Arbitrary-intensity-profiles measurementof laser beams by a scanning and rotating slit
Jose Soto
By taking advantage of the mathematical analogy of optical power passing through a narrow slit and theattenuation of an x-ray beam passing through a biological tissue, and by applying to the optical case theprojection and reconstruction algorithms of computed tomography, one can determine in detail theintensity profile of an arbitrary laser beam by properly combining the position and orientation of theslit. Until now, to the best of my knowledge, the scanning slit has been limited to the measurement ofGaussian or nearly Gaussian beams.
Introduction
Several scanning techniques have been developed forthe measurement of the intensity profiles of laserbeams either by the use of a pinhole or a slit or byvarious schemes involving a straightedge. Thus bymeans of an 11-pm pinhole in combination with aphotodiode, Shayler was able to determine the inten-sity distribution of a focused laser beam.' Becausethe pinhole was small compared with the beamdiameter, off-axis irregularities were resolved. In adifferent approach, McCally2 has shown that by scan-ning a Gaussian beam with a narrow slit, one canproduce a one-dimensional nearly Gaussian distribu-tion with a 1/e width close to that of the two-dimensional laser beam, thus making the determina-tion of this parameter easy. In one of its versionsthe straightedge technique was implemented by Ar-naud et al. by means of a rotating chopper and aresistor-capacitor differentiator so that a single-axisprofile of Gaussian beams could be obtained. Thisprofile was shown directly on the screen of an oscillo-scope.3 Another implementation of the straightedgemethod was made by Mauck for profiling the Gauss-ian beam of a Q-switched laser. For this purpose heused a scanning razor blade, and the differentiationwas made numerically.4
As we can see, with the pinhole method being theonly exception, the other methods are restricted tothe measurement of the parameters of TEMOO Gauss-
The author is with the Instituto de Fisica, Universidad Au-t6noma de Puebla, Apartado Postal J-43, Puebla 72570, Mexico.
ian beams. In this paper a more general slit methodis proposed for the measurement of arbitrary inten-sity structures of laser beams. Although this newmethod is probably as slow as the scanning pinholemethod, the optical power passing through the slitcan be 2 orders of magnitude higher than in thepinhole case.
Theory
The basis for the optical technique proposed here isthe mathematical equivalence of the way in which theoptical power passing through the slit is calculatedand also the way in which the attenuation of an x-raybeam, as with those used in computed tomography, iscalculated when it passes through a biological tissueor another absorbing material. In the optical case, ifthe slit is sufficiently narrow, the transmitted poweris approximately equal to the line integral of theintensity distribution I(x, y) of the optical beam alongthe straight slit multiplied by the width Aw of the slit.Similarly the total attenuation of the x-ray beam isequal to the line integral of the attenuation coefficient.L(r) of the tissue along the beam. In computedtomography pL(r) is determined in a two-dimensionalslice from a series of properly distributed line inte-grals of ,u(r). The value of each integral is given byits experimentally measured ratio I/Io of the outputand input intensities of the x-ray beam.5-8
The analogy between the two cases is shown graphi-cally in Fig. 1. Although the x-ray beams have afinite width, mainly dictated by the size of the detec-tor, they are represented by straight lines. In theoptical case it is convenient to represent the slit by itscentral line. According to Fig. 1, these lines arecharacterized by the two parameters 0 and t, and theirequation is x cos 0 + y sin 0 = t.
Fig. 1. Line integrals along the straight line x cos 0 + y sin 0 =t: (a) with respect to pL(x, y) for obtaining the attenuation of anx-ray beam; (b) with respect to I(x, y) for obtaining the opticalpower passing through a slit.
Once the mathematical analogy between these twoproblems is made evident, it is possible to proceed tothe determination of I(x, y). This is done by place-ment of the slit in a sequence of positions correspond-ing to those of the x-ray beams when they are used toobtain the projections of ,u(x, y) and by use of one ofthe reconstruction algorithms of computed tomogra-phy (CT).
