FIELD OF THE INVENTION
This invention relates generally to absorption spectroscopy and, in
particular, is directed to the use of a stable, high-finesse optical resonator for
ring-down cavity spectroscopy which incorporates Brewster's angle prism retroreflectors.
BACKGROUND OF THE INVENTION
Referring now to the drawing, wherein like reference numerals refer
to like elements throughout, Fig. 1 illustrates the electromagnetic spectrum on
a logarithmic scale. The science of spectroscopy studies spectra. In contrast with
sciences concerned with other parts of the spectrum, optics particularly involves
visible and near-visible light--a very narrow part of the available spectrum which
extends in wavelength from about 1 mm to about 1 nm. Near visible light includes
colors redder than red (infrared) and colors more violet than violet (ultraviolet).
The range extends just far enough to either side of visibility that the light can
still be handled by most lenses and mirrors made of the usual materials. The wavelength
dependence of optical properties of materials must often be considered.
Absorption-type spectroscopy offers high sensitivity, response times
on the order of microseconds, immunity from poisoning, and limited interference
from molecular species other than the species under study. Various molecular species,
but especially simple molecules such as water, can be detected or identified by
absorption spectroscopy. Thus, absorption spectroscopy provides a general method
of detecting important trace species. In the gas phase, the sensitivity and selectivity
of this method is optimized because the species have their absorption strength concentrated
in a set of sharp spectral lines. The narrow lines in the spectrum can be used to
discriminate against most interfering species.
In many industrial processes, the concentration of trace species in
flowing gas streams must be measured and analyzed with a high degree of speed and
accuracy. Such measurement and analysis is required because the concentration of
contaminants is often critical to the quality of the end product. Gases such as
N2, O2, H2, Ar, and He are used to manufacture
integrated circuits, for example, and the presence in those gases of impurities
such as water--even at parts per billion (ppb) levels--is damaging and reduces the
yield of operational circuits. Therefore, the relatively high sensitivity with which
water can be spectroscopically monitored is important to manufacturers of high-purity
gases used in the semiconductor industry. Various impurities must be detected in
other industrial applications.
Spectroscopy has obtained parts per million (ppm) level detection
for water in high-purity gases. Detection sensitivities at the ppb level are attainable
in some cases. Accordingly, several spectroscopic methods have been applied to such
applications as monitoring water content in gases, including: absorption measurements
in traditional long pathlength cells, photoacoustic spectroscopy, frequency modulation
spectroscopy, and intracavity laser absorption spectroscopy. These methods have
several features, discussed in U.S. Patent No. 5,528,040 issued to Lehmann, which
make them difficult to use and impractical for industrial applications. They have
been largely confined, therefore, to laboratory investigations.
In contrast, cavity ring-down spectroscopy (CRDS) has become an important
spectroscopic technique with applications to science, industrial process control,
and atmospheric trace gas detection. CRDS has been demonstrated as a technique for
the measurement of optical absorption that excels in the low-absorbance regime where
conventional methods have inadequate sensitivity. CRDS utilizes the mean lifetime
of photons in a high-finesse optical resonator as the absorption-sensitive observable.
Typically, the resonator is formed from a pair of nominally equivalent,
narrow band, ultra-high reflectivity dielectric mirrors, configured appropriately
to form a stable optical resonator. A laser pulse is injected into the resonator
through a mirror to experience a mean lifetime which depends upon the photon round-trip
transit time, the length of the resonator, the absorption cross section and number
density of the species, and a factor accounting for intrinsic resonator losses (which
arise largely from the frequency-dependent mirror reflectivities when diffraction
losses are negligible). The determination of optical absorption is transformed,
therefore, from the conventional power-ratio measurement to a measurement of decay
time. The ultimate sensitivity of CRDS is determined by the magnitude of the intrinsic
resonator losses, which can be minimized with techniques such as superpolishing
that permit the fabrication of ultra-low-loss optics.
