Fixed mount wavefront sensor

A rigid mount and method of mounting for a wavefront sensor. A wavefront dissector, such as a lenslet array, is rigidly mounted at a fixed distance relative to an imager, such as a CCD camera, without need for a relay imaging lens therebetween.

BACKGROUND OF THE INVENTION 
1. Field of the Invention (Technical Field) 
The present invention relates to methods and apparatuses for beam 
characterization and for construction of practical wavefront sensors for 
beam characterization, metrology, and other applications. 
2. Background Art 
In many instances where a laser beam is needed, it is important to know 
something about the laser beam quality. The beam quality affects how the 
beam will propagate, as well as how tightly it will focus. Unfortunately, 
beam quality is a somewhat elusive concept. Numerous attempts have been 
made to define beam quality, stretching back almost to the invention of 
the laser. In practice, any one of these measures will have some flaw in 
certain situations, and many different measures are often used. Among 
these is the M.sup.2 parameter (space-beamwidth product). 
The irradiance (or intensity) and phase distribution of a laser beam are 
sufficient for determining how the beam will propagate or how tightly it 
can be focused. Most of the beam quality measurements rely on 
characterizing the beam from only the irradiance distribution, since 
obtaining this is a comparably straightforward process. However, if both 
the irradiance and phase distribution could be obtained simultaneously, 
then all the information would be available from a single measurement. 
In general, phase is measured with an interferometer. Interferometers are 
sensitive instruments that have been extensively developed. They can be 
used to measure laser beams by using a shearing or filtered Mach-Zehnder 
arrangement, and can produce the desired irradiance and phase 
distribution. Unfortunately, these systems rapidly become complex, and are 
slow, unwieldy, sensitive to alignment, as well as being expensive. 
A Shack-Hartmann wavefront sensor is an alternative method for measuring 
both irradiance and phase. Such sensors have been developed by the 
military for defense adaptive optics programs over the last 25 years. This 
sensor is a simple device that is capable of measuring both irradiance and 
phase distributions in a single frame of data. The advent of micro-optics 
technology for making arrays of lenses has allowed these sensors to become 
much more sophisticated in recent years. In addition, advances in charge 
coupled device (CCD) cameras, computers and automated data acquisition 
equipment have brought the cost of the required components down 
considerably. With a Shack-Hartmann wavefront sensor it is relatively 
straightforward to determine the irradiance and phase of a beam. This 
allows not only the derivation of various beam quality parameters, but 
also the numerical propagation of the sampled beam to another location, 
where various parameters can then be measured. 
M.sup.2 has become a commonly used parameter to generally describe 
near-Gaussian laser beams. It is especially useful in that it allows a 
prediction of the real beam spot size and average irradiance at any 
successive plane using simple analytic expressions. This allows system 
designers the ability to know critical beam parameters at arbitrary planes 
in the optical system. Unfortunately, measuring M.sup.2 is somewhat 
difficult. To date, obtaining M.sup.2 has generally required measurements 
of propagation distributions at multiple locations along the beam path. 
Although efforts have been made to obtain this parameter in a single 
measurement, these still suffer from the need to make simultaneous 
measurements at more than one location. The present invention permits 
calculation of the parameter using only a single measurement at a single 
location. 
The following references relate to development of the present invention: A. 
E. Siegman, "New developments in laser resonators", SPIE Vol.1224, Optical 
Resonators (1990), pp.2-14; H. Weber, "Some historical and technical 
aspects of beam quality", Opt.QuantElec. 24 (1992), S861-864; M. W. 
Sasnett, and T. F. Johnston, Jr., "Beam characterization and measurement 
of propagation attributes", SPIE Vol. 1414, Laser Beam Diagnostics (1991), 
pp. 21-32; D. Malacara, ed., Optical Shop Testing, John Wiley & Sons, 
Inc., 1982; D. Kwo, G. Damas, W. Zmek, "A Hartmann-Shack wavefront sensor 
using a binary optics lenslet array", SPIE Vol.1544, pp. 66-74 (1991); W. 
H. Southwell, "Wave-front estimation from wavefront slope measurements", 
JOSA 70 (8), pp.993-1006 (August, 1980); J. A. Ruff and A. E. Siegman, 
"Single-pulse laser beam quality measurements using a CCD camera system", 
Appl.Opt., Vol.31, No.24 (Aug. 20, 92) pp. 4907-4908; Gleb Vdovin, 
LightPipes: beam propagation toolbox, ver.1.1, Electronic Instrumentation 
Laboratory, Technische Universiteit Delft, Netherlands, 1996; General 
Laser Analysis and Design (GLAD) code, v. 4.3, Applied Optics Research, 
Tucson, Ariz., 1994; A. E. Siegman, "Defining the Effective Radius of 
Curvature for a nonideal Optical Beam", IEEE J. Quant.Elec., Vol.27, No.5 
(May 1991), pp.1146-1148; D. R. Neal, T. J. O'Hern, J. R. Torczynski, M. 
E. Warren and R. Shul, "Wavefront sensors for optical diagnostics in fluid 
mechanics: application to heated flow, turbulence and droplet 
evaporation", SPIE Vol. 2005, pp. 194-203 (1993); L. Schmutz, "Adaptive 
optics: a modern cure for Newton's tremors", Photonics Spectra (April 
1993); D. R. Neal, J. D. Mansell, J. K Gruetzner, R. Morgan and M. E. 
Warren, "Specialized wavefront sensors for adaptive optics", SPIE Vol. 
2534, pp. 338-348 (1995); MATLAB for Windows, v. 4.2c.1, The MathWorks, 
Inc., Natick, Mass., 1994; and J. Goodman, Introduction to Fourier Optics, 
McGraw-Hill, (New York, 1968). 
The present invention is of a wavefront sensor that is capable of obtaining 
detailed irradiance and phase values from a single measurement. This 
sensor is based on a microlens array that is built using micro optics 
technology to provide fine sampling and good resolution. With the sensor, 
M.sup.2 can be determined. Because the full beam irradiance and phase 
distribution is known, a complete beam irradiance and phase distribution 
can be predicted anywhere along the beam. Using this sensor, a laser can 
be completely characterized and aligned. The user can immediately tell if 
the beam is single or multimode and can predict the spot size, full 
irradiance, and phase distribution at any plane in the optical system. The 
sensor is straightforward to use, simple, robust, and low cost. 
In addition to beam characterization, there are a wide variety of 
applications for wavefront sensors. These include metrology of surfaces, 
transmissive media, or other objects, measurement of turbulence or 
inhomogenous media, and static or dynamic measurement of surface or object 
deformation. These applications benefit in advances to basic sensor 
technology and can take advantage of many of the features, strengths, and 
objects of the methods and apparatuses described herein. 
