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BACKGROUND OF INVENTION
1. Field of Use
This invention relates generally to wavelength selective filtering of
light. More specifically, this invention concerns holographic filters used
in spectroscopic and spectral splitting applications.
2. Description of the Prior Art
Devices used to filter out or inhibit the transmission of certain
wavelengths of light are in widespread use, and have several particularly
important applications. One important application is that of
spectrophotometers which are able to determine the chemical composition of
a material by illuminating it with a broad wavelength range of light
waves. In such devices, light is generated by a source and dispersed and
collimated by an imaging system. A slit is moved within the system which
determines the wavelength of light to be measured.
The quality of a spectroscopic device depends primarily on two factors.
Resolution of the device determines the width of the spectral line that
the spectrophotometer will be able to discern. Resolution is directly
dependent upon the size of the slit which is usually better than one
nanometer in high quality devices. The maximum optical density ("O.D.") is
another critical quality parameter of spectroscopic devices.
Spectrophotometers of good quality are able to measure O.D.'s of up to 4.0
and sometimes more. O.D.'s of up to 4.0 are difficult to measure because
they represent four orders of magnitude attenuation of the incident light
intensity. To measure such a large range of O.D. requires that the noise
in the system from all sources be very low. Noise is picked up from every
conceivable source including ambient light and the electronics in the
spectroscopic system itself.
One type of spectrophotometer is the absorption spectrophotometer wherein
the composition of the specimen is determined by measuring the light
absorbed by the specimen. A logarithmic scale is used to plot absorption
in order to accommodate the several orders of magnitude of O.D. O.D. is
represented by the following equation
O.D.=-log.sub.10 T
where T represents the transmission coefficient of the material sample. For
example, if T equals 10.sup.-4, O.D. equals 4.0 Referring to FIG. 1, O.D.
is plotted against wavelength .lambda.. It can be seen that where the
curve in FIG. 1A reaches its maximum near the wavelength .lambda..sub.O,
the material is optically dense and indicates that the specimen in the
spectrophotometer has absorbed the range of wavelengths surrounding
.lambda..sub.0. The counterpart to the O.D. versus .lambda. curve is
illustrated in FIG. 1B. FIG. 1B plots T, the transmission coefficient
against wavelength .lambda.. It can be seen that all wavelengths of the
light source are transmitted by the specimen except for the range of
wavelengths about the wavelength .lambda..sub.O. Note that the typical
measurement of light absorbed by a specimen is a rather complex curve as
can be seen in FIG. 1C. Simplified curves such as those in FIGS. 1A and 1B
will be used throughout for simplicity.
There exists a certain class of spectroscopy called laser spectroscopy. In
laser spectroscopy a laser beam having wavelength .lambda..sub.0 is used
rather than some wider source beam. In laser spectroscopy the laser beam
is incident upon a scattering medium of interest which, according to its
chemical composition, will scatter the laser beam into multiple beams some
of which have the same wavelength as the incident beam and some of which
have a different scattered wavelength. Referring now to FIG. 2 a laser
beam .lambda..sub.0 is incident upon scattering medium S. The incident
beam is scattered into a multiplicity of light waves each having a
scattered wavelength .lambda..sub.S. It can be seen that the wavelength of
some of the light waves .lambda..sub.S is equal to .lambda..sub.0 and the
wavelength of other .lambda..sub.S light waves is not equal to
.lambda..sub.O. The scattered light waves whose wavelength is
.lambda..sub.0 can be said to have undergone elastic scattering. The
scattered light waves whose wavelength is not equal to .lambda..sub.O, can
be said to have undergone inelastic scattering.
Elastic scattering means that the scattered light photons have the same
energy as the incident light photons. Elastic scattering is by far the
stronger of the two scattering effects and thus the scattered energy to be
measured is usually heavily biased toward the .lambda..sub.s
=.lambda..sub.0 light waves. On the other hand, inelastically scattered
light photons usually have less energy than the .lambda..sub.0 light
photons. The energy of these light photons can be described by the
following equation
E.sub.p =hf=hc/.lambda.
where E.sub.p is photon energy, h is Planck's constant, f is the frequency
in Hertz of the light wave, c is the velocity of light in a vacuum, and
.lambda. is the wavelength of the light in a vacuum. Ep for the inelastic
scattering case is less than E.sub.p for elastic scattering.
