Download Deviation from Snell`s law for beams transmitted

January 1, 2002 / Vol. 27, No. 1 / OPTICS LETTERS
7
Deviation from Snell’s law for beams transmitted near the
critical angle: application to microcavity lasers
H. E. Tureci and A. Douglas Stone
Department of Applied Physics, P.O. Box 208284, Yale University, New Haven, Connecticut 06520-8284
Received July 30, 2001
We show that when a narrow beam is incident upon a dielectric interface near the critical angle for total
internal ref lection it will be transmitted into the far field with an angular def lection from the direction
predicted by Snell’s law, because of a phenomenon that we call “Fresnel filtering.” This effect can be quite
large for the parameter range that is relevant to dielectric microcavity lasers. © 2002 Optical Society of
America
OCIS codes: 140.3410, 140.4780, 230.3990, 230.5750, 260.2110, 260.5740.
A promising approach to making high-Q optical microcavities and microlasers is to base them on totally internally ref lected modes of dielectric microstructures.
This approach is currently under intense investigation with resonators and lasers based on a range of
shapes: disks, cylinders, spheres,1 deformed cylinders and spheres,2 – 7 squares,8 and hexagons.9 Many
different mode geometries, e.g., whispering-gallery
modes,1 bow-tie modes,2,3 and triangle6,7 and square8
modes, have been observed in such resonators. Typically these modes correspond to ray trajectories that
are incident upon the boundary of the resonator above
or at the critical angle to achieve adequately high
Q values and may correspond to periodic ray orbits
(POs), which are stable, unstable, or marginally stable.
The natural method for predicting how such a mode
will emit or scatter light is to apply Snell’s law to the
ray orbit and follow the refracted ray into the far field.
For a ray that is incident at the critical angle, this
method would imply emission in the direction tangent
to the boundary at the emission point. However, in
several recent experiments, very large deviations from
this expectation were observed.2,7 We show below
that such observations may be explained as arising
from the angular spread in the resonant mode around
the PO. The variation in Fresnel ref lection with incident angle filters out the angular components, which
are totally internally ref lected and preferentially
transmits those that are far from totally internally
ref lected, leading to a net angular rotation of the
outgoing radiation from the tangent direction. We
call this effect Fresnel filtering (FF).
The effects occurs for a bounded beam with an
arbitrary cross section that is incident from a semiinfinite medium of index n into vacuum, although it
will be quantitatively altered in a resonator because
of the curvature and (or) the finite length of the
boundary. We thus begin with the planar example,
which we solve analytically, before presenting numerical results for quadrupolar asymmetric resonant
cavities (ARCs).10 There is a large body of literature
on ref lection of a beam from a dielectric interface
near or above the critical angle, as the ref lected beam
exhibits the Goos–Hänchen lateral shift as well as
0146-9592/02/010007-03$15.00/0
other nonspecular phenomena.11 However, only a
few of these works addressed the transmission12,13 of
the beam, and they tended to focus on the evanescent
effects in the near f ield; none appear to have identif ied
the FF effect.
For simplicity, we consider a two-dimensional planar
interface that separates two semi-infinite regions with
a relative index of refraction, n. Consider a beam Eia
that is incident from the denser region, with a central
incidence angle ui . We take the beam to be Gaussian,
with a minimum beam waist w (which we use to scale
all lengths) at a distance z0 from the interface. The
basic effect is independent of the nature of the input
beam as long as it is focused and has significant
angular spread. The corresponding Snell emission
angle ue (which is in general complex) is given by
n sin ui 苷 sin ue . Si : 共xi , zi 兲 and Se : 共xe , ze 兲 refer to
coordinates tied to the incident and refracted beams,
respectively (see the inset of Fig. 1). We consider
linearly polarized beams; the corresponding beam
Fig. 1. Angular far-f ield intensity distributions I 共f兲 苷
jEe 共f兲j2 for (dotted curve) critical incidence on a planar interface with n 苷 1.56, D 苷 8.82, z0 苷 5, with
the Gaussian model [(GM) Eq. (4)]. Solid curve, exact
quasi-normal mode with diamond geometry at nk0 r0 艐 90,
for a quadrupolar ARC with e 苷 0.1 and n 苷 1.56.
Dotted –dashed curve, chiral version of diamond resonance
(see text). Inset, coordinates and variables for the GM
calculation.
© 2002 Optical Society of America
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OPTICS LETTERS / Vol. 27, No. 1 / January 1, 2002
fields, Ea , a 苷 TM, TE, then denote the electric (TM)
and the magnetic (TE) f ields normal to the plane of
incidence.