A key advantage of the technologies disclosed herein is their inherent 
stability and robustness. This results from the extremely compact, robust, 
and rigid sensors that can be constructed at low cost. This is a 
significant advantage for a host of applications, including beam 
characterization and those mentioned in the preceding paragraph. In some 
cases, a robust, compact sensor enables an application otherwise 
impossible. Furthermore, the techniques needed for constructing such 
sensors are not readily apparent, with many subtleties being involved in 
design aspects that would not be apparent even to those highly skilled in 
the art. 
SUMMARY OF THE INVENTION (DISCLOSURE OF THE INVENTION) 
The present invention is of a method and apparatus for characterizing an 
energy beam (preferably a laser), comprising a two-dimensional wavefront 
sensor comprising a lenslet array and directing the beam through the 
sensor. In the preferred embodiment, the wavefront sensor is a 
Shack-Hartmann wavefront sensor. Wavefront slope and irradiance 
(preferably at a single location along the beam) are measured, wavefront 
slope distribution is integrated to produce wavefront or phase 
information, and a space-beamwidth product is calculated (preferably by, 
as later defined, the gradient method, the curvature removal method, or 
the Fourier propagation method). A detector array is employed, such as a 
charge coupled device (CCD) camera, a charge inductive device (CID) 
camera, or a CMOS camera, rigidly mounted behind the wavefront sensor, 
ideally at the focal point of the lenslet array. Shims are used to adjust 
spacing between the wavefront sensor and the detector array, following 
computation of shim size and placement to properly adjust the spacing. The 
sensor is calibrated, preferably against known optically induced wavefront 
curvature or tilt, and most preferably by generating a reference beam and 
computing one or more spot positions (using a computation such as the 
center-of-mass computation, matched filter computation, or correlation 
computation). 
The invention is additionally of a method of fabricating micro optics 
comprising: generating a digital description of the micro optic; 
fabricating a photomask; lithographically projecting the photomask's 
pattern onto a layer of photoresist placed on a substrate; etching the 
photoresist layer and the substrate until all photoresist has been 
removed; and applying this method to fabricating lenslet arrays for 
Shack-Hartmann wavefront sensors. 
The invention is also of a means for constructing practical wavefront 
sensors for a wide variety of applications. Techniques are disclosed for 
the design and construction of compact, rigid, and robust sensors. More 
specifically, the present invention is also of a mount for, and method of 
mounting, a wavefront dissector comprising rigidly mounting a wavefront 
dissector relative to an imager at a fixed distance without a relay 
imaging lens between the wavefront dissector and the imager. In the 
preferred embodiments, a region between the wavefront dissector and the 
imager is sealed, such as with an o-ring, epoxy, or glue. Mounting is 
preferably done employing one or more of the following: a mechanical mount 
connected directly to the imager; threads, preferably with a locking 
mechanism; shims; and/or integration of the wavefront dissector directly 
onto imager optics. The wavefront dissector is preferably a lenslet array. 
The present invention is additionally of a wavefront sensor comprising a 
wavefront dissector, an imager, and a rigid mount such that the wavefront 
dissector is rigidly held relative to the imager at a fixed distance 
without a relay imaging lens between said wavefront dissector and said 
imager. Preferably, the region between the wavefront dissector and the 
imager is sealed. 
A primary object of the present invention is provide a means for laser beam 
characterization using only a single measurement at a single location, 
which is also the primary advantage of the invention. 
Another object of the present invention is to provide such a means that is 
compact, robust, and insensitive to vibration. 
Other objects, advantages and novel features, and further scope of 
applicability of the present invention will be set forth in part in the 
detailed description to follow, taken in conjunction with the accompanying 
drawings, and in part will become apparent to those skilled in the art 
upon examination of the following, or may be learned by practice of the 
invention. The objects and advantages of the invention may be realized and 
attained by means of the instrumentalities and combinations particularly 
pointed out in the appended claims.

DESCRIPTION OF THE PREFERRED EMBODIMENTS (BEST MODES FOR CARRYING OUT THE 
INVENTION) 
The present invention is of a two-dimensional (preferably Shack-Hartmann) 
wavefront sensor that uses micro optic lenslet arrays to directly measure 
the wavefront slope (phase gradient) and amplitude of the laser beam. 
Referring to FIG. 1, this sensor uses an array of lenslets 21 that 
dissects the beam 23 into a number of samples. The focal spot locations 24 
of each of these lenslets (measured by a detector array 22) is related to 
the incoming wavefront slope over the lenslet. By integrating these 
measurements over the laser aperture, the wavefront or phase distribution 
can be determined. Because the power focused by each lenslet is also 
easily determined, this allows a complete measurement of the irradiance 
and phase distribution of the laser beam. Furthermore, all the information 
is obtained in a single measurement. Knowing the complete scalar field of 
the beam allows the detailed prediction of the actual beam's 
characteristics along its propagation path. In particular, the 
space-beamwidth product, M.sup.2, can be obtained in a single measurement. 
The irradiance and phase information can be used in concert with 
information about other elements in the optical train to predict the beam 
size, shape, phase and other characteristics anywhere in the optical 
train. For purposes of the specification and claims, "characterization" 
means using information gathered about an energy beam to predict 
characteristics of the beam, including but not limited to size, shape, 
irradiance and phase, anywhere in the train of the beam. 
The time-independent electric field of a coherent light beam directed along 
the z-axis can in general be described by its complex amplitude profile, 
E(x, y; z)=.vertline.E(x, y; z).vertline.exp[i.phi.(x, y; z)]. 
The phase 
##EQU1## 
where .phi.(x, y) is the wavefront or optical path difference referenced 
to the wavefront on the z-axis. The wavefront is also defined as the 
surface normal to the direction of propagation. Due to rapid temporal 
oscillations at optical frequencies, it is not possible to directly 
measure the electric field. However, by using a Shack-Hartmann wavefront 
sensor, one can indirectly reconstruct a discrete approximation to the 
time independent electric field at a given plane normal to the z-axis. 
A Shack-Hartmann sensor provides a method for measuring the phase and 
irradiance of an incident light beam. The sensor is based on a lenslet 
array that splits the incoming light into a series of subapertures, each 
of which creates a focus on a detector (usually a CCD camera) (see FIG. 
1). The wavefront of the incoming beam is defined as a surface that is 
normal to the local propagation direction of the light. Hence distorted 
light will have a wide collection of propagation directions and the 
separate lenslets will focus the light into different positions on the 
detector. By determining the position of each of these focal spots the 
wavefront slope over the lenslet can be measured. The wavefront itself 
must be reconstructed by integrating these wavefront slope measurements. 