The technological challenge of filtering in laser beam spectroscopy arises
from the fact that it is the inelastically scattered wave that contains
more information about the chemical structure of the material under test
and consequently is the desired signal. Scattered light waves having
energy E.sub.P equal to the energy of the .lambda..sub.0 light wave thus
constitute noise and must be filtered out, along with the multiplicity of
other noise sources so that the desired inelastically scattered signal
energy can be measured with accuracy. Filters are needed to block the
.lambda..sub.0 wavelength light wave. The most common are Raman filters
used for Raman spectroscopy applications.
Wavelength selective optical filters have basically been of two types,
absorption dyes and dielectric multilayers. The advantages of absorption
dyes as wavelength selective optical filters is primarily their high
angular acceptance of nearly .+-.90.degree.. This means that light
incident upon the filter at most any angle will be filtered. The
disadvantages of absorption dyes stem from the fact that absorption dyes
have their own chemical structure and thus their own absorptive
characteristics which can affect the absorption measurement from the
specimen. Additionally, absorption dyes have rather broad bandwidths and,
consequently, have low wavelength selectivity, i.e. bandwidth is usually
higher than 20 nm. Furthermore, absorption dyes have secondary maxima due
to their sophisticated chemical composition that can be confused with the
absorption lines of the specimen. Finally, the disadvantage of an
absorption dye stems from what is usually its strong point, broad angular
acceptance. This strong point can be a disadvantage where the specimen is
tested for angular selectivity.
The second type of known wavelength selective optical filter is the
dielectric multilayer. Dielectric multilayer filters operate on the
principle of Bragg interference. Bragg interference filters operate on the
principle that for certain wavelengths near .lambda..sub.O, the reflected
waves interfere constructively with each other and so have a high
reflectivity for wavelengths in the vicinity of .lambda..sub.O. For other
wavelengths, the reflected waves interfere destructively. Dielectric
multilayer filters are usually used as reflection filters. A dielectric
multilayer filter is shown in FIG. 3A having alternating dielectric layers
made from materials A and B.
There is another type of filter called a Fabry-Perot etalon, however, which
is a transmission filter based on interference principles. Fabry-Perot
etalon filters transmit some wavelengths and reject all others in
contradistinction to reflection type filters which reflect only certain
wavelengths and transmit all others. A Fabry-Perot etalon (transmission)
filter is shown in FIG. 3B and has a first coating C' comprising
dielectric multilayers (not shown) similar to those in FIG. 3A separated
from a second coating C" comprising similar alternating dielectric layers.
In essence, the Fabry-Perot etalon is a combination of two highly parallel
multilayer dielectric coatings and operates by causing interference of the
light waves between the two coatings.
A transmission plot for a dielectric multilayer filter is shown in FIG. 4A
which illustrates that the filter transmits all wavelengths except those
around .lambda..sub.O. A transmission plot for a Fabry-Perot etalon
transmission filter is shown in FIG. 4B which illustrates that the filter
transmits wavelengths around .lambda..sub.01, .lambda..sub.02,
.lambda..sub.03 . . . .lambda..sub.On.
The wavelength selectivity of dielectric multilayer filters is directly
dependent upon the number of layers in the filter. The critical importance
of this is fully discussed infra. Vacuum deposition is used to produce
these filters, by evaporating layer after layer of alternating dielectric
materials. Each layer adds to the cost of the filter. Furthermore, the
cost of physically larger dielectric multilayer filters becomes
prohibitive due to the size of the required vacuum chamber in which the
filters are made.
