U.S. patent application number 09/953625 was filed with the patent office on 2002-05-23 for high omnidirectional reflector.
Invention is credited to Fan, Shanhui, Fink, Yoel, Joannopoulos, John D., Winn, Joshua N..
Application Number | 20020060847 09/953625 |
Document ID | / |
Family ID | 22124348 |
Filed Date | 2002-05-23 |
United States Patent
Application |
20020060847 |
Kind Code |
A1 |
Joannopoulos, John D. ; et
al. |
May 23, 2002 |
High omnidirectional reflector
Abstract
A reflector, a method of producing same and a method of creating
high omnidirectional reflection for a predetermined range of
frequencies of incident electromagnetic energy for any angle of
incidence and any polarization. The reflector includes a structure
with a surface and a refractive index variation along the direction
perpendicular to the surface while remaining nearly uniform along
the surface. The structure is configured such that i) a range of
frequencies exists defining a photonic band gap for electromagnetic
energy incident along the perpendicular direction of said surface,
ii) a range of frequencies exists defining a photonic band gap for
electromagnetic energy incident along a direction approximately
90.degree. from the perpendicular direction of said surface, and
iii) a range of frequencies exists which is common to both of said
photonic band gaps. In an exemplary embodiment, the reflector is
configured as a photonic crystal.
Inventors: |
Joannopoulos, John D.;
(Belmont, MA) ; Fan, Shanhui; (Somerville, MA)
; Winn, Joshua N.; (Somerville, MA) ; Fink,
Yoel; (Cambridge, MA) |
Correspondence
Address: |
Samuels, Gauthier & Stevens, LLP
Suite 3300
225 Franklin Street
Boston
MA
02110
US
|
Family ID: |
22124348 |
Appl. No.: |
09/953625 |
Filed: |
September 14, 2001 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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09953625 |
Sep 14, 2001 |
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09634099 |
Aug 8, 2000 |
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09953625 |
Sep 14, 2001 |
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09253379 |
Feb 19, 1999 |
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60075223 |
Feb 19, 1998 |
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Current U.S.
Class: |
359/584 |
Current CPC
Class: |
G02B 2006/12104
20130101; B82Y 20/00 20130101; G02B 27/0012 20130101; G02B 6/1225
20130101; G02B 6/122 20130101; G02B 5/0825 20130101 |
Class at
Publication: |
359/584 |
International
Class: |
G02B 001/10 |
Goverment Interests
[0002] This invention was made with government support under
9400334-DRM awarded by the National Science Foundation. The
government has certain rights in the invention.
Claims
What is claimed is:
1. A method of producing a reflector which exhibits high
omnidirectional reflection for a predetermined range of frequencies
of incident electromagnetic energy for any angle of incidence and
any polarization, comprising: configuring a structure with a
surface and a refractive index variation along the direction
perpendicular to said surface while remaining nearly uniform along
the surface, said structure configured such that i) a range of
frequencies exists defining a photonic band gap for electromagnetic
energy incident along the perpendicular direction of said surface,
ii) a range of frequencies exists defining a photonic band gap for
electromagnetic energy incident along a direction approximately
90.degree. from the perpendicular direction of said surface, and
iii) a range of frequencies exists which is common to both of said
photonic band gaps.
2. The method of claim 1, wherein step iii) comprises a range of
maximum frequencies that exists in common to both of said photonic
band gaps.
3. The method of claim 1, wherein ranges of frequencies exist
defining photonic band gaps for electromagnetic energy incident
along directions between 0.degree. and approximately 90.degree.
from the perpendicular direction of said surface.
4. The method of claim 1, wherein said structure is configured as a
photonic crystal.
5. The method of claim 4, wherein said structure is configured as a
one dimensionally periodic dielectric structure.
6. The method of claim 4, wherein said periodic dielectric
structure comprises periodic units each having two or more
layers.
7. The method of claim 6, wherein said periodic units comprise
layers of silicon and silicon dioxide.
8. The method of claim 6, wherein said periodic units comprise
layers of GaAs and Al.sub.xO.sub.y.
9. The method of claim 6, wherein the zone for high omnidirectional
reflection is 9 2 c = a cos ( - A - 2 A + 2 ) d 1 n 1 + d 2 n 2 - a
cos ( - B - 2 B + 2 ) d 1 n 1 2 - 1 + d 2 n 2 2 - 1 where A n 2 n 1
+ n 1 n 2 , B n 2 n 1 2 - 1 n 1 n 2 2 - 1 + n 1 n 2 2 - 1 n 2 n 1 2
- 1 .