There are several steps in wavefront sensor data reduction. First the 
sensor is placed in a reference beam and data is acquired with a camera 
for calibration. Since there are a large number of focal spots in the 
field, the image must be divided into a set of small windows, each 
centered on a focal spot peak, with one window per lenslet. Once the 
windows have been found, a centroid is computed using a center-of-mass 
algorithm: 
##EQU2## 
With pixels indicated by the i,j indices, a sum is made over the pixels in 
each window (W.sub.l, where l indicates a particular lenslet) of the 
irradiance-weighted locations. (When not mentioned explicitly, similar 
equations hold for the y-axis.) This results in a reference set of 
centroids, .rho..sub.x,l .vertline..sub.REF and .rho..sub.y,l 
.vertline..sub.REF. The sensor is now ready for acquisition and 
measurement of actual data. Note that the reference beam need not be a 
collimated beam, as long as its characteristics are known; results are 
then deviations from this reference. 
The first step in analyzing real data is the same as that for the reference 
data. The data is acquired and digitized and then centroids are computed 
using the windows calculated in the reference step. A typical image is 
shown in FIG. 2. Once these centroids have been obtained, and with the 
lenslet to CCD distance, L, known, the wavefront slopes can be calculated: 
##EQU3## 
FIG. 3 displays an example of this calculation for an expanding beam. 
The final step is the wavefront reconstruction. This is the solution of the 
gradient equation, 
##EQU4## 
where the data provides sampled values for the wavefront gradient, 
##EQU5## 
Here .theta..sub.x and .theta..sub.y are the measured slope data. The 
reconstruction proceeds by finding a set of .phi..sub.l values that obey 
the above equations. Commonly used methods include least-squares 
procedures and marching methods. Southwell teaches a variety of methods 
for solving Eq. 4. 
One method that has advantages in that it takes account of the irradiance 
distribution as well as the phase slopes is known as the modal 
reconstruction method. In this method the data is fit to the derivatives 
of an analytical surface described by an expansion in terms of a set of 
basis functions. One simple case is the use of a polynomial expansion. 
Thus the phase might be described by 
EQU .phi.=.alpha..sub.00 +.alpha..sub.10 x+.alpha..sub.01 y+.alpha..sub.11 xy+ 
. . . +.alpha..sub.ij x.sup.i y.sup.j. (5) 
This description uses normal polynomials in x and y. Different basis sets 
may also be used, such as Hermite and Zernike functions. The derivatives 
of the phase are then easily determined by 
##EQU6## 
with a similar expression for the y-derivative. Eq. 6 is then fit to the 
wavefront slope data using a least-squares method. Since Eq. 5 determines 
the wavefront phase in terms of these a.sub.ij (with an arbitrary constant 
of integration, a.sub.00, which is usually set equal to zero), the 
complete wavefront has been determined. The irradiance for each lenslet is 
determined by the denominator of Eq. 1. FIGS. 4 and 5 illustrate a typical 
phase and irradiance distribution obtained by this method. 
The above provides a complete measurement of the beam irradiance and phase, 
sampled by the lenslets. This measure is from a single time and location. 
It can be used for calculation of other parameters of interest, such as 
M.sup.2 as discussed below. In addition, the reconstructed wavefront can 
be numerically propagated to another location using a standard propagation 
code (e.g., LightPipes or GLAD) or other propagation method. 
Shack-Hartmann wavefront sensors have been used for many years as sensors 
for adaptive optics in military high energy laser and atmospheric 
compensation. However, recently they have been applied to measurement 
applications in thermal flow, turbulence and surface measurement. While 
some of these early sensors were one dimensional in order to make high 
bandwidth measurements, recently fully two-dimensional sensors have been 
developed. 
One of the chief limitations on making wavefront sensors is the fabrication 
of an appropriate lenslet array. Early lenslet arrays were either 
individually ground and polished lens segments that were assembled 
together, or were fabricated with step and repeat processes. With the 
advent of micro (continuous, diffractive, or binary) optics technology, 
the methods for fabricating lenslet arrays have greatly improved. This 
technology is discussed in detail in parent U.S. patent application Ser. 
No. 08/678,019. 
Micro optics technology is the application of integrated circuit 
manufacturing technology to the fabrication of optics. Swanson, et al. 
(U.S. Pat. No. 4,895,790) developed a process for the fabrication of micro 
optics known as binary optics. In this process, described in FIG. 12, a 
series of high contrast masks are used to construct the desired surface 
profile. Through accurate alignment of each new mask to the structure 
fabricated in the previous step, the desired optic may be approximated by 
a binary structure, much like a binary number can be used to represent 
higher values, even though only ones and zeros are used. 
There have been many additional means developed for fabricating micro 
optics. The photolithography etching processes may be used, but it is 
helpful to reduce the requirements for multi-mask alignment and the number 
of required masks. Through the use of special gray mask materials, such as 
High Energy Beam Sensitive (HEBS) glass (U.S. Pat. No. 5,078,771, to 
Che-Kuang Wu), the desired structure may be encoded as optical density 
variations in the mask. This allows a single mask, with a single exposure, 
to be used to fabricate the entire structure. In the present invention, 
micro optic fabrication method or methods may be used, however the gray 
mask process may have advantages of resolution and ease of fabrication in 
the present invention. The steps needed to create an optic using the gray 
mask process of the invention are as follows and are shown in FIG. 21: 
1. The design of the optic is developed using a series of computer programs 
to describe the desired lenslet shape and profile. These include a code to 
define the shape and placement that solves the exact Huygens-Fresnel 
equations for a lens, a diffractive analysis code, a photomask layout 
tool, and various other elements as needed to produce a complete digital 
description of the lens or lens array. 
2. A photomask is fabricated using the digital data described above whose 
optical density is a direct function of the desired final optic surface 
profile height. This mask may be fabricated through a number of methods, 
including the use of e-beam sensitive material, variable thickness metal 
or other coating layer, or though other techniques as appropriate. 
3. A thin layer of photoresist is spun onto the substrate (which may be 
made of fused silica or other appropriate optical material). The mask 
pattern is transferred to this layer by uv contact or projection 
lithography. Once the photoresist is developed, it assumes a surface 
profile shape similar, and directly related through a known function to, 
the shape of the desired lens. 
4. The substrate and photoresist is etched using chemical, reactive-ion, 
ion-milling, or other etching process that etches both materials until all 
of the photoresist has been removed. At this point, a replica of the lens 
surface profile has been produced in the substrate. 
This series of steps can result in lens profiles that are produced with no 
alignment between successive steps, a single etch step, and a much 
smoother profile. With this method, extremely high precision lens arrays 
can be made. They have an extremely precise surface profile, with features 
down to 1 micrometer and 100% fill factor. Furthermore, they can be 
arranged in many different configurations to compensate for other effects 
in the optical system, as taught in parent application Ser. No. 
