Reviewer 1
I'm sending you herewith my comments on the Ignatovich ms.
Because this is a situation where the Ignatovichs and I have a fundamental technical disagreement,
I do not believe I should submit this as a formal review, or at least I should not express any opinion
on whether this ms should or should not be published, leaving that to your editorial judgement and
the opinions of other independent referees.
Rather I am simply expressing my views on the technical matter itself, for whatever
use these may be to you. In addition, I do not think I should do this anonymously,
and I am therefore copying this response to the Ignatovichs also.
The major portion of this ms (19 of the 23 equations) is devoted to a straightforward
analysis of the Fresnel reflection coefficient at a dielectric interface when the
lower-index reflecting medium may have either loss or gain. Although I have not
checked every step of the derivation, this is in essence a "textbook problem" and
I have no reason to believe that their analytical results are not correct.
There is, however, a major disagreement between us at one step in the analysis.
At a point in the general region of Eqs. (11) through (13) in the ms, one must
take the square root of a certain quantity which becomes complex-valued for either
a lossy or gainy reflecting medium; and one must make a choice of which sign one
chooses for this square root.
For the case of a lossless or lossy medium, the conventional choice of sign is
the one which makes the fields decay away exponentially with distance into the
reflecting medium, a condition commonly referred as making the fields in this
region evanescent. The authors of this ms believe that one should make the
same choice for the case of a gainy medium, while I am convinced that the
opposite choice is the physically required or physically meaningful solution in this case.
My belief is that the real physical constraint on the solution in both
lossy and gainy cases is that the solution to be chosen should correspond
to energy or signals which have arrived from the higher-index medium
traveling on outward at a small but finite angle (as the equations
indicate) into the lower-index medium, being attenuated or amplified
as they go. I would note that this is in fact the condition that is
met by the "evanescent" lossy solution, where the direction of energy
flow is slightly outward away from the interface, with the fields
decaying in amplitude with distance as they travel outward because they
are traveling in a lossy rather than a gainy medium. The same condition
should also govern in the gainy case.
This solution seems to me to be mandated by simple causality, as well
as giving physically reasonable answers for many other specific waveguide
and other specific situations I have examined.
Whether this opinion be right or wrong, however, I must also note
that what I believe to be exactly the same analysis and discussion
of Fresnel reflectivity from a gainy or lossy medium as in this ms
has already been presented, and the same conclusions as in this ms
have been argued, in an OSA journal publication some seven years ago, i.e.
J. Fan, A. Dogariu, and L.-J. Wang, "Amplified total internal
reflection," Opt. Expr., vol. 11, pp. 299--308, (2003).
Abstract: Totally internal reflected beams can be amplified
if the lower index medium has gain. We analyze the reflection
and refraction of light, and analytically derive the expression
for the Goos-Hänchen shifts of a Gaussian beam incident on a
lower-index medium, both active and absorptive. We examine the
energy flow and the Goos-Hänchen shifts for various cases. The
analytical results are consistent with the numerical results. For
the TE mode, the Goos-Hänchen shift for the transmitted beam is
exactly half of that of the reflected beam, resulting in a "1/2" rule.
As always, slightly different notation and terminology has been used,
but the basic problem of Fresnel reflection and its analysis is
straightforward, and I believe this earlier publication and the
authors' current ms present essentially identical results. Unless
the present authors can point to genuine differences between this
analysis and theirs, I believe the content and conclusions of this
earlier publication are essentially the same as theirs. (This ms is
also referenced in my recent OPN article, with an implicit indication
that I disagree in exactly similar fashion with its conclusions.)
The current authors also make brief mention of a proposed experimental
test involving a sphere (or cylinder?) embedded in a dye medium. On
this I would comment that a planar waveguide or cylindrical fiber
would seem to me a simpler structure for such a test; that a finite-thickness
gain layer is in fact very different from an unbounded gain layer for such a
test in that it has two reflecting surfaces and will thus display regenerative
effects; and in any case these ideas have already been discussed and even
tested in several publications listed below.
Yours truly, Tony Siegman
REFERENCES
N. Periasamy and Z. Bor, "Distributed feedback laser action in an optical
fiber by evanescent field coupling," Opt. Commun., vol. 39, pp. 298-302, (1981).
N. Periasamy, "Evanescent wave-coupled dye laser emission in optical fibers,"
Appl. Opt., vol. 21, p. 2693, (August 1982).
G. J. Pendock, H. S. Mackenzie, and F. P. Payne, "Dye-Lasers Using Tapered
Optical Fibers," Appl. Opt., vol. 32, pp. 5236--5242, (1993).
H. Fujiwara and K. Sasaki, "Lasing of a microsphere in dye solution," Japan.
J. Appl. Phys., vol. 38, pp. 5101--5104, (1999).
H. J. Moon, Y. T. Chough, and K. An, "Cylindrical microcavity laser based on
the evanescent-wave-coupled gain," Phys. Rev. Lett., vol. 85, pp. 3161--3164, (2000).
Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang,
"Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling," J. Korean
Phys. Soc., vol. 39, pp. 928--931, (2001).
Reviewer 2
I am rejecting this paper based on my comments below.
The paper consists of two sections. Section 1 is a poor treatment of
electromagnetic wave reflection and and refraction at an interface. A better
treatment can be found in any standard optic textbook (e.g., Hecht).
Unfortunately, it has little bearing on solving for the mode structure of a
dielectric sphere. Solving for the mode structure of the resonances of a
dielectric sphere in vacuum is a classic problem in electricity and magnetism,
and the resulting field distributions have been known for some time (e.g.,
Stratton: Electricity and Magnetism.)
There are many mistakes and misleading assumptions used throughout the paper.
For example, in section 1 the authors state that "Maxwell's equations
require continuity of the electric field E_s at the interface..." However,
while the tangential component of the electric field must be continous at
the surface to satisfy the boundary conditions, there is a discontinuity in
the radial component of the electric field at the dielectric boundary. The
authors state that they are limiting themselves to the TE case, however,
the TM modes are the interesting ones for the Whispering Gallery Modes (WGM)
of microspheres, since their electric field vectors are predominantly radial.
Section 2 uses the treatment from Section 1 and applies it to the WGMs of
a dielectric sphere without using the proper treatment for a sphere where
the mode structure would be based on spherical Hankel functions given the
boundary conditions. There are several treatments of this in the
literature already (e.g., PHYSICAL REVIEW A 67, 2003 033806). It is
crucial that the proper mode structure be accounted for with the proper
boundary condition, especially if it is to be applied in a case where
gain will be present. In that case, there will also be time dependencies
that may need to be addressed for a proper treatment.
The only new items in the entire paper consist of a few sentences of
conjecture on the amplification of light using the WGMs of a dielectric
sphere immersed in a gain medium. This conjecture is not based on a
proper treatment of the electromagnetic modes, and there is no testable
prediction of the effect. The paper amounts to a comment on an idea and
needs serious expansion with a far more careful treatment before it could
possibly be considered for any peer reviewed journal.