10. The method of claim 6, wherein the layer thickness of materials
of first and second layers with respective indices of refraction
defined with respect to the ambient are chosen such that
.DELTA..omega. is greater than zero.
11. The method of claim 1, wherein said structure is configured
with a continuous variation in refractive index.
12. The method of claim 1, wherein said structure is configured as
an aperiodic dielectric structure.
13. The method of claim 1, wherein said reflector exhibits greater
than 99% reflectivity.
14. A high omnidirectional reflector which exhibits reflection for
a predetermined range of frequencies of incident electromagnetic
energy for any angle of incidence and any polarization, comprising:
a structure with a surface and a refractive index variation along
the direction perpendicular to said surface while remaining nearly
uniform along the surface, said structure configured such that i) a
range of frequencies exists defining a photonic band gap for
electromagnetic energy incident along the perpendicular direction
of said surface, ii) a range of frequencies exists defining a
photonic band gap for electromagnetic energy incident along a
direction approximately 90.degree. from the perpendicular direction
of said surface, and iii) a range of frequencies exists which is
common to both of said photonic band gaps.
15. The method of claim 14, wherein item iii) comprises a range of
maximum frequencies that exists in common to both of said photonic
band gaps.
16. The reflector of claim 14, wherein ranges of frequencies exist
defining photonic band gaps for electromagnetic energy incident
along directions between 0.degree. and approximately 90.degree.
from the perpendicular direction of said surface.
17. The reflector of claim 14, wherein said structure is configured
as a photonic crystal.
18. The reflector of claim 17, wherein said structure is configured
as a one dimensionally periodic dielectric structure.
19. The reflector of claim 17, wherein said periodic dielectric
structure comprises periodic units each having two or more
layers.
20. The reflector of claim 19, wherein said periodic units comprise
layers of silicon and silicon dioxide.
21. The reflector of claim 19, wherein said periodic units comprise
layers of GaAs and Al.sub.xO.sub.y.
22. The reflector of claim 19, wherein the zone for high
omnidirectional reflection is 10 2 c = a cos ( - A - 2 A + 2 ) d 1
n 1 + d 2 n 2 - a cos ( - B - 2 B + 2 ) d 1 n 1 2 - 1 + d 2 n 2 2 -
1 where A n 2 n 1 + n 1 n 2 , B n 2 n 1 2 - 1 n 1 n 2 2 - 1 + n 1 n
2 2 - 1 n 2 n 1 2 - 1 .
23. The reflector of claim 19, wherein the layer thickness of
materials of first and second layers with respective indices of
refraction defined with respect to the ambient are chosen such that
.DELTA..omega. is greater than zero.
24. The method of claim 14, wherein said structure is configured
with a continuous variation in refractive index.
25. The method of claim 14, wherein said structure is configured as
an aperiodic dielectric structure.
26. The method of claim 14, wherein said reflector exhibits greater
than 99% reflectivity.
27. A method of creating high omnidirectional reflection for a
predetermined range of frequencies of incident electromagnetic
energy for any angle of incidence and any polarization, comprising:
providing a structure with a surface and a refractive index
variation along the direction perpendicular to said surface while
remaining nearly uniform along the surface, said structure
configured such that i) a range of frequencies exists defining a
photonic band gap for electromagnetic energy incident along the
perpendicular direction of said surface, ii) a range of frequencies
exists defining a photonic band gap for electromagnetic energy
incident along a direction approximately 90.degree. from the
perpendicular direction of said surface, and iii) a range of
frequencies exists which is common to both of said photonic band
gaps.
28. The method of claim 27, wherein item iii) comprises a range of
maximum frequencies that exists in common to both of said photonic
band gaps.
29. The method of claim 27, wherein ranges of frequencies exist
defining photonic band gaps for electromagnetic energy incident
along directions between 0.degree. and approximately 90.degree.
from the perpendicular direction of said surface.
30. The method of claim 27, wherein said structure is configured as
a photonic crystal.
31. The method of claim 30, wherein said structure is configured as
a one dimensionally periodic dielectric structure.
32. The method of claim 30, wherein said periodic dielectric
structure comprises periodic units each having two or more
layers.
33. The method of claim 32, wherein said periodic units comprise
layers of silicon and silicon dioxide.
34. The method of claim 32, wherein said periodic units comprise
layers of GaAs and Al.sub.xO.sub.y.
35. The method of claim 32, wherein the zone for high
omnidirectional reflection is 11 2 c = a cos ( - A - 2 A + 2 d 1 n
1 + d 2 n 2 - a cos ( - B - 2 B + 2 ) d 1 n 1 2 - 1 + d 2 n 2 2 - 1
where A n 2 n 1 + n 1 n 2 , B n 2 n 1 2 - 1 n 1 n 2 2 - 1 + n 1 n 2
2 - 1 n 2 n 1 2 - 1 .