08/648,019. 
The other necessary item to make a wavefront sensor is a detection device, 
preferably a CCD, CID, or CMOS camera. Off-the-shelf cameras, which are 
low cost, yield excellent results. The camera is interfaced to a frame 
grabber for data acquisition into the computer. Once data is acquired, the 
analysis proceeds along the lines described above. Other detection means 
may be used to take advantage of various detector technologies to improve 
or modify dynamic range, sensitivity, frequency response, spectral 
sensitivity, and so forth. 
In the preferred embodiment, the lenslet array is mounted directly in front 
of the detector, as appropriate to the application, in a rigid assembly 
(preferably at the focal point of the lenslet array) with no optical or 
other elements located between the lenslet array and the detector. In this 
arrangement, the sensor head is extremely compact and lightweight. This 
means that the sensor can be mounted on common optical mounts, or easily 
incorporated into other optical systems. This is a significant advantage 
in some cases where there are severe restrictions on space and weight. The 
resulting sensor design is extremely rugged and robust, and has no moving 
parts. This allows use in non-ideal environments. By coupling with 
electronic shuttering or pulsed light sources, the sensor can be used in 
high vibration environments, such as industrial production-line 
environments, that would otherwise preclude the use of sensitive optical 
instruments. For many applications, small size, weight, and vibration 
insensitivity allow measurements to be made that were not possible 
otherwise. 
Accurate wavefront slope measurements require that the lenslet array be 
located a precise, known distance from the detector. There are a number of 
means to achieve mechanically rigid, precision spacing. This spacing must 
be precisely and rigidly controlled, and must be adjusted through a 
calibration step to a known, predetermined value. Therefore a means is 
needed for positioning, measuring, and adjusting this lenslet position. 
Preferred embodiments of the present invention provide an enhanced, 
practical sensor that is compact, robust, and insensitive to vibration. It 
must be noted that the utility of the rigid mount embodiments is not 
limited to only measurement of laser or energy beams. By directly affixing 
the lenslet array in front of the detector, with no intervening optics, a 
number of advantages are achieved. The combined instrument can be made 
extremely compact and lightweight. In most cases, the combined instrument 
is barely larger than the CCD camera itself. This enables a large number 
of applications where the sensor head can be placed into an existing 
optical system. For example, a small telescope can be tested by mounting 
the sensor head directly onto the telescope eyepiece. The small size of 
this arrangement is the only thing that makes this possible because the 
telescope eyepiece and mount cannot support any significant load. Any 
sensor that has internal optics will be much larger. When built as a total 
instrument, this means that the optical system will need to be aligned to 
the sensor that is fixed on some optical table. The size of such an 
instrument is determined by the internal optical system. While such 
optical systems can be made somewhat compact with sophisticated optics and 
opto-mechanics, the system will still be significantly longer and heavier 
than a direct lenslet array mount. 
A rigid mount will greatly reduce sensitivity to vibration. In some of the 
prior art, the lenslet array has been mounted on an adjustable stage that 
allows several degrees of freedom. This allows adjustment for alignment in 
x and y, rotation, and focus. In addition, lenslet array tip and tilt have 
often been adjustable. These adjustments require the use of 
opto-mechanical systems that are much larger and more complicated than a 
robust mounting scheme. More importantly, they are sensitive to vibration. 
While the use of such a mount allows for adjustment of the lenslet array 
relative to the camera, it means that vibration, inadvertent adjustment or 
other force may cause a misalignment of the system. Thus vibration can 
easily induce conditions requiring realignment, new reference files, or 
worse. 
A rigid mount also preferably completely seals the optical path between the 
lenslet array and the detector. Therefore, turbulence, dust, and other 
effects will be minimized. With an adjustable mount this seal is very 
difficult to maintain. This is very important since it is desirable to use 
a glassless CCD which may be permanently damaged by dust. This seal may 
also be used to protect CCD imagers that may be damaged by moisture or 
exposure to oxygen. 
Furthermore, the mount acts as a heat sink for the camera and eventually 
brings the lenslet array, mount and CCD camera to the same temperature. 
This reduces any temperature gradients that would otherwise induce 
turbulence. If the camera is calibrated in this condition, this greatly 
improves the overall stability of the instrument, because minimal changes 
in temperature induced expansion would be expected. 
A properly designed rigid mount will serve to enhance the stability of the 
CCD/lenslet assembly. In many CCD cameras the CCD imager is mounted in a 
socket that is soldered to a printed circuit (PC) card. The PC card is 
mounted with screws to the camera body. Thus, the only connection that 
determines the position of the CCD imager is through the PC card, which 
may be made of fiberglass, phenolic or some other flexible material. A 
properly designed rigid mount will preferably include replacing the camera 
front plate so that a firm contact is achieved between the camera front 
plate/lenslet mount assembly and the frame of the CCD imager. 
To make an effective rigid mount, a number of objects should preferably be 
achieved. The mount should completely encapsulate the lenslet array. It 
should position the lenslet array precisely at the required distance from 
the CCD. It should completely seal the CCD imager against dust and other 
foreign objects. It should mount against the frame of the CCD imager to 
provide a complete rigid and stiff assembly. The mount should have 
provision for changing the spacing of the lenslet array relative to the 
CCD imager without compromising the stability and robustness of the 
system, and it should provide a means for fixing the rotation of the 
lenslet array relative to the CCD imager. 
Previous implementations of wavefront sensors have been mostly research 
oriented in nature. It is common practice to place a lenslet array in 
front of a detector and so build a Shack-Hartmann wavefront sensor. 
However, one is immediately faced with issues such as alignment, lenslet 
to CCD spacing, and lenslet array rotation. In the prior art, these 
problems are routinely solved by one of two methods: 
1. Mount the camera and lenslet array in separate adjustable mounts. 
2. Use a relay-imaging lens to image the plane of the focal spots (or other 
plane as desired) onto the detector. 
While both of these methods have some advantages, they do not offer the 
advantages of the robust rigid assembly of the invention. Most wavefront 
sensors that have been built for adaptive optics research use a 
combination of these two methods. In the research environment, this allows 
for flexibility in the use and development of the instrument. 
There are two implementations of commercial sensors that both employ the 
relay imaging lens method. In the WaveScope.TM., built by Adaptive Optics 
Associates, a magnetic mount is provided for the lenslet array and a relay 
lenslet is mounted directly on the camera. The camera/lens assembly is 
mounted on an adjustable stage that may be moved relative to the lenslet 
array for calibration or other purpose. This system is over 1/2 m in 
length, and clearly is not a compact, robust, rigid assembly. This 
instrument is described in part by U.S. Pat. Nos. 4,490,039, 4,737,621, 
and 5,629,765. In the Zeiss wavefront sensor, the Detect.TM. 16, 32 or 64, 
is much more compact. However, it also uses an internal relay imaging 
lens. While this system may achieve some of the advantages in terms of 
compactness, the use of the relay lens introduces cost, complexity and may 
reduce the accuracy of the final system due to aberrations of the internal 
relay lens. While low aberrations lenses could be used in this 
configuration, this would likely result in a much larger package and 
higher cost, thereby obviating the compactness advantage. 