An additional disadvantage of this type of filter is that the rectangular
periodic distribution of its refractive index n creates unwanted harmonics
and secondary maxima. Certain of the harmonics, particularly the second
harmonic, can be suppressed, but other harmonics and secondary maxima
remain which can affect the performance of the filter. The secondary
maxima, similar to those shown in FIG. 4A, are especially problematic from
the standpoint of spectroscopic system accuracy because they can be
confused with the absorption spectral lines characterizing the chemical
structure of the sample. FIG. 5A shows the variation of the refractive
index n for a typical multilayer dielectric filter having alternating
dielectric layers A and B. It can be seen that the refractive index n is a
rectangular function over the several layers of the filter, and has an
average refractive index n, and the grating constant .LAMBDA.. The grating
constant can be described by the equation
.LAMBDA.=.lambda./2n
Dielectric filters can be made to have a more sinusoidal variation of
refractive index n, but the cost of such filters, called rugate filters,
is extremely high. FIG. 5B depicts the sinusoidal refractive index profile
of a rugate multilayer dielectric filter.
Due to the disadvantages of both the absorption dye and multilayer
dielectric filters, and especially the high cost of the latter, there is a
need for a filter for use in spectroscopic applications that has extremely
high rejection, high wavelength selectivity, high angular acceptance,
minimized secondary maxima, and which can be manufactured at low cost.
SUMMARY OF THE INVENTION
A filter for use in spectroscopic applications comprising a volume hologram
is presented. Specifically, a volume hologram recorded with Bragg planes
according to state of the art techniques which operates according to the
Bragg interference principle is disclosed. The Bragg planes in the
holographic filter of the present invention can be recorded to satisfy
virtually any design requirement imposed upon the filter such as for Raman
filter, Lippmann filter, curved Bragg surface filter, and multiplexed
filter applications. The Bragg planes of the holographic filter may be
recorded in a one step recording process and can be varied to concentrate,
focus, or direct the unwanted wavelength components of the laser light
source where desired.
The holographic filter of the present invention can be manufactured at low
cost and can obtain high optical densities of 4.0, 5.0, and 6.0, large
controllable bandwidths and wavelength selectivities, refractive index
profiles to meet a broad range of frequency, amplitude and bias
requirements, and with extremely high resolution and stability. Most
importantly the holographic filter of the present invention may have
extremely narrow wavelength rejection characteristics (i.e., the space
under the O.D.-.LAMBDA. curve can be made very narrow). Furthermore, the
strength of the rejection coefficient, O.D., can be made extremely high,
and the secondary maxima of the holographic filter can be suppressed
considerably and/or compressed to lessen their effect. In a single
holographic filter of the present invention, the the average index of
refraction, n, can be varied as a function of z, making the holographic
filter of the present invention not only low cost but highly flexible.
BRIEF DESCRIPTIONS OF THE DRAWINGS
FIGS. 1A and 1B show the relationship between optical density (O.D.) and
the transmission coefficient T versus wavelength .lambda. respectively and
FIG. 1C is a typical plot of O.D. versus .lambda. measured by a
spectrophotometer;
FIG. 2 illustrates the scattering of a laser beam by a scattering medium;
FIGS. 3A and 3B show respectively, a multilayer dielectric reflection
filter and a Fabry-Perot etalon filter;
FIGS. 4A and B show respectively, the reflection coefficient R for a
multilayer dielectric filter and the transmission coefficient T for a
Fabry-Perot etalon filter versus .lambda.;
FIGS. 5A and B show the quadratic periodic nature of the refractive index n
for a multilayer dielectric filter and the sinusoidal nature of the
refractive index n for a rugate multilayer dielectric filter;
FIG. 6 is a schematic of a holographic filter of the present invention and
the variation in the refractive index n;
FIG. 7 is a schematic of a Lippmann hologram undergoing recording;
FIG. 8 shows the desired light wave .lambda..sub.2 and the light waves
.lambda..sub.0 and .lambda..sub.1 to be filtered out in a plot of O.D.
versus .lambda.;
FIGS. 9A, B, C, and D are exemplary schematics of holographic optical
elements in the form of filters of the present invention having varying
Bragg plane structures: Lippmann, slanted, diverging in reflection, and
focusing in transmission, respectively;
FIGS. 10A and B are schematics of a polychromatic beam incident a filter of
the present invention in a spectral splitting application and its O.D.
curve;
FIG. 11A is a schematic of a holographic Fabry-Perot etalon filter and FIG.
11B is its transmission characteristic T versus .lambda..
