the text)
4) The authors should also add units for color bar in Figure 11 and Figure 13.
Units have been added
5) I also recommend the authors to cite resources used in section 2. ‘Fundamentals for two levels switching’.
There are no references, because the section 2 is based on the classical one-dimensional wave propagation equation with plane wave’s solution in a homogeneous medium, while other equations were obtained by us.
Author Response File: 
Author Response.docx
Reviewer 2 Report
In this manuscript the authors present a novel scheme for implementing nonreciprocal phonon emission in nanobeams exploiting the coherent superposition between two mechanical sources. Firstly, they provide a simplified 1D model for highlighting the underline concept. Following they reports numerical simulation for a straight phonon waveguide with virtual sources, and finally they report on the results for structured nanobeams with defects, which can support optical and mechanical cavities, and which are already studied by the authors in Ref. [30]. The authors show that a proper control of the phase relation between the phonon emission by two cavities, allows the control of the direction of the mechanical emission.
The manuscript is clearly written and readable. The results are sound, and presented in a pedagogical way. The effectiveness of their approach is well supported. However I do see a weakness in their manuscript. The authors base they scheme on the coherent emission of two phonon sources, which allows the proper interference for "reflecting" or "transmitting" phonons. In fact, the initial toy-model assumes two distant coherent sources. They claim that a possible way to excite the two cavities is either through locally piezoelectric materials, or by exploiting optomechanical effect. For strengthening the manuscript the authors should comment on the possible dephasing mechanisms in the phonon emissions, and how the efficiency of their scheme is affected.
Below, few suggestions that might be considered for a revision.
1. Line 12: what do the authors mean with "time phase shift"? Time delay that in frequency space is a phase shift?
2. Line 65-66: The authors should specify they consider coherent emission for reaching eq. (2) and (3).
3. Line 120: in the caption of figure 2, "Figure 2" is not bold as for the other figures.
4. Figure 2: the authors might consider, for a better readability, to plot their results in a 2D contour plot (P2 as function of "Temporal phase" and "Spatial phase").
5. Line 141: in the caption of figure 3, "Figure 3" is not bold as for the other figures.
6. Line 182: It is not clear whether "emitted" is referred to "phonons" or to "phonon modes". Phonon modes are the frameworks (eigenfunctions) over which quanta of energy might be excited. The authors should consider to revise such sentence.
7. In Figure 7, 8 and 9 the authors report the outputs of the two ports normalised to the results obtained with a straight nanobeam. I miss the point here. I understand the straight nanobeam simulation as a "test bed" for the simulation of structured nanobeams, which, however, cannot be really used for the lack of cavity confinement. Also the "port output difference" range runs on the order of 10^4. Does this amplification have a physical meaning?
Author Response
Thank you very much for your useful comments that improved our manuscript.
The answer to the reviewer comments:
- Line 12: what do the authors mean with "time phase shift"? Time delay that in frequency space is a phase shift?
We have corrected “time phase shift” to “temporal phase shift” to make the term consistent with the rest of the paper. In our article, we are dealing with wave propagation described by the factor , where we call the first term in the exponent argument the temporal phase shift, and the second is related to the spatial phase shift .
- Line 65-66: The authors should specify they consider coherent emission for reaching eq. (2) and (3).
We have changed the sentence
“Let us first consider a single homogeneous nanobeam along the x-axis with two harmonic sources of longitudinal displacement located at x1 and x2 (see Figure 3a) with different phases , and positive real amplitudes (, ) that produce guiding waves of wave vector . In this case, the displacement can be presented in the form:”
to
“To demonstrate the effect of unidirectional phonon emission, let us first consider the simpler case of a single homogeneous nanobeam along the x-axis with two harmonic sources of longitudinal displacement located at the points x1 and x2 (see Figure 3a). Longitudinal displacement sources are characterized by positive real amplitudes (, where is the number of the source) and phases, , (we will call them temporal phase shifts) and produce guiding waves with a wave vector . Assuming the sources are coherent, the displacement in the nanobeam can be presented in the form:”
- Line 120: in the caption of figure 2, "Figure 2" is not bold as for the other figures.
We corrected the typo
- Figure 2: the authors might consider, for a better readability, to plot their results in a 2D contour plot (P2 as function of "Temporal phase" and "Spatial phase").
We followed the referee’s suggestion and presented the figure in a 2D contour plot
- Line 141: in the caption of figure 3, "Figure 3" is not bold as for the other figures.
We corrected the typo
- Line 182: It is not clear whether "emitted" is referred to "phonons" or to "phonon modes". Phonon modes are the frameworks (eigenfunctions) over which quanta of energy might be excited. The authors should consider to revise such sentence.
Since the phonon crystal is finite in the system under consideration, the modes are not completely localized (they are quasi-localized modes or leaky modes), i.e. due to leakage, we can emit phonons into the nanobeam region outside the resonators.
Moreover, we have changed the sentence
“Therefore, to analyze the coherent emission of phonons, we consider the coupling of two phonon modes, emitted from two coupled optomechanical cavities inside a phoxonic nanobeam crystal9-11,30.”
to
“Therefore, to analyze the coherent emission of acoustic waves, we consider the coupling of two phonon modes in two coupled optomechanical cavities inside a phoxonic nanobeam crystal9-11,30.”
- In Figure 7, 8 and 9 the authors report the outputs of the two ports normalised to the results obtained with a straight nanobeam. I miss the point here. I understand the straight nanobeam simulation as a "test bed" for the simulation of structured nanobeams, which, however, cannot be really used for the lack of cavity confinement. Also the "port output difference" range runs on the order of 10^4. Does this amplification have a physical meaning?
In our case, the source is given by a fixed longitudinal displacement provided by external sources, which in the case of systems with resonators requires a larger external power to provide such displacement in contrary to single source of displacement in the straight nanobeam.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
The authors addressed my concerns and modified the statement properly in the manuscript. Now I recommend to publish.