The present invention is a derivative of a Shack-Hartmann sensor, which is 
used to measure an incoming wavefront distribution. It should be noted 
that the wavefront dissector may be a lenslet array, difraction grating 
array, array of holes, or any other means for creating a set of spots or 
separated regions of light on the detector in a known manner, and 
"wavefront dissector" is so defined for purposes of the specification and 
claims. 
If the wavefront dissection element is arranged in a rigid and robust 
mount, there are a number of advantages. These include compactness, 
insensitivity to vibrations, freedom from calibration drift, enhanced 
accuracy, better temperature stabilization and easier operation. In order 
to achieve the advantages of a practical wavefront sensor, the sensor 
should be designed to be compact, rigid and robust. There are a number of 
means for achieving these ends. 
There are two preferred embodiments for this positioning. The first is 
shown in FIGS. 19, 22, and 23. In FIG. 19, the lenslet array 57, is 
mounted in an insert 53, which is custom fit to the size and shape of the 
particular lenslet array. Different inserts may be used that are matched 
to the lenslet array focal length and required position. The lenslet array 
57 may be glued or otherwise affixed to the insert 53. The insert/lenslet 
array assembly is positioned in a sensor body 52 which is firmly attached 
to the detector front plate 51. The sensor body is mounted with threaded 
50 or other means with considerable torque so that the camera front plate 
and the sensor body are an integral assembly, with no possibility of 
relative motion. Special tools may be required to accomplish this 
attachment step. The CCD or other detector 58 is mounted to the camera 
front plate 51 in a rigid manner with the use of shims or other means so 
that precise physical contact is maintained between the detector chip 
frame and the camera front plate. This assures that the sensor body will 
maintain precise physical spacing and alignment to the detector element. 
To mount the lenslet array/insert assembly to the sensor body 52 at a known 
spacing, the following elements are used. One or more shims 54 of a 
precise thickness and character are used between a step in the sensor body 
and the insert. The thickness and selection of these shims can be 
determined in a calibration step described below. A nylon or other low 
friction material 55 is used between the insert 53 and the retaining ring 
56 to prevent rotation during final assembly and tightening. The lenslet 
array 57 and detector 58 are rotationally aligned relative to one another 
by rotating the insert 53 relative to the sensor body 52. This is 
accomplished through the use of a special tool that is designed to 
interface to notches or tabs 59 on the insert 53. This rotation step may 
be accurately accomplished while monitoring the position of the focal 
spots electronically while rotating the insert/lenslet assembly. 
In some embodiments, it may be preferable to replace the front plate of the 
camera to assure this contact. It may also be desirable to provide for an 
o-ring seal or for fittings to allow purging with clean air or nitrogen. 
A similar concept can be used for different size cameras, wavelengths, or 
other operation. FIG. 22 is an example of a mount that is used for a 
large-format IR camera. The lenslet in this case is made from silicon 
because the instrument is designed for operation from 1.1-1.7 mm. The 
mount is similar in design and construction to the mount described in FIG. 
19, except scaled to the larger lenslet array and camera dimensions. The 
sensor body 72 screws into the camera faceplate 71 with large diameter 
threads. The camera faceplate has provision for an o-ring seal 79 to 
completely enclose the optical path between the lenslet array 77 and the 
detector 78. In other configurations, a port for a nitrogen purge line is 
also preferably provided. This assembly has the same basic concept with 
the insert 73, shims 74, slip ring 75, and insert retainer 76. 
While this design has the advantage of easy assembly and rotation, it is 
also possible to construct simpler, lower cost designs. FIG. 23 is an 
example of a configuration where the threads of the camera itself are used 
as the primary mounting elements for the lenslet array. To achieve all of 
the objectives of the rigid mount, a camera must be used that has a rigid 
mounting of the CCD chip 39 with CCD detector 34. The lenslet array 36 is 
mounted with epoxy or other means in an insert 37 which slips into the 
camera front mount. It may be either free to rotate, or arranged with 
threads to allow linear adjustment. Ideally it should contact the frame of 
the CCD chip 39. Shims 31 may be used to adjust the axial position. 
Notches 33 may be used to allow a tool to rotate the insert assembly. A 
retaining ring 32 fixes the elements and holds everything in alignment in 
conjunction with slip ring 38 and camera front plate 30. This arrangement 
has the advantage of fewer parts and simpler mechanical systems. However, 
it relies on the internal organization of the camera to be a rigid sealed 
assembly. This may be true for some cameras, but for some it is not 
adequate. 
While shims are discussed as a means for rigidly fixing the lenslet to 
detector spacing, other means are also useful. This may include the use of 
a threaded assembly with a locking mechanism or other means as 
appropriate. 
In another embodiment shown in FIG. 20, the lenslet array 88 is designed 
with mounting means directly on the lenslet substrate 83. In this 
embodiment, the lenslet array is arranged with a special mounting surface 
87 that is designed to interface to the frame for the detector 81. The 
detector sensitive surface 84 is mounted rigidly with epoxy, solder, or 
other means to the detector frame 81. The detector frame 81 has sufficient 
depth for the wire bonds 86. It also holds pins or other connection means 
85 for electrical connection to other circuits. Precision machining of the 
various components is used to assure the proper separation of components. 
The lenslet array substrate 83 is fixed to the detector frame 81 using UV 
cured epoxy or other means. The mounting surface 87 is designed such that 
there is sufficient space to allow for slight rotational alignments. The 
use of UV cured epoxy allows the position of focal spots to be monitored 
during an alignment step while the epoxy is in place but not yet set. Once 
the final position has been obtained, then the epoxy is set by application 
of UV light. This embodiment allows for an extremely compact, hermitically 
sealed and robust sensor, in configurations where the detector is not 
mounted in a separate mechanical fixture. 
In order to use these rigid assembly embodiments, the lenslet-to-CCD 
distance, L in Eq. 2, must be determined experimentally. To accomplish 
this objective (referring to FIG. 13), an optical system 10, comprising 
laser 16 and lenslet array 18 and detector array 20 (rigidly assembled to 
the lenslet array to form a wavefront sensor) as shown in FIG. 13 is 
preferably used to introduce various amounts of wavefront curvature in a 
known fashion. A pair of achromatic lenses 12 and 14 spaced 2 f apart are 
examples of devices which may be employed. By adjusting the position of 
the second lens 14 slightly (e.g., with a micrometer driven translation 
stage), data with known curvature as shown in FIG. 14 may be generated. 