At present, CRDS is limited to spectroscopic regions where high reflectivity
dielectric mirrors can be used. This has significantly limited the usefulness of
the method in much of the ultraviolet and infrared regions, because mirrors with
sufficiently high reflectivity are not presently available. Even in regions where
suitable dielectric mirrors are available, each set of mirrors only allows for operation
over a small range of wavelengths, typically a fractional range of a few percent.
Further, construction of many dielectric mirrors requires use of materials that
may degrade over time, especially when exposed to chemically corrosive environments.
Because these present limitations restrict or prevent the use of CRDS in many potential
applications, there is a clearly recognized need to improve upon the current state
of the art with respect to resonator construction.
The article by A. Pipino et al., "Evanescent wave cavity ring-down
spectroscopy with a total-internal reflection minicavity," Rev. Sci. Instrum. 68
(8) (Aug. 1997), presents one approach to an improved resonator construction. The
approach uses a monolithic, total internal reflection (TIR) ring resonator of regular
polygonal geometry (e.g., square and octagonal) with at least one convex facet to
induce stability. A light pulse is totally reflected by a first prism located outside
and in the vicinity of the resonator, creating an evanescent wave which enters the
resonator and excites the stable modes of the resonator through photon tunneling.
The absorption spectrum of matter located at the totally reflecting surfaces of
the resonator is obtained from the mean lifetime of a photon in the monolithic resonator,
which is extracted from the time dependence of the signal received at a detector
by out coupling with a second prism (also a totally reflecting prism located outside,
but in the vicinity of, the resonator). Thus, optical radiation enters and exits
the resonator by photon tunneling, which permits precise control of input and output
coupling. A miniature-resonator realization of CRDS results and the TIR-ring resonator
extends the CRDS concept to condensed matter spectroscopy. The broadband nature
of TIR circumvents the narrow bandwidth restriction imposed by dielectric mirrors
in conventional gas-phase CRDS. The work of A. Pipino et al. is only applicable
to TIR spectroscopy, which is intrinsically limited to short overall absorption
pathlengths, and thus powerful absorption strengths. In contrast, the present invention
provides long absorption pathlengths and thus allows for detection of weak absorption
It is also possible to build a resonator out of two Brewster's angle
roof prisms with crossed axes, as described in Gould et. al., "Crossed Roof Prism
Interferometer," Appl. Opt., Vol. 1, 533-34 (1962). The advantage of this resonator
is that it remains aligned for any small angle deviation of the prisms. The disadvantage
is that the Brewster's angle of one of the prisms must be set by construction, i.e.,
the Brewster's angle cannot be adjusted for wavelength by rotation of the prism.
There are applications (e.g., at specific wavelengths) where the robust alignment
of such a resonator is sufficiently desirable that the loss of the ability to tune
the Brewster's angle can be tolerated. The inability to adjust Brewster's angle,
however, restricts its application. Furthermore, the resonator described by Gould
et. al. is not optically stable, and thus cannot be used to produce a low-loss resonator,
due to diffraction.
US 4,161,436 discloses a light amplifier apparatus and the use of
optical elements at Brewster's angle for the purpose of polarization and reducing
losses in passing a light beam along a path.
To overcome the shortcomings of the known approaches to improved resonator
construction, a new high-finesse resonator (or optical resonator) for CRDS is provided.
An object of the present invention is to replace the conventional dielectric mirrors
with Brewster's angle prism retroreflectors, thereby providing an improved resonator.
A related object is to circumvent the narrow bandwidth restriction of conventional
dielectric mirrors used in CRDS. Another related object is to expand the variety
of potential applications for CRDS.
It is still another object of the present invention to provide a resonator
which incorporates materials that do not degrade significantly over time, even in
chemically corrosive environments. An additional object is to enable "tuning," or
alignment, of the resonator by rotating the prisms of the resonator. Yet another
object of the present invention is to provide an innovative CRDS resonator design
that achieves a low intrinsic energy loss and a well-defined relationship between
photon decay time and absorption.