FIG. 12 is a schematic of the rejection characteristics of a multiplexed
Lippmann holographic filter;
FIGS. 13A and B and C show varying refractive index profiles for
holographic filters of the present invention having varying amplitude,
bias, and grating constant respectively wherein n is plotted against z.
FIG. 14 shows the wave vector k and grating vector K for a light wave in a
medium for the on-Bragg condition;
FIG. 15 shows the relationship between .rho. coupling efficiency, and
.upsilon., the coupling constant, for a rejection holographic filter of
the present invention;
FIGS. 16A and B show fully uniform Kogelnik holographic filters of the
present invention;
FIG. 17 shows the relationship between .rho. and .xi., and the turning
points .+-..upsilon.;
FIG. 18 is a table showing numerical values for T (thickness), N (number of
layers), and .DELTA..lambda. (bandwidth);
FIG. 19 shows a sin curve and the location of the turning points, first
zero, and first secondary maxima of the equation defining .xi.;
FIGS. 20A and B are graphs showing .rho. and .lambda. where the secondary
maxima are uncompressed and compressed respectively;
FIGS. 21A and B show a filter of the present invention having a varying
grating constant .LAMBDA. and its O.D. curve;
FIG. 22 is a schematic of one embodiment of the present invention utilizing
a curved filter;
FIG. 23 is another embodiment of the invention;
FIG. 24 is a schematic of a compound holographic filter of the present
invention;
FIG. 25 is yet another embodiment of the present invention.
DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring to FIG. 6, a holographic Lippmann filter 20 of the present
invention is depicted. The holographic filter 20 is manufactured in
accordance with state-of-the-art recording, exposure, and processing
techniques. Bragg planes 22 are recorded in the volume of the holographic
filter 20. The grating constant A is shown as the distance between Bragg
planes 22. The thickness T of the holographic filter is also shown. The
refractive index n varies as shown by curve 24 and .DELTA.n represents the
amplitude of the refractive index .DELTA.n. Typical values for the lower
limit of n for example, might be 1.49 and its upper limit might be, for
example, 1.51 with an average n, n, of 1.50; then .DELTA.n=0.01.
Referring now to FIG. 7 the recording of a holographic Lippmann filter of
the present invention is explained. The holographic material 26 to be
recorded, such as DCG, DCG polymer grafts and/or composites, PVA, PMMA,
and such materials as DuPont photopolymer or Polaroid DMP-128, is set up
adjacent to a mirror 28 and exposed to laser light of wavelength .lambda..
The laser light incident upon the holographic material 26 is also incident
upon the mirror 28 which reflects the laser energy and creates a standing
wave pattern in the holographic material 26. The electromagnetic energy
present in the standing wave is shown by the plot 30, E versus z. The
spacing of the Bragg planes, .LAMBDA., is determined by the wavelength of
the laser light and the average refraction index n of the material to be
recorded according to
.LAMBDA.=.lambda./2n.
The sinusoidal variation of the electromagnetic energy as shown in plot 30
causes the refractive index n of the material to change in direct relation
to the standing wave of electromagnetic energy in the material 26. The
variation in refractive index n is shown in plot 32 of n versus z. It can
be seen that refractive index n varies as does the electromagnetic energy
E. Where the electromagnetic energy E in the standing wave in the material
26 is high, so is the refractive index n and where energy E is low, the
refractive index n is low as well. Thus, the recording process may create
in the holographic material to be recorded in the material 26 a sinusoidal
variation in the refractive index of the material. As is shown infra, the
refractive index of the material can be varied broadly during recording to
yield variations in the spacing of the Bragg-planes, .LAMBDA., the shape
of the refractive index variation (sinusoidal to quasi-rectangular), the
amplitude, .DELTA.n, of the index of refraction in the material, and its
bias, n, from one end of the material to the other. The more advanced HOE
(holographic optical element) filters may be recorded by using standard
two beam interference techniques described in R. Collier, et. al., Optical
Holography, Academic Press (1971).