Aberrations in the lenses can be dealt with by referencing the wavefront 
sensor with light that passes through the same lenses at exactly 2 f 
spacing. It is not necessary to use two lenses in this configuration. A 
rotating wedge, a single lens with a point source and an accurate tip/tilt 
stage, or other means for introducing low order wavefronts with known 
character can be employed. 
To calibrate the sensor by determining the exact separation L of lenslet 
and CCD, data is acquired as a function of the position of lens 14. A 
typical summary of this data is presented in FIG. 15, as a plot of 
measured wavefront curvature versus input curvature. The slope of this 
line is related to the exact distance between the lenslet array and the 
detector. Using this information, this distance can be adjusted to produce 
an exact match between nominal lens focal length and camera to lens 
spacing, preferably through the use of shims or other means that maintain 
the rigid nature of the wavefront sensor. Setting L=f (where f is the 
lenslet focal length) produces the smallest spot size, allowing the 
largest dynamic range on the sensor. This positioning procedure allows for 
an accurate determination of L. Typical post-positioning data after adding 
the appropriate shims is shown in FIG. 16. This procedure allows the 
wavefront sensor to be accurately calibrated even though a rigid mounting 
system for the lenslet array is used. Of course, the same procedure may be 
used to set the distance to any desired value. 
One commonly used parameter for characterizing laser beam quality is the 
space-beamwidth product, or M.sup.2 parameter. Let the complex electrical 
field distribution of a beam directed along the z-axis be given by E(x, y, 
z), with the corresponding spatial frequency domain description of the 
beam, P(s.sub.x,s.sub.y,z) given by its Fourier transform, 
P(s.sub.x,s.sub.y,z)=.Fourier.{E(x, y, z)}. Beam irradiances in each 
domain are then defined by I(x, y, z).tbd..vertline.E(x Y, 
z).vertline..sup.2 and by 
P(s.sub.x,s.sub.y,z).tbd..vertline.P(s.sub.x,s.sub.y,z).vertline..sup.2. 
The M.sup.2 parameter is then defined by: 
EQU M.sub.x.sup.2 =4.pi..sigma..sub.x.sbsb.o .sigma..sub.s.sbsb.x(7) 
where .sigma..sub.x is irradiance weighted standard deviation at position z 
in t x-direction, defined by 
##EQU7## 
and .sigma..sub.s.sbsb.x the spatial-frequency standard deviation of the 
beam along the x-axis 
##EQU8## 
Note that .sigma..sub.s.sbsb.x.sup.2 is not a function of z, and can be 
obtained using the Fourier transform of the electric field. 
(The first moments of the beam along the x-axis and the s.sub.x -axis are 
indicated by x and s, respectively. The spot size of the beam is W.sub.x 
(z).tbd.2.sigma..sub.x. The corresponding y-axis quantities hold for 
.sigma..sub.y.sbsb.o, .sigma..sub.s.sbsb.y, etc., mutatis mutandis, 
throughout this description. In addition, the normalizing factor in the 
denominators shall be indicated by P=.intg..intg.I(x, y, 
z.sub.1)dxdy=.intg..intg.P(s.sub.x,s.sub.y, z)ds.sub.x ds.sub.y.) 
In the case of a paraxial beam in the z-direction, with an arbitrary 
reference plane (z.sub.1), the irradiance weighted standard deviation will 
have an axial distribution given by, 
EQU .sigma..sub.x.sup.2 (z)=.sigma..sup.2 (z.sub.1)-A.sub.x,1 
.times.(z-z.sub.1)+.lambda..sub.s.sbsb.x.sup.2 .sigma..sub.s.sbsb.x.sup.2 
.times.(z-z.sub.1).sup.2 (10) 
where A.sub.x,1 is given by the function 
##EQU9## 
The beam waist, or location of minimum irradiance variance, is obtained 
from Eq. 10: 
##EQU10## 
Substituting back into Eq. 10 yields the relationship 
##EQU11## 
M.sub.x.sup.2 follows immediately from Eq. 7. 
These formulas form the basis for defining the space-beamwidth product, 
M.sup.2 To calculate M.sup.2 from discrete irradiance and phase 
measurements requires appropriate processing of the data. The present 
invention provides three exemplary methods of equal validity, dependent 
upon experimental parameters such as instrument noise, resolution, and 
dynamic range, or depended upon wavefront and irradiance distribution 
characteristics. The three methods may be summarized as gradient method, 
curvature removal method, and multiple propagation method, and are next 
discussed. 
It should be noted that Eqs. 13 and 11 are not derived from series 
expansions in the vicinity of the beam waist, but are analytical 
derivations dependent only upon the paraxial wave equation, the paraxial 
propagation assumption, and the Fourier transform relationships between 
the complex electric field amplitude (E(x, y, z)) and the 
spatial-frequency beam description (P(s.sub.x, s.sub.y, z)). 
Gradient Method. As shown previously, it is possible to obtain a discrete 
description of the beam electric field amplitude and phase in a given 
plane normal to the z-axis. As part of the measuring process, discrete 
values for 
##EQU12## 
are also obtained. By means of the above formulae and standard numerical 
integration techniques one can then obtain values for M.sup.2 and the 
waist locations, Z.sub.ox,y. 
The sequence is as follows, and is referred to as the gradient method. (See 
FIG. 6.) From the Shack-Hartmann sensor, the distribution of irradiance, 
I(x, y, z.sub.1) and wavefront slope, 
##EQU13## 
are obtained. From these, the electric field, 
##EQU14## 
is calculated. The spatial-frequency electric field distribution, 
P(s.sub.x, s.sub.y, z.sub.1), is derived using a Fourier transform 
algorithm, such as the fast Fourier transform (FFT). From these the 
irradiance distributions in both domains, I(x, y, z.sub.1) and P(s.sub.x, 
s.sub.y, z.sub.1), are obtained, whence numerical values for the 
variances, .sigma..sub.x.sup.2 (z.sub.1) and .sigma..sub.s.sbsb.x.sup.2 
are calculated. Concurrently, the integral of Eq. 11 is computed by using 
the directly measured values of 
##EQU15## 
with the results being used in Eqs. 13 and 12 to produce the waist 
location and irradiance variance. The waist irradiance standard deviation 
and the spatial-frequency standard deviations immediately yield the 
M.sup.2 parameter per Eq. 7. 