In the angular-spectrum representation,14 the
incident beam in Si coordinates will consist of a
superposition of plane waves of the same frequency
v with a Gaussian distribution of transverse wave
vectors nko s, where s 苷 sin Dui , k0 苷 v兾c0 is the wave
vector in vacuum and Dui is the deviation angle of the
plane-wave component from ui :
∑ µ ∂2
D
E0 D Z `
s2
ds exp 2
Eia 共xi , zi 兲 苷 p
2 p 2`
2
∏
1 iD共sxi 1 czi 兲 ,
(1)
p
where c 苷 1 2 s2 and the dimensionless width parameter D 苷 nk0 w.
The beam on the z . 0 side of the interface in polar coordinates 共r, f兲 attached to the interface (after
refraction) is then given by the integral:
E0 D Z `
Eea 共r, f兲 苷 p
dsTa 共s兲G共s兲
2 p 2`
∑
∏
D
3 exp i
r cos共f 2 ue 2 Due 兲 .
n
(2)
Here Due is obtained from n sin共ui 1 Dui 兲 苷 sin共ue 1
Due 兲, and G 共s兲 is given by
∏
∑ µ ∂2
p
D
G共s兲 苷 exp 2
s2 1 iD 1 2 s2 z0 .
2
(3)
Evaluating this integral in the asymptotic far f ield
共r ! `兲 by use of the saddle-point method, we obtain
our GM of the FF effect:
q
1 2 s02 cos f
E0 D
p
Eea 共f兲 苷 r
D
n2 2 sin2 f
2i r
n
µ
∂
D
3 Ta 共s0 兲G共s0 兲exp i
r ,
(4)
n
where the transmission functions, evaluated at the
1
relevant
q saddle point, s0 共f兲 苷 /n共sin f cos ui 2
2
2
sin ui n 2 sin f 兲, are given by
p
2n n2 2 sin2 f
. (5)
p
p
Ta 关s0 共f兲兴 苷
m n2 2 sin2 f 1 n2 1 2 sin2 f
Here, m 苷 1 共n兲 for a 苷 TE 共TM兲. The relevant saddle
point arises from setting to zero the derivative of the
cosine in the exponent of Eq. (2); this saddle-point
value selects the angular component, which refracts in
the observation direction, f, by Snell’s law. However,
the amplitude factor obtained by Gaussian integration
around the saddle point shifts the maximum of the
outgoing beam away from the Snell direction. As
noted above, the effect occurs for narrow beams with
an arbitrary (non-Gaussian) wave-vector distribution
B共s兲; in such a case, the factor G 共s0 兲 in Eq. (4) is
replaced with B 共s0 兲 (see, e.g., Ref. 8.
Equation (4) gives the angular beam prof ile in the
far f ield, which is none zero for any incident angle ui ,
even ui . uc 苷 sin21 共1兾n兲. The key point is that the
angular maximum of this outgoing beam, fmax, is in
general not at the angle ue predicted by application
of Snell’s law to the central incident beam direction,
ui . Instead, because of the unequal transmission of
the different angular components, the beam direction
is shifted by an angle DuFF that corresponds to less
refraction than expected from Snell’s law. This angular def lection can be quite large for incidence near uc
in typical microcavity resonators; in Fig. 1, the dotted
curve is the result of Eq. (4) for critical incidence, for
which the Snell angle is f 苷 90± but fmax 苷 62±, giving
c
苷 28±. The far-f ield peak shift, DuFF , depends
DuFF
on the beam width, D, and on n; analysis of the stationary phase solution gives the result that, at ui 苷 uc ,
c
艐 共2兾tan uc 兲1兾2 D21兾2 ,
DuFF
(6)
c
which predicts that DuFF
艐 30± for the parameters
of Fig. 1.
Two technical points are in order here: First, there
may be branch-cut contributions to integration along
the steepest descent path, as found in analysis of the
ref lected beam. Such contributions are subdominant
with respect to the f irst-order asymptotic term derived
in Eq. (4). Second, there is another saddle point, s̃0 苷
cos 共ui 兲, which corresponds to angular components with
grazing incidence to the interface. Because the Fresnel transmission factor vanishes for such components,
s̃0 contributes to the integral only at order O 共r 23兾2 兲
and is important only very near f 苷 p兾2. We neglect
this contribution here.