Referring now to FIG. 8, a curve 34 depicting optical density (O.D.) versus
.lambda. for a holographic filter is presented. The curve 34 represents
the wavelengths that are rejected by the holographic filter. The central
wavelength of the rejected wavelengths is .lambda..sub.O. .lambda..sub.1
represents another wavelength not equal to .lambda..sub.0 that is also
rejected, and .lambda..sub.2 represents a wavelength that is not rejected
by the filter, i.e., is transmitted. Since the specimen usually has a very
sophisticated chemical structure, there is a broad set (usually continuous
set) of wavelengths to be measured (say, .lambda..sub.2 ', .lambda..sub.2
", .lambda..sub.2 "', etc.), and .lambda..sub.2 is only one of them. Each
of these wavelengths, however, should be outside the bandwidth of
rejection. In a laser beam spectroscopy system where an incident light
beam .lambda..sub.0 is scattered by a scattering medium, .lambda..sub.0
and .lambda..sub.1 would represent elastic or virtually elastic scattering
of the laser light beam. .lambda..sub.2 represents inelastic scattering.
It is the signal having wavelength .lambda..sub.2, a wavelength outside
the range of wavelengths near .lambda..sub.O, that is the signal of
interest and represents information about the chemical structure of the
specimen. All other signals may be filtered out.
Referring now to FIGS. 9A, B, C and D, it can be seen that the positioning
of the Bragg surfaces in a holographic filter of the present invention can
be varied during the recording process to direct either the desired signal
(.lambda..sub.2) or the undesired signal (.lambda..sub.O, .lambda..sub.1)
in particular directions. Those more advanced volume holograms are usually
called holographic optical elements (HOE's). They combine spectral
filtering operations with focusing, diverging, imaging, etc. FIG. 9A
represents a Lippmann-Bragg holographic filter mirror 36 having Bragg
planes 38 parallel to the surface of the mirror. An incoming laser beam 40
is incident upon the mirror 36 and is in part reflected and in part
transmitted by the holographic filter mirror 36. The reflected portion 42
comprising light waves .lambda..sub.0 and .lambda..sub.1 of the incoming
laser beam 40 represents elastic scattering. Inelastic scattering is
represented by light wave 44 of the incident laser beam that is
transmitted through the holographic filter mirror 36. The inelastically
scattered portion has wavelength .lambda..sub.2 outside of the range of
rejection of the mirror 36. Note that the angle of the rejected beam 42
with respect to the holographic filter mirror 36 is the same as the angle
of the incoming laser beam 40 with respect to the holographic filter
mirror 36.
Referring now to FIG. 9B, an incoming laser beam 46 having wavelength
.lambda..sub.0 is incident a non-Snellian (slanted) holographic filter 48.
A portion 50 of incoming laser beam 46 is rejected by elastic or
quasi-elastic scattering and comprises the wave components .lambda..sub.0
and .lambda..sub.1. Since the Bragg planes are not parallel to the surface
of the hologram neither of the reflected waves satisfy Snell's law.
Another portion 52 of the incoming laser beam 46 is inelastically
scattered, is transmitted, and has a wavelength of .lambda..sub.2.
Referring now to FIG. 9C, a diverging (in reflection) holographic filter is
depicted having an incoming laser beam 54 with wavelength .lambda..sub.0
at normal incidence to a HOE mirror 56. Elastic scattering of the light
beam is represented by light rays 58 having wavelengths of .lambda..sub.0
and .lambda..sub.1 within the rejection range as shown by the O.D. versus
.lambda. plot in FIG. 8. They are reflected in the form of a divergent
beam. Inelastic scattering is represented by light rays 60 having
wavelength .lambda..sub.2 outside the rejection region of the holographic
filter 56.
Referring now to FIG. 9D, a concentrating (in transmission) holographic
filter of the present invention is depicted. It can be seen that the
incoming laser beam 62 is elastically scattered in the direction of light
ray 64 (.lambda..sub.00) and the inelastically scattered light waves are
represented by light rays 66 (.lambda..sub.2). Not only are light rays 66
scattered in a direction different than that of the elastically scattered
light ray 64, the inelastically scattered light rays are focused to a
particular point in space by virtue of the curved Bragg surfaces 68
recorded in the holographic filter 70. Such focusing may facilitate
detection and measurement of the inelastically scattered light waves which
contain information about the chemical structure of the specimen under
test.