36. The method of claim 32, wherein the layer thickness of
materials of first and second layers with respective indices of
refraction defined with respect to the ambient are chosen such that
.DELTA..omega. is greater than zero.
37. The method of claim 27, wherein said structure is configured
with a continuous variation in refractive index.
38. The method of claim 27, wherein said structure is configured as
an aperiodic dielectric structure.
39. The method of claim 27, wherein the omnidirectional achieved is
greater than 99%.
40. A method for producing an all dielectric omnidirectional
reflector which exhibits omnidirectional reflection that is greater
than 95% for a predetermined range of frequencies of incident
electromagnetic energy of any angle of incidence and any
polarization comprising: providing a structure with a surface and a
refractive index variation along the direction perpendicular to the
said surface while remaining nearly uniform along the surface said
surface configured such that (i) a range of frequencies exists
defining a reflectivity range which is higher than 99% for EM
energy incident along the perpendicular direction of the said
surface, (ii) a range of frequencies exists defining a reflectivity
range which is higher than 99% for EM energy incident a direction
approximately 90.degree. from the perpendicular direction of the
said surface, and (iii) a range of frequencies exists which is
common to both of said reflectivity ranges.
41. The method of claim 40, wherein the reflectivity is greater
than 96%
42. The method of claim 40, wherein the reflectivity is greater
than 97%
43. The method of claim 40, wherein the reflectivity is greater
than 98%
44. The method of claim 40, wherein the reflectivity is greater
than 99%
Description
[0001] This application claims priority from provisional
application Ser. No. 60/075,223 filed Feb. 19, 1998.
BACKGROUND OF THE INVENTION
[0003] The invention relates to the field of photonic crystals, and
in particular to a dielectric high omnidirectional reflector.
[0004] Low-loss periodic dielectrics, or "photonic crystals", allow
the propagation of electromagnetic energy, e.g., light, to be
controlled in otherwise difficult or impossible ways. The existence
of photonic bandgap in certain photonic crystals has given rise to
the possibility that a photonic crystal can be a perfect mirror for
light from any direction, with any polarization, within a specified
frequency range. Within the frequency range of photonic bandgaps,
there are no propagating solutions of Maxwell's equations inside a
periodic medium. Consequently, a wave-front with a frequency within
the gap which is incident upon the surface of such a crystal would
be completely reflected.
[0005] It is natural to assume that a necessary condition for such
omnidirectional reflection is that the photonic crystal exhibit a
complete three-dimensional photonic band-gap, i.e., a frequency
range within which there are no propagating solutions of Maxwell's
equations. Such a photonic crystal would require periodic
variations in dielectric constant in all three dimensions. These
crystals, if designed for infrared or optical light, are difficult
to fabricate, since the spatial periods must be comparable to the
wavelength of operation. This is the reason why, despite heroic
experiments involving advanced lithographic methods or
self-assembling microstructures, most of the proposals for
utilizing photonic crystals are in early stages of development.
SUMMARY OF THE INVENTION
[0006] It is therefore an object of the invention to provide a
dielectric structure that acts as a perfect mirror by exhibiting
high omnidirectional reflection of energy regardless of
polarization and incident angle.
[0007] It is a further object of the invention to provide a
one-dimensionally periodic photonic crystal structure, such as
multi-layer films, that can exhibit complete reflection of
radiation in a given frequency range for all incident angles and
polarizations.
[0008] Accordingly, the invention provides a reflector, a method of
producing same and a method of creating high omnidirectional
reflection for a predetermined range of frequencies of incident
electromagnetic energy for any angle of incidence and any
polarization. The reflector includes a structure with a surface and
a refractive index variation along the direction perpendicular to
the surface while remaining nearly uniform along the surface. The
structure is configured such that i) a range of frequencies exists
defining a photonic band gap for electromagnetic energy incident
along the perpendicular direction of said surface, ii) a range of
frequencies exists defining a photonic band gap for electromagnetic
energy incident along a direction approximately 90.degree. from the
perpendicular direction of said surface, and iii) a range of
frequencies exists which is common to both of said photonic band
gaps. In one exemplary embodiment the reflector is configured as a
photonic crystal.
