Correcting inherent aberrations in the gratingrhomb beam sampler
Michael F. Becker
The inherent aberrations are analyzed for the case of a two-grating rhomb laser beam sampler used in highenergy laser systems. Although plane wave fronts are sampled without aberration, spherical or more com-plex wave fronts suffer 1-D phase and displacement deviations. An inverse filter description is developedthat employs the angular spectrum concept for the incident and sampled beams. An inverse filter is easilysynthesized and may be used to deconvolve the aberrations from a sampled data set. In addition, an optimi-zation was performed in order to minimize phase errors in the sampled beam and to develop design criteria.Some practical examples are given which show that, in an optimized system, the aberrations are often negli-gible, and deconvolution is seldom necessary.
1. IntroductionThe two-grating rhombic beam sampler has been
used often in sampling and in obtaining amplitude andphase maps of high energy laser beams. A first gratingof low diffraction efficiency samples the beam, while asecond grating of identical groove spacing corrects forbeam ellipticity. Although plane waves are sampledwithout aberration, curved wave fronts suffer pathlength distortion. Only the weak sampled beam is af-fected, where phase errors of hundreds of waves may beintroduced when nonplane wave beams are sampled bya poorly designed system. These residual 1-D aberra-tions inherent in such a system have never been ana-lyzed; in fact, their presence is seldom acknowledged.Due to the new generation of high energy laser systemsbeing designed or under construction, there is a timelyneed for the analysis of this problem.
In this paper, the grating rhomb beam sampler isanalyzed (Sec. II), and an inverse filter method thatallows sampled data to be corrected for the aberrationsintroduced at the rhomb sampler is presented (Sec. III).In addition, a design optimization procedure is given(Sec. IV), thereby the aberrations may be reduced to lessthan a tenth of a wave phase error in certain cases. Inan optimized system deconvolution of the aberrationswill seldom be necessary.
The author is with University of Texas at Austin, Department of.Electrical Engineering, Austin, Texas 78712.
11. Basic Equations and AnalysisThe basic configuration of the two-grating rhomb
beam sampler is shown in Fig. 1. Gratings G1 and G2separated by a distance L have the same groove spacingbut may have different groove depths or diffractionefficiencies. Typically, G1 will be extremely lightlyruled. Grating Gi separates the incoming beam intotwo beams: one is specularly reflected at an angle 'I(zeroth grating order), and the other is diffracted at themth grating order to an angle 0 m and on to strike G2.At G2, the sampled beam is diffracted a second time atthe -mth order and exits at an angle T. Hence, theoutput sampled beam is parallel to the input beam forall angles of incidence.
The grating rhomb is most easily analyzed if it istransformed into an equivalent transmission gratingdevice as shown in Fig. 2. The sampled beam deviationangle in the region between the gratings is now T - ,and the output sampled beam is separated by Ax fromthe transmitted beam.
The analysis of this device proceeds from the gratingequation
sinO = mX/d + sinL (1)
and from the geometry. Based on the various trianglesinvolved one can write
Ax = AC sin( - Om) (2)
AC = LicosOm. (3)
Combining Eqs. (2) and (3), we find an expression forthe beam deviation Ax as a function of the incidenceangle,
Fig. 1. Configuration of a typical grating rhomb using reflectivegratings.
Al 1 [sin('P - Om) I - + tanI Is I--* (7)
L cosOm I cOm - COS'
The path difference or phase angle difference is clearlya complicated transcendental function of incidenceangle. Beam wave front aberrations will be created fornonplanar waves.
In addition, the total path phase angle may be writtenas a function of the path length AC + CB' as
= 2r(AC + CB')/X,
2 =2-7rL 1 + tan (sin ( O m)) (9)X cosom, coso, ii
Knowing the path phase angle, it will later be possibleto deconvolve the aberrations of the grating rhomb fromthe sampled data.
It is immediately evident from Eqs. (1) and (4) that raysat different incidence angles suffer different lateraldisplacements. Thus, perfectly plane-wave beams aresampled without distortion, while curved wave frontbeams suffer aberration in one dimension as they aresampled. The transmitted beam remains unaffectedin either case. The distortion of the sampled beamresembles a cylindrical aberration because convergingrays at different angles will focus at different loca-tions.
A more tractable analysis to apply to coherent opticalsystems is based upon wave front distortions or pathlength phase differences rather than ray displacements.In this case, beam distortion will manifest itself whenthe optical path length for the sampled beam is a non-linear function of incidence angle. This approach lendsitself to a plane wave decomposition of any arbitraryinput wave.'