SUMMARY OF THE INVENTION
To achieve these and other objects, and in view of its purposes, the
present invention provides a stable resonator for a ring-down cavity spectroscopy
cell having an optic axis as claimed in claim 1. Preferred embodiments are defined
in the subclaims. The resonator includes two Brewster's angle retroreflector prisms,
each having a plurality of total internal reflection surfaces. The prisms are disposed
in alignment along the optic axis of the resonator. One or both of the prisms can
be rotated independently so that light rays enter and leave a surface of the prism
nearly at Brewster's angle to the normal of the prism surface. This feature maintains
alignment between the prisms and allows the resonator to be tuned. One of the total
internal reflection surfaces of at least one of the prisms is a curved surface (either
a ground and polished curved surface or a surface curved by the addition, through
optically contacting or gluing, of a plano-convex lens to the surface).
In a preferred embodiment, each of the prisms has an apex angle of
about 135° minus Brewster's angle, a second angle of about 90°, and a third angle
of about 135° minus two times Brewster's angle.
It is to be understood that both the foregoing general description
and the following detailed description are exemplary, but are not restrictive, of
BRIEF DESCRIPTION OF THE DRAWING
The invention is best understood from the following detailed description
when read in connection with the accompanying drawing. It is emphasized that, according
to common practice, the various features of the drawing are not to scale. On the
contrary, the dimensions of the various features are arbitrarily expanded or reduced
for clarity. Included in the drawing are the following figures:
DETAILED DESCRIPTION OF THE INVENTION
- Fig. 1 illustrates the electromagnetic spectrum on a logarithmic scale;
- Fig. 2 illustrates total internal reflection in a prism;
- Fig. 3 illustrates deviation of light as it passes through a prism;
- Fig. 4 illustrates how a corner reflector (retroreflector) returns light in
exactly its original direction;
- Fig. 5 illustrates an unpolarized light beam incident upon a glass surface;
- Fig. 6 is a side view of a lens, showing meridional rays and depicting how an
off-axis object suffers astigmatism;
- Fig. 7 is a top view of the lens shown in Fig. 6, showing sagittal rays and
depicting how an off-axis object suffers astigmatism;
- Fig. 8 illustrates the improved resonator for CRDS using two Brewster's angle
retroreflector prisms in accordance with the present invention;
- Fig. 9A is a top view of the preferred prism used in the resonator shown in
- Fig. 9B is a back view of the prism of Fig. 9A;
- Fig. 10 shows how light incident rays enter and leave the prism, constructed
in accordance with the present invention, nearly at Brewster's angle to the normal
of the prism surface (with angles calculated for a prism made of fused silica);
- Fig. 11 depicts one of the total internal reflection surfaces on one prism ground
with a curvature according to the present invention; and
- Fig. 12 shows a plano-convex lens optically contacted or glued to a prism surface
according to the present invention; and
Presented immediately below is an introductory summary of the general
principles of modem optics relevant to the present invention. The summary is intended
to provide context for a complete understanding of the invention. Those who are
skilled in the art may proceed to the next section.
I. General Principles
When light travels from a first medium to a more optically dense second
medium, the light is refracted toward the normal. Light approaching a rarefied medium
from a dense medium is refracted away from the normal. There exists an angle, called
the critical angle, Θc, such that for all angles of incidence greater
than this angle, all of the light is reflected and none is transmitted. This effect
is called total internal reflection (TIR) and occurs inside a material that is optically
more dense than the material outside the boundary.
A prism is one type of refractive and reflective device. As shown
in Fig. 2, a prism 10 is a wedge of optical material that can either refract or
totally reflect light, depending on the angle of incidence. The 45° glass prism
shown in Fig. 2 is especially useful because incident light 12 entering normal to
one face will totally reflect out the other face, having changed direction by 90°.
Total reflection occurs because the light strikes the inner surface at 45 °, which
is greater than the critical angle of about 41 ° for glass. The line "N" represents
a line normal (perpendicular) to a surface.