BRIEF DESCRIPTION OF THE DRAWINGS
[0009] FIG. 1 is a schematic block diagram of an exemplary
embodiment of a high omnidirectional reflector in accordance with
the invention;
[0010] FIG. 2 is a graph of the first three bands of an exemplary
multilayer film quarter-wave stack;
[0011] FIG. 3 is a graph showing the projected band structure for a
quarter-wave stack with n.sub.1=1, n.sub.2=2;
[0012] FIG. 4 is a graph showing the projected band structure for a
quarter-wave stack with the same ratio n.sub.2/n.sub.1=2 and
n.sub.1=1.7, n.sub.2=3.4 (.alpha.=1.7), and d.sub.1=0.67a,
d.sub.2=0.33a, where a is the period;
[0013] FIG. 5 is a graph of the calculated spectra for a
quarter-wave stack of ten films (n.sub.1=1.7, n.sub.2=3.4) for
three angles of incidence; and
[0014] FIG. 6 is a contour plot of the range-midrange ratio for the
frequency range of high omnidirectional reflection, as n.sub.1 and
n.sub.2/n.sub.1 are varied, for the maximizing value of
d.sub.1/a.
DETAILED DESCRIPTION OF THE INVENTION
[0015] FIG. 1 is a schematic block diagram of an exemplary
embodiment of a high omnidirectional reflector 100 in accordance
with the invention. The reflector 100 is a one-dimensionally
periodic photonic crystal having an index of refraction that is
periodic in the y-coordinate, perpendicular to a surface 101, and
consists of a repeating stack of dielectric slabs 102, 104, which
alternate in thickness from d.sub.1 to d.sub.2 (in the illustrated
embodiment d.sub.1=d and d.sub.2=1-d) and an index of refraction
from n.sub.1 to n.sub.2. In the illustrated embodiment, d.sub.1 and
d.sub.2 are assumed to be in the unit of period a. Only a few
periods of such a periodic system are illustrated. For a
quarter-wave stack, n.sub.1d.sub.1=n.sub.2d.sub.2. The stacks are
fabricated in a conventional manner on a substrate 106, e.g.,
silicon.
[0016] FIG. 1 also shows two orthogonal polarizations of incident
light. An s-polarized wave has an electric field E perpendicular to
the plane of incidence and a magnetic field B parallel to the plane
of incidence. A p-polarized wave has an electric field parallel to
the plane of incidence and a magnetic field perpendicular to the
plane of incidence. Since the medium is periodic in the y-direction
(discrete translational symmetry) and homogeneous in the x- and
z-directions (continuous translational symmetry), the
electromagnetic modes can be characterized in Bloch form by a wave
vector k. In particular, ky is restricted to the first Brillouin
zone -.pi./a<k.sub.y<.pi./a, and k.sub.x and k.sub.z are
unrestricted. One can suppose that k.sub.z=0, k.sub.x.gtoreq.0 and
n.sub.2>n.sub.1 without loss of generality. The allowed mode
frequencies .omega..sub.n for each choice of k constitute the band
structure of the crystal. The continuous functions
.omega..sub.n(k), for each n, are the photonic bands.
[0017] FIG. 2 is a graph of the first three bands of an exemplary
multilayer film quarter-wave stack with n.sub.1=1, n.sub.2=2, as a
function of k.sub.y, for the special case k.sub.x=0 (normal
incidence). The thicknesses were chosen to be d.sub.1=0.67 and
d.sub.2=0.33. For k.sub.x=0, there is no distinction between s- and
p-polarized waves. There is a wide frequency gap between the first
and second bands. This splitting arises from the destructive
interference of the waves which are transmitted and reflected at
each interface. It will be appreciated that the frequency has been
expressed in units of c/a, where c is the speed of light in the
ambient medium and a=d.sub.1+d.sub.2.
[0018] Any one-dimensional photonic crystal, as defined by a
varying index function n(y) that in the illustrated case is
periodic will have a non-zero gap for k.sub.x=0. Within it there
are no propagating modes, so a wave with its frequency falling in
the range of the gap, if incident normal to the surface of such a
crystal, will be reflected.
[0019] For k.sub.x>0 (an arbitrary direction of propagation) it
is convenient to examine the projected band structure, which is
shown in FIG. 3 for the same medium as in FIG. 2, a quarter-wave
stack with n.sub.1=1, n.sub.2=2. To make this plot, first the bands
.omega..sub.n(k.sub.x, k.sub.y) for the structure were computed,
using a numerical method for solving Maxwell's equations in a
periodic medium. For each value of k.sub.x, the mode frequencies
.omega..sub.n, for all possible values of k.sub.y were plotted.
Thus, in the gray regions there are electromagnetic modes for some
values of k.sub.y, whereas in the white regions there are no
electromagnetic modes, regardless of ky. The s-polarized modes are
plotted to the right of the origin, and the p-polarized modes to
the left. Frequencies are reported in units of c/a.