It should be realized that virtually any Bragg plane structure can be
recorded in holographic material in order to scatter either the wanted or
unwanted light waves in any desired direction or manner. By way of
example, if the spectroscopy of interest is Raman spectroscopy, the filter
of the present invention for such an application may preferably be a Raman
filter. If the particular application requires a filter having Bragg
planes parallel to the surface, a Lippmann holographic filter of the
present invention may preferably be used and if slanted Bragg planes are
required a non-Snellian holographic filter of the present invention may be
used. If focusing, imaging, or diverging operations are required of the
filter, the interference fringes of the filter (Bragg surfaces as opposed
to planes) can be curved. Such filters are called HOEs. Filter
applications requiring the filter to reject two or more different Bragg
wavelengths may be satisfied by a filter of the present invention having
two sets of interference fringes, this type of filter being a multiplex
holographic filter. A broad family of varying HOE and multiplex filters of
the present invention are possible.
Furthermore, the incident beam that is filtered may be a laser beam of
monochromatic light, a beam of quasi-monochromatic light, or polychromatic
light including solar radiation. Filters used with polychromatic light may
preferably be used where spectral splitting is desired. A polychromatic
beam 71 is illustrated in FIGS. 10A and B wherein spectral band (A, B) 73
comprising wavelengths .lambda..sub.A <.lambda.<.lambda..sub.B is the
rejection band and can be located in UV, visible, or near-IR ranges or all
of them if a sandwich or multiplex filter is employed. By way of example
in the near-IR case, .lambda..sub.A =700 nm, .lambda..sub.B =1200 nm, and
.DELTA..lambda..sub.AB =500 nm.
Referring now to FIG. 11A, a Fabry-Perot etalon transmission holographic
filter of the present invention is depicted. The holographic filter is
designated 72 and has a first coating 74 comprising Bragg planes 80. The
transmission characteristics of the holographic Fabry-Perot etalon filter
72 are shown in FIG. 11B which plots O.D. versus .lambda.. It can be seen
that the holographic filter 72 transmits wavelengths around the central
wavelengths .lambda..sub.01, .lambda..sub.02, etc. The Fabry-Perot etalon
holographic filter may basically be a sandwich of two Lippmann filters of
the type shown in FIG. 7 or of any other type, coherently coupled. If the
rejection peaks designated 82 and 84 in FIG. 11B are recorded close enough
together in the holographic material, a transmission filter which
transmits the wavelengths in the region between .lambda..sub.01 and
.lambda..sub.02 can be fabricated.
FIG. 12 shows a multiplex holographic filter of the present invention
wherein the rejection peaks of the two combined Lippmann filters are close
enough together that the filter can be used as a transmission filter for
the wavelengths between the two peaks. Again, the extreme flexibility of
holographic fabrication would allow many different Bragg plane sets to be
recorded in the holographic material, each Bragg plane set having a
different diffraction constant .LAMBDA., such that a multiplicity of
wavelengths in different discrete wavelength locations could be rejected,
i.e., a series of rejection peaks would be present in an O.D. versus
.lambda. plot.
The advantages of holograms and holographic filters of the present
invention in particular are numerous as outlined below. The optical
efficiency of holographic filters can exceed 6.0. Typical O.D.'s for
multilayer dielectric filters are about 4.0. Thus, the rejection strength
for holographic filters is usually greater than that for multilayer
dielectric filters. The bandwidth .DELTA..lambda. may be controllable from
5 to more than 100 nm if necessary. The peak wavelength .lambda..sub.0 may
also be controllable and may be tuned within the range 250-3000 nm, or
broader. A wide variety of holographic materials can be used to make the
holographic filters including DCG, DCG polymer grafts and/or composites,
PVA, PMMA, DuPont Photopolymer, and Polaroid DMP-128, as well as many
other holographic materials. Additionally, polymer materials having a
polymer matrix such as PVA or PMMA and various sensitizers can be used.
Typically holographic materials are not transparent for wavelengths
greater than 3 microns due to absorption from water (or OH ions) at the
vicinity of that wavelength, but holographic materials not containing
water or having absorption lines other than at 3.mu. can be used.