Light energy striking an outer surface of the prism 10 at an angle,
shown in Fig. 3, is refracted in part, reflected in part by any internal surface,
and refracted again as it emerges as exiting light 14. It has deviated from its
original direction to emerge at a new angle. The general result is that the light
is bent partly back in the direction from which it came. The deviation depends on
the index of refraction of the prism, the angle of incidence, and on the angle in
the vertex of the prism. For a symmetrical arrangement of incident and exiting light,
12 and 14 respectively, the angle of deviation is a minimum. More complex prisms
use reflections to perform complex changes in image orientation. For example, the
comer-cube prism 10 of Fig. 4 has the geometric property of sending light back exactly
in the direction it came (i.e., to "retroreflect" the light).
Like all electromagnetic radiation, light is predicted by electromagnetic
theory to be a transverse wave: the directions of the vibrating electric and magnetic
vectors are at right angles to the direction of propagation (instead of parallel
to it, as in a longitudinal wave). The transverse wave also has the characteristic
that the vibrations of the electric vector are parallel to each other for all points
in the wave (i.e., the wave is oriented, or polarized). In reality, incoherent (non-laser)
light propagated in a given direction can consist of short, independent wavetrains
whose planes of vibration are randomly oriented about the direction of propagation.
Such light, although transverse, is unpolarized. Light can be partially or completely
polarized by reflection.
Fig. 5 shows unpolarized incident light 12 traveling in air and falling
on a glass surface 16. The glass has an index of refraction, n, of 1.5. The electric
vector for each wavetrain in the light can be resolved into two components. One
component is perpendicular to the plane of incidence, which is the plane of Fig.
5, and the other lies in the plane of incidence. The first component, represented
by the dots, is the S-polarization component (from the German "senkrecht," meaning
perpendicular). The second component, represented by the arrows, is the P-polarization
component (for parallel). On average, for completely unpolarized light, these two
components are of equal amplitude.
For glass or other dielectric materials, there is a particular angle
of incidence, called the polarizing angle (also called Brewster's angle, ΘB,
because it was found experimentally by David Brewster), at which the reflection
coefficient for the P-polarization component is zero. Thus, the light 18 reflected
from the glass, although of low intensity, is plane-polarized, with its plane of
vibration at right angles to the plane of incidence. The P-polarization component
at the polarizing angle is entirely refracted at angle of refraction Θr';
the S-polarization component is only partially refracted. Thus, the transmitted
light 20, which is of high intensity, is only partially polarized.
Because light is a wave, it does not abruptly vanish on the other
side of a boundary where there is total reflection. A damped non-propagating form
of the wave leaks past and appears along the boundary as an "evanescent wave." This
evanescent wave can be converted to a propagating wave if another surface is brought
very close to the interface, within a few wavelengths. This process is called "frustrated
total internal reflection. "
Materials often are optically anisotropic in their response to light.
In such materials, the response is different for the three independent directions
possible in the material; in contrast, isotropic materials show no directional preference.
For the purposes of this disclosure, materials are considered that have an identical
response in two of the three directions. The third (unique) direction is referred
to as the optic axis. In these materials, known as uniaxial, for light propagating
in any direction except along the optic axis, the light can be resolved into two
distinct waves with unique polarizations; one with the electric field oriented at
right angles to the optic axis (the ordinary wave), and the other with a component
of the electric field parallel to the optic axis (the extraordinary wave). These
waves of different polarization refract differently in the medium, having different
indices of refraction and, therefore, different speeds, which gives rise to a physical
separation of the light and is referred to as double refraction or birefringence.
Light that travels along the optic axis is always polarized at right angles to the
axis and is purely an ordinary wave. In the more general case, with different response
to light in the three spatial directions (biaxial systems), although more complex
in analysis, a similar birefringence occurs. Common birefringent materials include
calcite, crystalline quartz, and sapphire.