[0020] The shape of the projected band structure for a multilayer
film can be understood intuitively. At k.sub.x=0, the
normal-incidence bandgap of FIG. 2 is recovered. This range of
frequencies is enclosed by dashed lines. As k.sub.x>0, the bands
curve upwards in frequency, as the condition for destructive
interference shifts to shorter wavelengths. As
k.sub.x.fwdarw..infin., the frequency width of the gray regions
shrinks until they become lines. In this regime the modes are
largely confined to the slabs with the higher index of refraction.
For large k.sub.x they are very well confined and do not couple
between layers (independent of k.sub.y). They are approximately
planar waveguide modes, so the dispersion relation approaches
.omega.=ck.sub.x/n.sub.2 asymptotically.
[0021] One obvious feature of FIG. 3 is that there is no complete
bandgap. For any frequency, there exists a wave-vector and an
associated electromagnetic mode corresponding to that frequency.
The normal-incidence bandgap 300 (enclosed by the dashed lines) is
crossed by modes with k.sub.x>0. This is a general feature of
one-dimensional photonic crystals.
[0022] However, the absence of a complete band-gap does not
preclude omnidirectional reflection. The criterion is not that
there be no propagating states within the crystal; rather, the
criterion is that there be no propagating states that may couple to
an incident propagating wave. This is equivalent to the existence
of a frequency range in which the projected band structures of the
crystal and the ambient medium have no overlap.
[0023] The two diagonal black lines 302, 304 in FIG. 3 are the
"light lines" .omega.=ck.sub.x. The electromagnetic modes in the
ambient medium (air) obey
.omega.=c(k.sub.x.sup.2+k.sub.y.sup.2).sup.1/2, where c is the
speed of light in the ambient medium, so generally
.omega.>ck.sub.x. The whole region above the solid diagonal
"light-lines" .omega.>ck.sub.x is filled with the projected
bands of the ambient medium.
[0024] For a semi-infinite crystal occupying y<0 and an ambient
medium occupying y>0, the system is no longer periodic in the
y-direction (no translational symmetry) and the electromagnetic
modes of the system can no longer be classified by a single value
of k.sub.y. They must be written as a weighted sum of plane waves
with all possible k.sub.y. However, k.sub.x is still a valid
symmetry label. The angle of incidence .theta. upon the interface
at y=0 is related to k.sub.x by .omega.sin.theta.=ck.sub.x.
[0025] For there to be any transmission through the semi-infinite
crystal at a particular frequency, there must be an electromagnetic
mode available at that frequency which is extended for both y>0
and y<0. Such a mode must be present in the projected photonic
band structures of both the crystal and the ambient medium. The
only states that could be present in the semi-infinite system that
were not present in the bulk system are surface states, which decay
exponentially in both directions away from the surface, and are
therefore irrelevant to the transmission of an external wave.
Therefore, the criterion for high omnidirectional reflection is
that there are no states in common between the projected bands of
the ambient medium and those of the crystal, i.e., there exists a
frequency zone in which the projected bands of the crystal have no
states with .omega.>ck.sub.x.
[0026] It can be seen from FIG. 3 that there is such a frequency
zone (0.36c/a to 0.45c/a) for s-polarized waves. The zone is
bounded above by the normal-incidence bandgap, and below by the
intersection of the top of the first gray region with the light
line. The top edge of the first gray region is the dispersion
relation for states with k.sub.y=.pi./a.
[0027] The lowest two p-bands cross at a point above the line
.omega.=ck.sub.x, preventing the existence of such a frequency
zone. This crossing occurs at the Brewster angle
.theta..sub.B=tan.sup.-1(n.sub.2/n.- sub.1), at which there is no
reflection of p-polarized waves at any interface. At this angle
there is no coupling between waves with k.sub.y and -k.sub.y, a
fact which permits the band-crossing to occur. As a result, the
bands curve upwards more rapidly.
[0028] This difficulty vanishes when the bands of the crystal are
lowered relative to those of the ambient medium, by raising the
indices of refraction of the dielectric films. For example, by
multiplying the index of refraction n(y) by a constant factor
.alpha.>1, all of the frequencies of the electromagnetic modes
are lowered by the same factor .alpha..