Curvature Removal Method. The M.sup.2 of a laser beam is completely 
independent on its overall curvature. Hence, performing an operation on 
the beam that affects its curvature will not affect the resultant value of 
M.sup.2. Many previous methods for measuring M.sup.2 depend on this fact, 
in that a weak focusing lens is introduced, and the second moment of the 
beam measured at differing Z locations. The weak lens is used to assure 
that all of the light arrives at the detector and to reposition the beam 
waist such that measurements are made near this waist. In general, this 
gives the best sensitivity to the measurement process. 
Since a Shack-Hartmann wavefront sensor gives a complete measure of the 
irradiance and phase distribution of the light, the same operation can be 
performed without using a physical lens. Wavefront curvature may be added 
or subtracted from the digitally stored irradiance and phase distribution 
without affecting the M.sup.2 of the beam. This operation can then be 
performed as part of the numerical process of determining M.sup.2, without 
need to introduce a physical beam. 
To compute M.sup.2, information at the waist (denoted by the subscript 0 in 
equations 7-13) is needed. The waist is that plane that has infinite 
effective radius of curvature. With a wavefront sensor measurements may be 
made at another location, however. It is somewhat difficult to construct 
the location of the waist, and hence determine the second moment at the 
waist as required by equation 7. In the gradient method, this was the 
primary object: to use the gradient information (also produced by the 
wavefront sensor) to compute the location and size of the waist, so that 
M.sup.2 can be determined. However, since wavefront curvature has no 
affect on the M.sup.2 calculation, an artificial waist can be created by 
removing the average curvature from the beam. This can be done by fitting 
the wavefront to a polynomial with second order terms, such as in Eq. 5. 
These second order terms are related to the radius of curvature of the 
real beam R(z.sub.1). The wavefront corresponding to the fit can then be 
subtracted out of the data, and the M.sup.2 value computed through 
application of Eqs. 7-13, where the measurement plane is also the waist 
plane. 
This method, referred to as the curvature removal method, has several 
advantages. It is simple to implement, and requires a minimum of 
calculations to determine M.sup.2. The calculation of M.sup.2 does not 
rely on determination of the waist plane or the waist size, and is thus 
somewhat less sensitive to noise or other errors. However, often these are 
desirable parameters as well. Hence additional calculations are needed to 
calculate the waist distance and size. 
The real beam spot size propagation equation states: 
##EQU16## 
Furthermore, the real beam radius of curvature is given by 
##EQU17## 
R(z.sub.1) is known from the curvature removal step. 
Since the irradiance and phase of the beam is known (at an arbitrary plane 
z.sub.1), and the radius of curvature R(z.sub.1) was determined in order 
to remove curvature from the beam, all of the information is available 
that is needed for determining the waist size and location. Using Eq. 14 
the real beam can be propagated (numerically) back to the waist. Thus the 
waist size is given by, 
##EQU18## 
and its location by, 
##EQU19## 
This gives a complete description of the beam at both the waist and the 
measurement planes and a calculation of M.sup.2. 
One disadvantage of this method is that the calculation of waist size and 
location depend upon the M.sup.2 calculation. As long as an accurate value 
of M.sup.2 has been obtained, then these values are also accurate. 
However, it has been shown that M.sup.2 is extremely sensitive to noise 
far from the laser beam center, and from truncation of the field at the 
edge of the detector. In this case the inaccurate M.sup.2 values will also 
lead to inaccurate waist size and location values. In this respect the 
gradient method is better. The waist location is determined by the 
wavefront and wavefront gradients directly. Truncation or other errors 
will not have a strong effect of the waist size and location, although 
they will still affect M.sup.2 because of the second order moment 
calculation (Eq. 8). 
Fourier Propagation Method. Given a known irradiance and phase of the laser 
beam, the beam irradiance and phase distribution may be determined at 
another plane, Z, through the Fresnel integral: 
##EQU20## 
This equation may be written as the Fourier transform of the E field 
modified by the appropriate phase factor, or 
##EQU21## 
This expression may be discretized and the discrete Fourier Transform (or 
Fast Fourier Transform, FFT) used to calculate the results. It has been 
shown that the Fast Fourier Transform is an efficient algorithm which can 
be readily implemented on common computers. This efficient algorithm 
allows the E field to be calculated at a new Z location very quickly. 
Since the field can be determined at a new Z location, it is also 
straightforward to calculate the field at a number of locations, Z.sub.j. 
The irradiance distribution is calculated from the field as shown 
previously. The second order moment of the irradiance distribution can be 
calculated from the field at each of these locations (.sigma..sub.j). 
These second order moments should obey Eqs. 14 and 15. This equation can 
be fit, using a least squares method, to the measured values of 
.sigma..sub.j. Thus, the values of M.sup.2, W.sub.0 and Z.sub.0 can be 
determined. 
This method, referred to as the Fourier propagation method, has several 
advantages. It does not calculate any of the parameters with better 
accuracy than the others, as in the curvature removal method. All of the 
parameters are determined from the basic propagation of the light itself. 
It is also more independent of the irradiance distribution. Thus the 
defining equation are extremely simple and robust. However, it does rely 
on an accurate Fourier propagation. This can be difficult because for 
sampling, aliasing, and guard band issues. These problems are mitigated 
through care in the design of the propagation algorithm, and because the 
integrals are generally performed for the least stressing case of Eq. 19, 
that is for propagation over long distances or near the focus of a 
simulated lens. It may be advantageous to add a simulated lens to the 
calculation. In that case the first phase factor in Eq. 20 cancels out, 
and minimum aliasing occurs. It should also be noted that, since the 
wavefront gradients are also known, an appropriate grid may be selected 
algorithmically so that aliasing and other effects can be minimized. 
The invention is further illustrated by the following non-limiting 
examples. 
EXAMPLE 1 
In order to determine the sensitivity of the invention, a number of 
different modeled beams were created. This allowed for a check on the 
technique of the invention with known conditions, without having to 
consider the effects of noise or experiment errors. To this end, the laser 
beam was modeled with either a Gaussian or sech.sup.2 propagation profile, 
and the effect of various parameters was considered. The modeled beam was 
broken into the appropriate samples to model the lenslet array and 
detector, and the equations above were used to determine M.sup.2. For 
calculations to obtain beam characteristics, the integrals in Eqs. 8, 9, 
and 11 are replaced with discrete sums over validly measured values. All 
Fourier transforms are performed using standard discrete Fourier transform 
methods, and the fast Fourier transform (FFT) algorithm when possible. 
Elliptical Gaussian beams were modeled, adjusted in piston by setting the 
phase equal to zero on the z-axis. FIG. 7 details the results of modeling 
two Gaussian beams of differing waist size. For these beams, the M.sup.2 
parameter is unity. The smaller beam was propagated over several Rayleigh 
ranges, and the larger over a full Rayleigh range. In each case the 
invention correctly calculated the M.sup.2 parameter based solely on a 
sampling of the wavefront at a given (but unknown to the invention) 
distance from the waist. A similar computation was conducted with a 
Gaussian beam with a constant 1.3 milliradian tilt, or roughly one wave 
across the beam diameter. Again the invention correctly calculated a value 
of unity for the M.sup.2 parameter throughout the range tested. 