A lens 26 (or 22 in Figs. 6 and 7) maps each object point 28 into
an image point 30. In astigmatism, the rays from off-axis object points arrive at
different focal points. Consider the rays 32 from the top of the object shown in
side view in Fig. 6. Rays 32 are in a meridional plane and pass through the lens
26 asymmetrically. Meanwhile, in the top view of lens 26 shown in Fig. 7, another
set of rays 34 from the same point are in a sagittal plane and strike the lens 26
symmetrically. The focal points are separated for the two planes of rays, with the
focal point for the sagittal rays 34 located a farther distance from lens 26 than
for the meridional rays 32.
A simple way to test for astigmatism is to use a test pattern made
of dots. In the two different focal planes, meridional and sagittal, there will
be two different blurrings of the images of the pattern. In the meridional focal
plane, the dots blur tangentially while in the sagittal focal plane the dots blur
radially and form small arrows ("sagitta" is Latin for arrows) pointing toward the
axis. This astigmatism occurs for spherically symmetrical lenses. These effects
can be seen by this method only if the lens is free of other aberrations such as
spherical and coma. Spherical aberration results in marginal rays being focuses
closer to the lens than axial rays; coma is an aberration where slanted rays have
different focal points depending on which part of the lens they passed through.
II. The Resonator of the Present Invention
The present invention provides an improved resonator 100 for CRDS
based upon using two Brewster's angle retroreflector prisms 50, 52 made from a high
quality optical material. Fig. 8 is a schematic drawing of prisms 50, 52; optic
axis 54; and the expected optical path within each prism 50, 52. The polarizing
or Brewster's angle, ΘB, is shown relative to prism 50. The specific
angles of Fig. 8 are drawn assuming that the prisms 50, 52 are made from fused silica,
although (as will be discussed below) other materials could be used instead. Incident
light 12 and exiting light 14 are illustrated as input to and output from prism
52, respectively. The resonant optical beam undergoes two total internal reflections
without loss in each prism 50, 52 at about 45°, an angle which is greater than the
critical angle for fused quartz and most other common optical prism materials.
Resonator optical losses are caused principally by (1) scattering
due to imperfections and dirt at the surfaces of prisms 50, 52; (2) residual birefringence
in the optical material, due to either strain or misalignment of the optic axis
of the prism substrate material; (3) misalignment from parallelism of the coupling
surfaces of the prisms 50, 52; (4) deviation from Brewster's angle; and (5) internal
optical transmission loss in the prism substrates due to absorption or scattering.
Prisms 50, 52 can be constructed to provide low loss (i.e., less than 0.01 % per
round trip) over a wide range of the optical spectrum. In addition, some of the
most desirable materials for use as prism substrates, including but not limited
to fused silica, sapphire, and diamond, are materials that are extremely hard and
largely chemically inert, addressing the issue of hostile environments. Thus, resonator
100 for CRDS constructed from such prisms 50, 52 will meet and greatly expand the
range of applicability of CRDS.
III. The Prism Design of the Present Invention
The preferred design of prisms 50, 52 is illustrated in Figs. 9A and
9B. Taking it as an example, prism 50 has a first surface 1, a second surface 2,
a third surface 3, and a fourth surface 4. Fig. 9A is a top view of prism 50 and
shows the preferred length dimensions of surface 1 (25.8 mm), surface 2 (15 mm),
and surface 3 (19 mm). Fig. 9B is a back view of prism 50 and shows the preferred
height dimensions of surfaces 2, 3, and 4 (12.5 mm) and the preferred width of surfaces
3 and 4 combined (25.4 mm).
For prisms constructed of material with an index of refraction "n"
relative to the surrounding medium (i.e., n = n2 ÷ n1, where
n2 is the index of refraction of the prism and n1 is the index
of refraction of the medium surrounding the prism--typically air with n1
= 1), Brewster's angle, ΘB, is given by the arctangent of n. The
value of n for the example prism 50 shown in Figs. 9A and 9B is about 1.4607; ΘB
is about 55°36'. Prism 50 has a design center of about 0.532 µm. The apex angle
of prism 50 (Θ1) is set equal to 135° - ΘB and,
in the preferred embodiment, is about 79°24'. Angle Θ2 is preferably
about 90°. Angle Θ3 is set equal to 180° - 2ΘB
and, in the preferred embodiment, is about 68°48'.