[0029] FIG. 4 is a graph showing the projected band structure for
an exemplary quarter-wave stack with the same ratio
n.sub.2/n.sub.1=2 and n.sub.1=1.7, n.sub.2=3.4 (.alpha.=1.7), and
d.sub.1=0.67, d.sub.2=0.33. In this case there is a frequency zone
in which the projected bands of the crystal and ambient medium do
not overlap, namely from the point 400 (.omega.a/2.pi.c=0.21) to
the point 402 (.omega.a/2.pi.c=0.27). This zone is bounded above by
the normal-incidence bandgap and below by the intersection of the
top of the first gray region for p-polarized waves with the
light-line 404. While the illustrated embodiments of the invention
will be described utilizing a silicon-silicon dioxide materials
system, the invention can be fabricated with other materials
systems.
[0030] Between the frequencies corresponding to the points 400 and
402, there will be total reflection from any incident angle for
either polarization. For a finite number of films, the transmitted
light will diminish exponentially with the number of films. The
calculated transmission spectra, for a finite system of ten films
(five periods), are plotted in FIG. 5 for various angles of
incidence, e.g., from 0.degree. to approximately 90.degree.. The
calculations were performed using transfer matrices. The stop band
shifts to higher frequencies with more oblique angles, but there is
a region of overlap which remains intact for all angles.
[0031] FIG. 5 is a graph of the calculated spectra for a
quarter-wave stack of ten films (n1=1.7, n2=3.4) for three angles
of incidence. The solid curves correspond to p-polarized waves, and
the dashed curves correspond to s-polarized waves. The overlapping
region of high reflectance corresponds to the region between the
points 400 and 402 of FIG. 4. While the illustrated embodiment
describes the characteristics of a structure having a ten-layer
film of silicon and silicon dioxide, it will be appreciated that a
reflector of the invention can be achieved with other multilayer
arrangements or other material systems with appropriate index
contrasts.
[0032] The criterion for high omnidirectional reflection (the
non-overlap of the projected bands of both crystal and ambient
medium) applies for a general function n(y) that is not necessarily
periodic. For the special case of a multilayer film it is possible
to derive an explicit form of the band structure function
.omega..sub.n(k.sub.x,k.sub.y) and use it to investigate
systematically the frequency zone of directional reflection, if
any, which results from a given choice of n.sub.1, n.sub.2, d.sub.1
and d.sub.2.
[0033] The graphical criterion for high omnidirectional reflection,
as shown in FIG. 4, is that the point 400 (the intersection of the
light line 404 and the first p-polarized band at k.sub.y=.pi./a) be
lower than the point 402 (the second band at k.sub.x=0,
k.sub.y=.pi./a). Symbolically, 1 p1 ( k x = p1 c , k y = a ) <
p2 ( k x = 0 , k y = a ) ( 1 )
[0034] where .omega..sub.pn(k.sub.x, k.sub.y) is the p-polarized
band structure function for the multilayer film. It will be
appreciated that the left side is a self-consistent solution for
the frequency .omega..sub.p1. The difference between these two
frequencies is the range of high omnidirectional reflection.
[0035] For a multilayer film, the dispersion relation
.omega..sub.n(k.sub.x,k.sub.y) may be derived by computing the
eigenvalues .LAMBDA. of the transfer matrix associated with one
period of the film at a particular frequency and incident angle.
When .LAMBDA.=exp(ik.sub.ya) with k.sub.y real, there is a
propagating mode at that frequency and angle. The dispersion
relation .omega..sub.n(k.sub.x,k- .sub.y) is governed by the
transcendental equation: 2 ( 1 + A 2 ) cos [ ( 2 + 1 ) ] - A 2 cos
[ ( 2 - 1 ) ] = cos ( k y a ) ( 2 )
[0036] Here .beta..sub.1,2=(d.sub.1,2/c){square
root}n.sub.1,2.sup.2-sin.s- up.2.theta. is defined for each film.
The polarization-dependent constant A is defined by: 3 A = ( r 1 -
r 2 ) 2 2 r 1 r 2 ( 3 ) r 1 , 2 = { n 1 , 2 2 - sin 2 ( s -
polarized ) n 1 , 2 n 1 , 2 2 - sin 2 ( p - polarized ) } ( 4 )
[0037] These results may be used to evaluate the criterion as
expressed in equation (1). The roots of equation (2) may be found
numerically, for a given k.sub.y and
.theta.=asin(ck.sub.x/.omega.). The frequency range (if any) of
omidirectional reflection, according to equation (1), is between
the first root of equation (2) for p-polarized waves with
k.sub.y=.pi./a and .theta.=.pi./2 (point 400 of FIG. 4), and the
second root for k.sub.y=.pi./a and .theta.=0 (point 402).