Referring again to Fig. 2, the optical path length maybe written as AC + CB' for the sampled beam and as ABfor the direct beam. The path length difference will becomputed using the direct beam as a reference, Al = AC+ CB' - AB. The various path segments may bewritten as
AC = given by Eq. (3),
fsin(TI - 0Om) tanNIiCB' = Ax tanI = L 1'sm) (5)
AB = L/(cosT). (6)
A dimensionless path length difference may be definedas AI/L and is given by
11. Deconvolution of Contaminated DataPropagation through the aberrating beam sampler
will be treated using the angular spectrum.",2 Thisgeneral approach will yield a method for deconvolvingthe aberrations in data already collected from a gratingrhomb system.
Given an arbitrary input wave amplitude and phasefunction U(x,y), its angular spectrum is defined as
(10)X X( = I [U(xy)],
where denotes the 2-D Fourier transform,3 and a and0 are the x - and y -axis direction cosines of a plane wavecomponent. Propagation of the angular spectrum isachieved by simply multiplying by the path phase factorexp(q5). The output angular spectrum is expressedas
(11)
where 0 is the path phase angle, 0 = 27r(1/X). The pathphase angle will in general be a function of the directioncosines a and A3.
The output wave amplitude and phase may now beexpressed as an inverse transform,
(12)
This process may easily be reversed in order to write theinput wave as a function of the measured output waveand the known aberration:
U(xy) = /-' jexp(-j0)JY[U'(x,y)]j. (13)
We see that the deconvolution corresponds to a Fourierdomain filter with a pure phase spectrum, exp(-jo),determined using Eq. (9).
Since most optical rhomb systems presently in usewould produce at least tens of waves of phase error fora moderately curved beam, some correction of the datashould be necessary. For a given output data set, thisprocess may be easily implemented using a digitalcomputer. The phase spectrum, exp(-jo), may beobtained from the physical configuration of the grating
Fig. 3. Optimum values of grating parameter mX/d vs beam inci-dence angle , for a spherical wave beam in a grating rhomb.
rhomb, and the filtering can then be efficiently per-formed using fast Fourier transform techniques. Theresult will be sampled phase and amplitude data of highaccuracy. Alternatively, the correcting algorithm maybe directly incorporated into the computational routineused in the phase and amplitude measuring device.
IV. Optimization of the Rhomb Beam SamplerAn optimization of the rhomb beam sampler was
undertaken with the objective of minimizing path phasedifferences over the beam aperture for the case ofspherical wave fronts. Because the treatment isachieved by a plane wave decomposition, the results alsoapply to other input waves containing the same angularcomponents. The arbitrary input wave may be con-sidered as either a decomposition of plane waves or adecomposition of spherical wave fronts.
The parameters for the optimization may be definedas (Fig. 2):
/ = beam axis incidence angle;x = displacement from beam axis in aberration
plane;R = incident wave front radius of curvature;T = wave front incidence angle, T = tan-'(x/R)
+ , andmX/d = grating parameter for order m and groove
spacing d.The optimization minimized variations in path lengthdifference AI/L, as given by Eq. (7), across the beamdiameter with respect to the incidence angle : and thegrating parameter mX/d.
The first result, shown in Fig. 3, gives the optimumm X/d for any particular value of /. The relationshipis very nearly linear for incidence angles less than 300.The trivial case of m X/d = 0 always results in a secondoptimum but was excluded.
In order to select between different values of /, theanalysis was extended. The analysis indicates that should be as small as possible in order to minimize pathlength deviations as shown in Fig. 4. In practice, a lowerbound on : is given by the requirement that the direct
beam be allowed to exit the device without interceptinggrating G2 (Figs. 1 and 2). The parameter xIR in Fig.4 is a measure of beam size quality, where x is themaximum value of beam radius, and R is the wave frontradius of curvature. The path length deviation, /L,is defined as (AI/L)max - l/L)min over the beam ap-erture.
The behavior of the beam quality vs and mX/d iscomplex, but several examples should illustrate someof its features. Figures 5 ahd 6 show the path lengthdifference across identical beams incident at = 450 ona grating rhomb. The magnitude of mX/d is at theoptimum value in Fig. 5 and decreases in the next figure.At an optimum grating parameter, the beam path lengthis always a symmetric function as in Fig. 5. All non-optimum cases resemble Fig. 6. For a given beam sizex/R, the path phase difference first increases as youleave an optimum mX/d but later decreases in order toreach a second minimum at mX/d = 0.