Fig. 10 shows that rays of incident light 12 enter prism 50, and leave
as rays of exiting light 14, nearly at Brewster's angle (within a small deviation,
δ) to the normal "N" of surface 1. This results in small but controlled reflection
loss for optical radiation with P-polarization with respect to the Brewster's angle
surface. The value of n for the example prism 50 shown in Fig. 10 is approximately
1.45047; ΘB is about 55°25'. Prism 50 has a design center of 1
µm. Any optical radiation in the S-polarization is rapidly damped due to large reflection
loss. The symbol "ω" characterizes the size of the spot generated by the light
beam; negligible "clipping" of the beam occurs. The spot size for the lowest order
mode can be calculated from standard optical resonator theory. For the prism 50
illustrated in Fig. 10, the apex angle (Θ1) is preferably about
79°35' (or 79.58°). Angle Θ2 is preferably about 90°. Angle Θ3
is set equal to about 69°10' (or 69.17°).
IV. Material of Construction
The choice of optimal material for use in the construction of the
prisms 50, 52 will depend upon the particular application. In order to allow for
polishing of the surfaces to the required tolerances, a "hard" and chemically stable
substrate material is needed. Also desirable is a material that has both low absorption
and scattering loss over the spectral region of interest. Although four substrate
materials are known to be suitable, namely fused silica, sapphire, calcium fluoride,
and diamond, the present invention is not limited to these specific materials.
Fused silica is an excellent material which is widely used in the
optics industry for construction of precision optical components. It has low absorption
loss over a wide range of wavelengths. Because it is a glass, however, fused silica
has frozen disorder on the molecular level that leads to significant Raleigh scattering
loss, especially in the ultraviolet region.
Single crystal sapphire substrates are available and can also be manufactured
to precision specifications. Sapphire has a wider spectral range of low absorption
loss than fused silica; the highest quality samples have almost negligible scattering
loss throughout the visible and into the near-ultraviolet region. Sapphire is a
birefringent material and, to prevent excess loss due to polarization rotation within
the resonator optics, the unique optic axis must be oriented along the axis perpendicular
to the plane in Fig. 9A. This can be done to the required tolerance. The natural
birefringence characteristic of sapphire is advantageous because the material is
less susceptible to losses from strain birefringence which typically are the result
of imperfect mechanical mounting of the prisms.
Sapphire is likely the material of choice for most applications. Calcium
fluoride is likely the material of choice for much of the infrared region where
both fused silica and sapphire have too high a bulk absorption loss. Diamond would
in many ways be the ideal substrate material, except for the high cost of the material
The use of "roof" retroreflectors renders a prism optical resonator
alignment insensitive to small rotation of the prisms around the roof line and makes
for a more robust alignment. Such a resonator can be constructed using Brewster's
angle roof prisms with crossed axes. The advantage of this resonator is that it
remains aligned for any small angle deviation of the prisms. The disadvantage is
that the Brewster's angle of one of the prisms must be set by construction, i.e.,
it cannot be "tuned" by rotation of the prism around the roof axis. The resonator
100 of the present invention avoids that disadvantage.
Resonators can be characterized by a quality factor, Q, defined as
the energy stored divided by the energy lost per cycle. Resonators with higher "Q"
values are better at conserving energy and thus lead to higher sensitivity in cavity
ring-down spectroscopy. According to the present invention, the resonator "Q" and
coupling are controlled by tilting the prisms 50, 52 to adjust the level of reflection
loss. The reflection loss per surface is determined by the Fresnel relations, and
is approximately 10-4ΔΘ2, where ΔΘ
is the deviation from Brewster's angle in degrees.