[0038] The frequency range has been calculated (when it exists) for
a comprehensive set of film parameters. Since all the mode
wavelengths scale linearly with d.sub.1+d.sub.2=a, only three
parameters need to be considered for a multilayer film: n.sub.1,
n.sub.2, and d.sub.1/a. To quantify the range of high
omnidirectional reflection [.omega..sub.1, .omega..sub.2] in a
scale-independent manner, the "range-midrange ratio" is defined as
(.omega..sub.2-.omega..sub.1)/[(1/2)(.omega..sub.1+.omega..-
sub.2)].
[0039] For each choice of n.sub.1 and n.sub.2/n.sub.1, there is a
value of dl/a that maximizes the range-midrange ratio. That choice
may be computed numerically. FIG. 6 is a contour plot of the
range-midrange ratio for the frequency range of high
omnidirectional reflection, as n.sub.1 and n.sub.2/n.sub.1 are
varied, for the maximizing value of d.sub.1/a (solid contours). The
dashed curve is the 0% contour for the case of a quarter-wave
stack. For the general case of an ambient medium with index
n.sub.0.noteq.1, the abscissa becomes n.sub.1/n.sub.0. This plot
shows the largest possible range-midrange ratio achievable with
n.sub.1 and n.sub.2 fixed.
[0040] An approximate analytic expression for the optimal zone of
high omnidirectional reflection may be derived: 4 2 c = a cos ( - A
- 2 A + 2 ) d 1 n 1 + d 2 n 2 - a cos ( - B - 2 B + 2 ) d 1 n 1 2 -
1 + d 2 n 2 2 - 1 where ( 5 ) A n 2 n 1 + n 1 n 2 , B n 2 n 1 2 - 1
n 1 n 2 2 - 1 + n 1 n 2 2 - 1 n 2 n 1 2 - 1 ( 6 )
[0041] Numerically this is found to be an excellent approximation
for the entire range of parameters depicted in FIG. 6 including the
case of a quarter-wave stack.
[0042] It can be seen from FIG. 6 that, for high omnidirectional
reflection, the index ratio should be reasonably high
(n.sub.1/n.sub.2>1.5) and the indices themselves be somewhat
higher (n.sub.1/n.sub.0.gtoreq.1.5) than that of the ambient
medium. The former condition increases the band splittings, and the
latter depresses the frequency of the Brewster crossing. An
increase in either factor can partially compensate for the other.
The materials should also have a long absorption length for the
frequency range of interest, especially at grazing angles, where
the path length of the reflected light along the crystal surface is
long.
[0043] For example, for light with a wavelength of 1.5 .mu.m,
silicon dioxide has n.sub.1=1.44 and silicon has
n.sub.2=3.48=2.42n.sub.1. From FIG. 6, this corresponds to a
range-midrange ratio of about 27%. Likewise, for
GaAs/Al.sub.2O.sub.3 multilayers (n.sub.1=1.75,
n.sub.2=3.37=1.93n.sub.1), the range-midrange ratio is about
24%.
[0044] In practice, the optimization of d.sub.1/a results in a gap
size very close to the gap size that would result from a
quarter-wave stacked with the same indices
d.sub.1/a=n.sub.2/(n.sub.2+n.sub.1). The 0% contour for
quarter-wave stacks is plotted in FIG. 6 as a dashed line, which is
very close to the optimized 0% contour.
[0045] With this in mind, an approximation to equation (2) may be
derived for films which are nearly quarter-wave stacks. In that
limit .beta..sub.2-.beta..sub.1.apprxeq.0, so the second cosine in
equation (2) is approximately 1. In this approximation the
frequency of the band edge at ky=.pi./a is: 5 1 1 + 2 a cos [ A 2 +
1 A 2 - 1 ] ( 7 )
[0046] using the same notion as in equations (3) and (4). This
frequency can be computed for the cases .theta.=0 and
.theta.=.pi./2. If the difference between these two frequencies is
positive, there will be omnidirectional reflection for any
frequency between them.
[0047] The invention demonstrates that, even though it is not
possible for a one-dimensional photonic crystal to have a complete
bandgap, it is still possible to achieve reflection of ambient
light regardless of incident angle or polarization. This happens
whenever the projected bands of the crystal and ambient medium have
no overlap within some range of frequencies.
[0048] This constraint is not unrealistic, even for the most common
sort of one-dimensional photonic crystal, the multilayer film. As
can be seen in FIG. 6, what is required is that the index ratio be
reasonably high (n.sub.2/n.sub.1>1.5) and the indices themselves
be somewhat higher than that of the ambient medium
(n.sub.1/n.sub.0>1.5). An increase in either factor can
partially compensate for the other. They should also have a
relatively long absorption length for the frequency range of
interest. Such materials, and the technology required to deposit
them in multiple layers, are conventional. To achieve high
omnidirectional reflection, therefore, it is not necessary to use
more elaborate systems such as multiple interleaving stacks,
materials with special dispersion properties, or fully
three-dimensional photonic crystals.