A final case is illustrated in Fig. 7 where / = 5° for anoptimum grating parameter. The path length devia-tions are significantly reduced in comparison with theprevious cases at /3 = 450.
It is instructive to consider a set of numerical casesin order to gain insight into the absolute magnitude ofthe phase errors in the grating rhomb. Let the beamdiameter be 10 cm (0• x < 5 cm), L = l m, X = 10 Am,and take the cases of : = 450 and : = 5°. Two wave-front radii of curvature will be considered: R = 10 m,a relatively good beam; and R = 1 m, a bad beam. Onlyoptimum grating parameters are treated. The wave-front errors for these cases are given in Table I. Theyemphasize the-fact that a should be chosen to be assmall as possible for a given beam size and gratingspacing.
Id
8L
PATHLENGTH
DEVIATIONAT
OPTIMUMmX _a Id,
16
0' 10' 20' 30' 40' 50'BEAM INCIDENCE ANGLE ,:
Fig. 4. Optimum rhombic beam sampler performance. Path lengthdeviation (AI/L)max - (Al/L)min vs beam incidence angle 3 is shown
Fig. 5. Path difference across a sampled beam for /= 450 and mX/d-1.106, an optimum case.
.052
Path DifferenceAl ~~~- .050
L
.048
.046
/Ga: 45o, m) = -.232
.044 (not optimum)
I I I ~~.042-.06 -.04 -0.2 0 .02 .04 .06
X/R
Fig. 6. Path difference across a sampled beam for /3 = 45° and mX/d= -0.232, a nonoptimum case.
Path DifferenceAl
- ~-3 L 8.63x10 -
8.62x10' --_ 3
8.61x10 -
8.60x103 -
/= 5',MX - 16
(optimum)
-.06 -.04 -0.2 0 .02 .04 .06X/R
Fig. 7. Path difference across a sampled beam for /3 = 5° and mA/d= -0.116, an optimum case.
Table 1. Wavelengths of Phase Error for an Optimized Grating Rhomb
R = 10m R = 1m
/3 = 50 0.034X's 2.2X's# = 450 3.6X's 220X's
V. ConclusionsThe two grating rhomb beam sampler as currently
used in high energy laser systems introduces 1-D aber-rations in nonplane wave sampled beams. There aretwo approaches to alleviate the effect of these aberra-tions. The first, described in Sec. III, enables datataken on an aberrating rhomb system to be deconvolvedfrom the path phase deviations using digital computa-tion techniques. A Fourier domain filtering process isrequired with a purely phase type filter. The filterphase function can be synthesized based on the gratingrhomb configuration as given by Eq. (9). In this way,data taken on an arbitrarily poor grating rhomb systemmay be restored.
The second approach is preventive in nature. Thegrating rhomb may be designed using the optimizationcriteria outlined in Sec. IV. The beam incidence angleshould be chosen to be as small as possible, and then thegrating parameter should be chosen using Fig. 3. Theperformance of such an optimized system can now bepredicted using the curves for various size-qualitybeams given in Fig. 4.
Based on a typical case of a 10-cm diam, 10-,umwavelength beam incident at / = 50, better thantenth-wave performance may be obtained for phasefronts greater than 8 m in radius. At this level of ab-erration, deconvolution will seldom be necessary. Thus,in many cases, optimization alone will reduce the effectsof aberrations to an acceptable level.
Finally, it is obvious that all the previous problemsmay be avoided in a four-grating system, two comple.mentary two-grating rhombs in series. All aberrationsin this system will cancel completely. It is also possiblethat an optimized three-grating system could be de-vised. However, due to the practical problems en-countered in fabrication and alignment, an optimizedor digitally deconvolved two-grating system seems tobe of more practical interest.
The author expresses his thanks to Jerome Knopp formany helpful discussions during the course of this workand for a careful reading of the manuscript.
This research was supported in part by the Depart-ment of Defense Joint Services Electronics Programthrough the Air Force Office of Scientific Research(AFSC) contract F 49620-77-C-0101.
References1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,
New York, 1968).2. W. T. Cathey, Optical Information Processing and Holography
(Wiley, New York, 1974), Chap. 8.3. The 2-D Fourier transform integral is defined as