The model was also tested on non-Gaussian beam profiles. FIG. 8 depicts the 
results for a beam with a hyperbolic secant squared propagation profile, 
which has a theoretical M.sup.2 of 1.058. The beam was modeled with a flat 
phase front at z=0, simulating a beam waist, and then propagated over the 
distance shown (roughly one Rayleigh range) using a commercial propagation 
program (LightPipes.TM.). 
As another check on the invention, beams with various levels and types of 
aberration were examined, as shown in FIG. 9 (M.sub.x.sup.2 values are 
shown). Four types of aberration were examined, based upon four Zernike 
polynomial aberration functions: astigmatism with axis at .+-.45.degree. 
(U.sub.20), astigmatism with axis at 0.degree. or 90.degree. (U.sub.22); 
triangular astigmatism with base on x-axis (U.sub.30); and triangular 
astigmatism with base on y-axis (U.sub.33). The invention correctly 
calculated an M.sup.2 value near unity for U.sub.22 astigmatism, as well 
as showing increasing values of M.sup.2 for increased U.sub.20, U.sub.30, 
and U.sub.33 astigmatism. 
Of concern in the use of the invention is the granularity of the 
reconstructed wavefront and the effect this would have on the computation 
of M.sup.2. This was tested by examining the results of the invention when 
sampling a modeled Gaussian beam at the waist. The invention correctly 
calculated the M.sup.2 parameter once information was available from 
several lenslets. Accuracy remained within a few percent until the beam 
size (2.sigma.) reached about 45% of the total aperture. (See FIG. 10.) At 
this point, in a zero noise environment, detectable energy from the beam 
just reaches to the edge of the aperture. Thus all beam energy outside the 
aperture is below the sensitivity threshold of the detector. However, once 
energy which would otherwise be detectable fell outside of the detector 
aperture, the value of the M.sup.2 parameter determined by the invention 
drops. We also found, as shown in FIG. 11, that there was no need to go to 
an extreme number of lenslets in order to obtain good results in a 
low-noise environment. It is important to note that this set of results 
are for a Gaussian beam at the waist, and as a result there were no beam 
aberrations. It is believed that the invention will correctly calculate 
M.sup.2 as long as the spatial structure of the aberration is larger than 
twice the lenslet spacing. 
EXAMPLE 2 
Once a wavefront sensor according to the invention was assembled and 
calibrated according to the invention, a series of laser beams were 
measured to experimentally determine M.sup.2 to test the methods of the 
invention. The reference beam was an expanded, collimated Helium--Neon 
(HeNe) laser. The laser source was a variety of different Helium--Neon 
lasers operated in different conditions. This way, a number of different 
lasers with different beam sizes and aberration content could be tested. 
Three basic laser sources were used in this example. The first was a low 
quality HeNe that is used as an alignment and test laser at WaveFront 
Sciences, Inc. This laser was attenuated with several neutral density 
filters in order to reduce the peak irradiance to a level that did not 
saturate the sensor. It was tested with a wavefront sensor constructed 
from a Cohu 6612 modified camera and a 2.047-mm focal length, 0.072-mm 
diameter lenslet array. This combination was aligned and calibrated using 
the principles of the invention. The resulting measurements are presented 
in FIG. 18 as a table of measured values. For this case the M.sup.2 values 
were 1.375 and 1.533 for x and y respectively. This matches well with the 
observations of the way this beam propagated. There was considerable 
non-Gaussian shape to the irradiance distribution and 0.038 m of phase 
aberration. The waist size, waist position, and real beam spot size are 
also shown. Since the laser was set up approximately 0.5 m from the 
wavefront sensor, the measured waist position of 0.44 (x) and 0.48 (y) are 
in good agreement. The waist size of 0.242 is in good agreement with the 
published specifications for this laser. 
The same laser was used with a 1 mr tilt introduced between the laser and 
the wavefront sensor. In this case, very similar values for M.sup.2, 
Z.sub.0, W and W.sub.0 were obtained. This is a good indication that tilt 
has little effect on the overall measurement. This is important because it 
means that even poorly aligned beams may be measured. 
In order to measure good beams, a series of experiments was conducted at 
the National Institute of Standards, using lasers with known good beam 
quality. A series of data sets were acquired at varying distances from the 
laser. A wavefront sensor with an 8.192-mm focal length, 0.144-mm diameter 
lenslet array was used for these measurements. Extreme care was taken not 
to aberrate the beam in the process of the measurement by using low 
reflectivity, but high quality mirrors as attenuators. The laser in this 
case had quite a small waist size, and hence the beam expanded quite 
rapidly. In FIG. 18 the NIST-HeNe1 and 5 data sets were measured 1650mm 
from the laser output mirror and the NIST HeNe 7 data set was measured at 
2400 mm from the HeNe laser output mirror. In this case all of the 
measurements had M.sup.2 in the 1.1-1.3 regime, except for the farthest 
from the laser (2.4 m). In this case the beam had overfilled the detector, 
so that it was slightly clipped in the vertical direction. This lead to 
higher M.sup.2 values (1.4) for this case. 
The final example was for a larger beam that was directed through and 
acousto-optic modulator. The same 8.192-mm, 0.144-mm wavefront sensor was 
used in this case. The M.sup.2 values were measured by the wavefront 
sensor to determine the quality of the beam after passing through this 
optic. A comparison of the beam both with and without the modulator 
allowed a determination of the effect of the modulator on the beam 
quality. FIG. 17 shows the irradiance and phase distributions for this 
case. In FIG. 18, the tabulated values for M.sup.2 for this case are 1.21 
and 1.29 (x and y respectively). This is in good agreement with 
propagation performance of this beam. While the wavefront was relatively 
flat for this case (0.012 m RMS WFE), the larger beam size and 
non-Gaussian beam shape lead to larger M.sup.2 values. 
In all of these examples the measured M.sup.2 values, as well as the 
calculations of waist position and waist size were in good agreement with 
expected values. 
The preceding examples can be repeated with similar success by substituting 
the generically or specifically described reactants and/or operating 
conditions of this invention for those used in the preceding examples. 
Although the invention has been described in detail with particular 
reference to these preferred embodiments, other embodiments can achieve 
the same results. Variations and modifications of the present invention 
will be obvious to those skilled in the art and it is intended to cover in 
the appended claims all such modifications and equivalents. The entire 
disclosures of all references, applications, patents, and publications 
cited above are hereby incorporated by reference.