Light rays undergo two internal bounces at prism surfaces 2 and 3,
and then leave the prism 50, 52 by transmission at surface 1. If angle Θ2
is constructed to be 90°, the input rays or incident light 12 and output rays or
exiting light 14 of the prism 50, 52 will be parallel but displaced if contained
in the plane of Fig. 9A. The angles of incidence of both the rays of incident light
12 and the rays of exiting light 14 are equal, and can be tuned by rotation of the
prism about the axis normal to the plane in Fig. 9A. It is understood that the prisms
50, 52 have been aligned such that the roof lines forming the 90° angles are normal
to the plane of Fig. 9A. As the prism 50, 52 is rotated, the angle of incidence
for the internal reflections will increase by the same angle on one surface, and
decrease by an equal amount on the other. In order to make these two total internal
reflection angles approximately equal, the apex angle of the prism (Θ1)
should be constructed to be equal to 135° - ΘB.
For prisms made of fused quartz, Brewster's angle varies from 55.5-57.1°
as the wavelength is varied from near-infrared to the onset of the vacuum ultraviolet
(200 nm) while the critical angle varies from 43.4° to 40.31°. As a result, one
pair of prisms 50, 52 can be designed to provide total internal reflection while
allowing the tilt to reach Brewster's angle over that range of wavelength. By selecting
angle Θ3 to be equal to 180° - 2ΘB, an optical
beam coupled into the resonator by reflection from surface 1 will propagate through
the crystal and also leave through surface 4 with an angle of incidence near Brewster's
angle. This will reduce the amount of light energy that is reflected inside the
prism that could be a source of unwanted stray light energy.
VI. Stability Control
Optical resonator 100 is formed from a pair of prisms 50, 52 which
act as retroreflectors. To form a stable optical resonator 100, and thus control
the diffraction of the optical beam as it bounces back and forth, at least one of
the total internal reflection surfaces on one prism is configured with a curvature.
Such a curved surface 60 is shown in surface 2 of prism 50 in Fig. 11.
To correct for the astigmatism produced both by the Brewster's angle
surface and reflection from the curved surface near 45°, the tangential curvature
of curved surface 60 must be
and the sagittal curvature (i.e., the curvature in the plane normal to that of
Fig. 11) must be
, where f is the desired effective focal length of the curved surface 60. The focal
length, f, is selected to be approximately equal to the separation distance between
the two prisms, 50, 52, which is on the order of 1 meter in the preferred embodiment,
to form a nearly half or folded confocal resonator 100.
Such an astigmatically compensated resonator 100 will have stable
resonant modes that are cylindrical symmetric, simplifying the design of the mode-matching
optics that are used to couple the radiation into the optical resonator 100. It
will be appreciated that the construction of such a prism 50 may be difficult because
it requires polishing and centering an astigmatic lens of precise curvature onto
one of the prism surfaces. A simple spherical surface ground into one prism surface,
such as surface 2, can be used with a curvature selected to give stability for rays
with sagittal deviation from the optic axis 54 of the resonator. The presence of
a focusing element inside the resonator 100 also compensates for small errors in
the manufactured angles and the positioning of the prisms, 50, 52, maintaining stability
and low loss despite small deviations of the optic axis 54. In the latter case,
the resonator eigenmodes will not be cylindrically symmetric.
Alternatively, as shown in Fig. 12, fabrication of prism 50 may be
simplified by following a two step procedure. First, the prism 50 is fabricated
with purely planar surfaces 1, 2, 3, and 4. Then a plano-convex lens 70 is made
of the same material as prism 50 and of the appropriate astigmatism. The plano surface
of the lens 70 is optically contacted to a prism surface (e.g., surface 2). When
optically contacted, the interface between the components disappears, eliminating
losses and providing optical performance equivalent to a monolithic (or integral,
or one-piece) structure. When working with near-infrared and visible wavelengths,
the lens 70 can be glued to the surface 2 of prism 50 with index-matching optical
cement 80 which is a much simpler procedure than optical contacting.