The holographic material for the holographic filter of the present
invention preferably meets three requirements: one, transparency in the
spectrum of interest; two, sensitivity to some wavelength of laser light
used for recording; and three, acceptable resolution. Resolution is
determined by the grating constant .LAMBDA. where .LAMBDA.=.lambda./2n.
For example, if A=1 micron and n equals 1.5, then .LAMBDA. equals 1/3
micron. Taking the new equation F=1/.LAMBDA. where .LAMBDA. equals 1/3
micron yields a holographic filter having three lines per micron or 3,000
lines per millimeter. If .lambda. is decreased to 0.5 microns, .LAMBDA.,
i.e., resolution, jumps to 6,000 lines per millimeter.
One of the most important advantages of holographic filters of the present
invention is the variability of the refractive index profile of the
filter. The refractive index of the holographic filter is determined
during recording as described supra. The refractive index profile of a
holographic filter can be made sinusoidal or quasi-rectangular according
to the needs imposed by the particular filter application. Holographic
filters can be recorded at low cost either in the sinusoidal or
quasi-rectangular refractive index profile unlike multilayer dielectric
filters where the cost for a rugate-type filter quickly becomes large. The
refractive index profile, whether it be sinusoidal or quasi-rectangular,
or in between, may be confirmed by analyzing the harmonics of the
holographic filter. The refractive index profile can also be varied by
varying its amplitude or height .DELTA.n. .DELTA.n is shown in FIG. 7 and
can be made to vary across the thickness T of the holographic filter 20
shown in that figure. Typically, the amplitude .DELTA.n may be between
0.001-0.2. This means that the change of refractive index can be made
quite extreme causing the material to act very differently from one
portion to the next.
Not only can the shape and amplitude of the refractive index profile be
changed, but its frequency can also be changed by changing the grating
constant .LAMBDA. by changing the spacing between Bragg planes across the
thickness T of the holographic filter 20. Finally, the average refractive
index n can also vary across the width T of the holographic filter 20 and
thus the rejection strength of the filter 20 can vary from one portion of
the filter to the next. FIGS. 13A and B and C show a few of the possible
variations in .DELTA.n, n (with some saturation effects), and .LAMBDA.,
respectively. These many variations, and others made possible through
varying recording of the holographic filter, makes the holographic filter
of the present invention far more flexible than any state-of-the-art
filter.
The reduction of secondary maxima and sidelobes may preferably be
accomplished during the course of all processing steps: coating, exposure,
and development, the latter including water swelling and alcohol
dehydration. These processing steps tend to be surface related. Reduction
of sidelobes and secondary maxima is primarily due, however, to
shrinking/swelling which creates a vertical nonuniformity of the grating
constant equivalent to grating chirp normal to the surface. This effect is
observed best in materials that are wet processed such as DCG and
DCG/polymer-grafts/composites. From a physical point of view, the
sinusoidal distribution of exposure (due to the presence of the standing
wave during recording) creates a periodic distribution of hardness
differential, then of material density differential, and finally
refractive index differential according to the Lorentz-Lorenz formula. B.
Born and E. Wolf, Principles of Optics, Pergamon Press (1980) incorporated
by reference herein.
Other advantages of the holographic filter of the present invention include
variability of the holographic coating thickness, T, between 1-100
microns, material absorption of only 0.2%/10.mu. (equivalent to 0.9dB/mm)
which can regulated up or down, and the ability to coat single or double
curvature substrates made of hard material such as glass or soft materials
such polycarbonate, acrylic, or foil coatings. All the coatings can be
encapsulated for protection. Resolution of the holographic filter in
accordance with the present invention, as discussed above, is typically
6,000 lines per millimeter for .lambda.=0.5.mu. or more. Environmentally,
the holographic gratings of the present invention can withstand
temperatures of -800.degree. C.-+200.degree. C. and have laser damage
thresholds of greater than 10J/cm.sup.2. Mechanically there is no
possibility of interface damage and their elasticity is good. They are
also usually resistant to nuclear radiation.