[0049] The optical response of a particular dielectric multilayer
film can be predicted using the characteristic matrix method. In
this method, a 2.times.2 unitary matrix is constructed for each
layer. This matrix represents a mapping of the field components
from one side of the layer to the other. To successfully predict
the optical response of a multilayer film the characteristic matrix
for each layer needs to be calculated. The form of the
characteristic matrix for the j.sup.th layer is 6 m g ( ) j = [ cos
j - i p j g sin j - ip j g sin j cos j ] ( g = TE , TM ) j = kh j n
j 2 - snell ( ) 2 snell ( ) = n 0 sin 0 p j g = { n j 2 - snell ( )
2 g = TE n j 2 - snell ( ) 2 n j 2 g = TM ( 8 )
[0050] where n.sub.j is the index of refraction, and h.sub.j is the
thickness of the J.sup.th layer, .theta..sub.0 is the angle between
the incident wave and the normal to the surface and n.sub.0 is the
index of the initial medium, e.g., air.
[0051] The matrices are then multiplied to give the film's
characteristic matrix 7 M g ( ) = j = 1 N m j g ( g = TM or TE ) (
9 )
[0052] which in turn can be used to calculate the reflectivity for
a given polarization and angle of incidence, 8 R g ( ) = ( M 11 g (
) + M 12 g ( ) p 1 g ) p 0 g - ( M 21 g ( ) + M 22 g ( ) p 1 g ) (
M 11 g ( ) + M 12 g ( ) p 1 g ) p 0 g + ( M 21 g ( ) + M 22 g ( ) p
1 g ) 2 ( 10 )
[0053] where p.sup.g.sub.0 contains information about the index of
the medium and angle of incidence on one side of the multilayer
film and p.sup.g.sub.1 contains information about the index of the
medium and angle of incidence on the other.
[0054] To construct a reflector exhibiting a reflectivity R of a
minimal prescribed value for all angles of incidence and both
polarizations one needs to (1) satisfy the criteria for
omnidirectional reflection, and (2) solve equation (10) for
.theta.=89.9.degree., g=TM and R.TM. (89.9)=R.
[0055] Although the invention has been illustrated by using
multilayered films, the invention as described can apply generally
to any periodic dielectric function n(y), or even an aperiodic
dielectric function n(y). What is required is that n(y) leads to
photonic bandgaps along various directions such that there exists a
zone of frequencies in which the projected bands of the dielectric
structure and ambient media do not overlap. Such a requirement can
also be satisfied by a photonic crystal with two- or
three-dimensionally periodic index contrasts, which have incomplete
bandgaps.
[0056] However, the absence of a complete bandgap does have
physical consequences. In the frequency range of high
omnidirectional reflection, there exist propagating solutions of
Maxwell's equations, but they are states with .omega.<ck.sub.x,
and decrease exponentially away from the crystal boundary. If such
a state were launched from within the crystal, it would propagate
to the boundary and reflect, just as in total internal
reflection.
[0057] Likewise, although it might be arranged that the propagating
states of the ambient medium do not couple to the propagating
states of the crystal, any evanescent states in the ambient medium
will couple to them. For this reason, a point source of waves
placed very close (d<.lambda.) to the crystal surface could
indeed couple to the propagating state of the crystal. Such
restrictions, however, apply only to a point source and can be
easily overcome by simply adding a low index cladding layer to
separate the point source from the film surface.
[0058] Many potential applications are envisioned for such a high
omnidirectional reflector or mirror. For example, in the infrared,
visible, or ultraviolet regimes, high omnidirectional reflectors
could serve as a frequency-selective mirrors for laser beams or
highly-reflective coatings on focusing instruments. These would be
effective for light that is incident from any angle, instead of
just a finite range around a fixed design angle.
[0059] The invention can also be utilized in coatings with infrared
mirrors to keep heat in or out of the items coated, e.g., walls,
windows, clothes, etc. The mirrors can be cut into small flakes and
mixed with paint or fabrics to allow for application to the desired
items.
[0060] The reflector of the invention could be used in improving
thermo-photovoltaic cells that trap waste heat and convert it into
energy. The reflector of the invention can also be made to reflect
radio waves and thus can be used to boost performance of radio
devices such as cellular telephones.
[0061] Although the present invention has been shown and described
with respect to several preferred embodiments thereof, various
changes, omissions and additions to the form and detail thereof,
may be made therein, without departing from the spirit and scope of
the invention.
* * * * *