Clearly, the same FF effect will occur in emission
from dielectric resonators, and its magnitude will be
similar to the planar case when the typical radius of
curvature is much larger than w. As an example, we
investigate the effect of FF on the far-f ield emission
pattern of quadrupolar ARCs,2,6,7,10 dielectric cylinders
with a cross section given by r共fw 兲 苷 r0 共1 1 e cos 2fw 兲.
We study the exact numerically generated quasi-bound
TM modes of a resonator with 10% 共e 苷 0.1兲 deformation for different values of the refractive index n,
focusing on resonances based on the stable four-bounce
(diamond) PO. The numerical method used is a variant of the “scattering quantization” for billiards.15 If,
as in this case, the relevant orbit is stable and we
neglect leakage, then it is possible to construct approximate modes that are piecewise Gaussian on each
segment of the PO. One find that the effective
p beam
waist in each segment will scale as D 苷 j nk0 r0 ,
where j is a constant that is dependent only on the stability-matrix eigenvectors of that particular segment
and k0 is the real part of the quantized eigenvalue
of the mode. In Fig. 2(a) we plot one representative
quasi-bound mode at n 苷 1.56 (the index is chosen so
that ui 苷 uc for this mode); the corresponding far-f ield
angular intensity is plotted in Fig. 1.
January 1, 2002 / Vol. 27, No. 1 / OPTICS LETTERS
Fig. 2. (a) Field-intensity plot (gray scale) for a diamond
resonance of the quadrupole at critical incidence for the
points at fw 苷 0, p, calculated numerically at nk0 r0 艐 90,
n 苷 1.56, and e 苷 0.1. Note that there is negligible
emission from the upper and lower bounce points at
fw 苷 690± because they are above the critical angle.
(b) Chiral counterpart of this exact resonance.
9
that the critical angle is scanned through the PO incidence angle ui 艐 39±. To remain as close as possible
to our GM with fixed D, we have chosen the resonances
so that nk0 r0 is approximately constant. In Fig. 3, the
numerical resonance peak is compared with the calculated value fmax from Eq. (4) and with fmax predicted
by Snell’s law. One can see that maximum deviation
takes place at uc .
In conclusion, we have shown that the transmission
direction of a narrow beam through a plane dielectric
interface can be quite different from the direction predicted by application of Snell’s law to the incident beam
direction. This effect is important for predicting the
emission patterns from resonances based on POs in
microcavities. This is true even when the size of the
resonator, r0 , is much larger than the wavelength, and
one might have expected ray optics to be quite good.
Specifically, the effective
beam waist for stable resop
nances scales as D ~ nk0 r0 , so from Eq. (6) the deviac
tion angle at critical incidence uFF
~ 共nk0 r0 兲21兾4 and
hence may be large for nk0 r0 ⬃ 102 103 , as in recent
experiments on semiconductor ARC lasers.2,3,7
We acknowledge helpful discussions with H.
Schwefel, N. Rex, and R. K. Chang. This work was
supported by National Science Foundation grant
DMR-0084501. H. Tureci’s e-mail address is hakan.
[email protected].
References
Fig. 3. Comparison of peak angular far-field values
fmax for varying critical angle uc 苷 sin21 共1兾n兲. Diamonds, exact resonances at nk0 r0 艐 90; solid curve, GM
calculation with D 艐 8.82; dashed curve, Snell’s law
c
prediction: sin fmax 苷 n sin ui , where ui 艐 39±. DuFF
,
deviation from Snell’s law at uc 苷 ui .
One can see the rapid oscillations due to interference, but in Fig. 1 it is clear that the maximum of the
intensity is displaced from f 苷 90± as expected because
of FF. To compare the size of the effect with that of
the GM it is convenient to eliminate the interference
by calculating the chiral resonance, shown in Fig. 2(b).
This is the original resonance with the negative
angular-momentum components projected out, hence
generating a unidirectional beam. Plotting this chiral
resonance in Fig. 1 (dotted – dashed line) gives the
diamond resonance a smooth envelope without the
oscillations. Regarding this chiral resonance as a
beam that is incident at fw 苷 0 on the boundary, with
an angle of incidence ui 艐 39±, we can compare the
FF shift with that of the GM. From Fig. 1, one can
see that the resonance emission peaks at fmax 艐 66±,
whereas the GM gives a similar envelope that is
slightly shifted, with fmax 艐 62±.
To evaluate systematically the FF effect, we have
calculated the far-f ield peaks of the set of diamond
resonances while varying the index of refraction, so
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