The number of layers N that can be recorded in a holographic filter of the
present invention analogous to the layers in a multilayer dielectric
filter can be described by the equation
N=T/.LAMBDA.=2nT/.lambda.
where N equals the number of layers, T equals the thickness of the
holographic coating, and .LAMBDA. is the grating constant. For example, if
n equals 1.55 and T equals 20.mu. and .lambda. equals 0.5.mu., then N
equals 124 which can be achieved at low cost compared with a 124-layer
multilayer dielectric filter of the state of the art.
A theoretical relationship can be drawn between multilayer dielectric
filters and the holographic filters of the present invention. This should
be useful for designers of multilayer dielectric filters. Taking Abele's
theory governing multilayer dielectric structures at normal incidence
defined as follows:
##EQU1##
where R.sub.2N equals reflectivity for 2N number of layers, n.sub.1,
n.sub.L equal the refractive indices of the surrounding media, and
n.sub.2, n.sub.3 equal the refractive indices of alternate layers. Taking
the limit of R.sub.2N as N.fwdarw..infin. and for .DELTA.n/n<<1, we get
the following equation:
lim R.sub.2N =tanh.sup.2 (.upsilon..sub.79)
where
.upsilon..sub.A =N .DELTA.n/n
and
.DELTA.n=/n.sub.3 -n.sub.2 /
and
n=1/2(n.sub.3 +n.sub.2)
Thus, applying the approximation for N.fwdarw..infin. and .DELTA.n/n, we
get a formula which is analogous to Kogelnik's equation (for the
fundamental frequency of the periodic refraction index distribution) which
is the basic equation governing uniform holograms.
In order to determine analytically holographic bandwidths, secondary
maxima, etc. Kogelnik's theory adapted to the Lippmann-Bragg case must be
studied. Referring to the following equation,
##EQU2##
Eq. 1 defines diffraction efficiency .rho. of a holographic filter of the
present invention for the general case, where .xi. is the off-Bragg
parameter defined by the equation
##EQU3##
and where .upsilon. is the coupling constant (which represents diffraction
power) defined by the equation
##EQU4##
where T is the thickness of the hologram, .DELTA.n is the amplitude of the
refractive index n, .lambda..sub.80 is the wavelength at normal incidence,
and .theta.' is the angle of incidence in the medium. Taking equation 1,
if .xi. is set equal to 0, the equation for .rho. becomes
.rho.=tanh.sup.2 .upsilon. (Eq. 4)
which represents efficiency for the Kogelnik's case which applies to purely
phase (non-absorptive) reflection holograms. The on-Bragg condition
(.xi.=0) is defined as the condition where each angle of incidence has a
particular wavelength for which the Bragg condition is satisfied and each
wavelength has a particular angle of incidence for which the Bragg
condition is satisfied and can be illustrated as shown in FIG. 14
depicting the vector k of the light wave in the medium where
##EQU5##
According to FIG. 14, the Bragg condition can be satisfied by the pair of
vectors k', k.sub.o '), (k", k.sub.o "), etc. directed as in FIG. 14, and
satisfying Eq. 5, for the same grating vector K. If the angle of incidence
is kept constant and the wavelength .lambda. is changed or, vise versa,
.xi..noteq.0 and the off-Bragg case exists.
For the on-Bragg case of .xi.=0 the equation .eta.=tanh.sup.2 .upsilon.can
be plotted as shown in FIG. 15 which plots .eta. against .upsilon.. Thus,
it can be seen that as .eta. approaches the theoretical limit of 1,
.upsilon., the coupling constant or diffraction power, increases. .eta.
will be equal to roughly 99% of its full value of 1 when .upsilon.=.pi..
The Kogelnik theory, it will be remembered, assumes a fully uniform
structure, i.e., a structure where the Bragg planes are equidistant from
each other as can be seen in FIGS. 16A and B depicting the diffraction
constant .LAMBDA. and where the average refractive index n and the
refractive index modulation .DELTA.n are the same everywhere.
Assuming the Lippmann case, a correspondence exists between diffraction
efficiency .eta. and light rejection as shown in the following equation
R=.eta.,
where R is reflectivity and .eta. is diffraction efficiency. The analogous
equation for transmission is
T=1-R-A/S
where T is transmissivity, R is reflectivity, and A/S represents absorption
and scattering in the holographic material. Given the presence of the A/S
factor, this equation can be rewritten as T<1-R and | | |