At this point it is convenient to briefly review theway CT works. The simplest case of CT is that inwhich the line integrals are parallel and equallyspaced and form an angle 0 with they axis so that theprojection Po,(t) of ,u(x, y) is obtained, as shown in Fig.2. The function Pe(t) is known as the Radon trans-form of p.(x, y). 5 ,6
These parallel projections are repeated at manyviewing angles until a convenient set of projectiondata is obtained. A parallel projection could bemeasured by movement of the x-ray source and thedetector along parallel lines on opposite sides of theobject. Another type of projection is possible if asingle source is placed in a fixed position relative to aline of detectors. This is known as the fan beamprojection because the line integrals are measuredalong fans. This case is not considered here.
The mathematical basis for the reconstruction of
X
Fig. 2. In computed tomography, parallel projections are taken bymeasurement of the absorption of a set of parallel x-ray beams for anumber of different angles. These angles go from 0 to nearly1800.
>i(x, y) from its projections in CT is the central-slicetheorem, which says that the one-dimensional Fou-rier transform of the projection P8 (t) is mathemati-cally identical to a slice, at an angle 0, of the two-dimensional Fourier transform of the object p,(x,y)itself.5-7 It follows that, given the projection data, itshould be possible to synthesize the whole two-dimensional Fourier transform of the object, fromwhich the object is reconstructed by performance ofan inverse Fourier transform. While this provides asimple conceptual model of tomography, in practicethe approach currently being used in almost allapplications is the filtered backprojection algorithm,which has been shown to be extremely accurate andeasy to implement.6-8
Backprojection is the opposite of projection in thesense that a two-dimensional distribution is gener-ated by smearing of each one-dimensional projectionuniformly over all space at the same angle that theprojection was made. The combination of manybackprojections creates a new two-dimensional imageb(x, y), known as the summation image.6 However,the summation image is not the same as the originalobject p,(x, y); instead, it is equal to (x, y) convolvedtwo dimensionally with a point spread function pro-portional to 1/r. By appropriate filtering of thesummation image one can significantly improve thequality of the reconstruction. Since summation andfiltering are both linear operations, one can reversethe order and filter the projections before summingthem.6 In this case the filtering is now a one-dimensional operation since it is applied to the one-dimensional function P(t). This method requiresthat each projection be convolved with a one-dimen-
sional filter function h(t), yielding the filtered projec-tion Q6(t); that is,
Q0(t) = f PO(t')h(t - t')dt'. (1)
These filtered projections are then backprojected andsummed to yield the improved reconstructed image.
A filter function commonly used is the abrupt-cutoff filter, defined in the frequency domain by
H(i) = fI for i~ s emaH() otherwise (2)
where (-max, max) represents the frequency rangeoutside of which the Fourier transforms of the projec-tions do not contain any energy. The filter functionh(t) is given by the inverse Fourier transform of H(e)and is
h(t) = H(e)exp(+jzrrt)dt
sin 2iTnmt (sin Trlmaxt 2
= max 27rrmaxt - m8rXmaxt(3)
For the filtered backprojection algorithm the dis-crete projection data are filtered by means of adiscrete version of the convolution process of Eq. (1),which can be written as6 ,7
N-1
Q(t.) = At I h(t. -t)PO(tk),k=O
n = 0,1, 2,... ,N- 1, (4)
where At is the sampling interval, t, = nAt, and PO(tk)is supposed to be zero outside the index range k =0, . . , N - 1. The discrete form h(tn) of the filterfunction is given by Eq. (3) evaluated at the samplepoints
1
4(At)2 '
h(tn) = 0,
1
-h2sr2(At)2
n = 0
n = 2, 4, 6,....
n = +1, +3, ±5...
One of the great advantages of the filtered backprojec-tion algorithm is that the reconstruction procedurecan be started as soon as the first projection isobtained and filtered according to Eq. (4). This canspeed up the reconstruction of the image consider-ably.
For reconstruction of the image a grid is definedand the filtered backprojections are arithmeticallyadded at the intersecting points of the grid, as shownin Fig. 3. According to this figure, for a given angle0k of the backprojection to the point (xi, yj) of the grid,
Fig. 3. Image is reconstructed at the crossing points of the grid bybackprojection of the filtered projection Qek(t) for all the projectionangles Ok.
there corresponds the value tj = xi cos 0k + yj sin okof the t parameter, and the filtered projection Qok con-tributes to this point its value at tij. Since in generalthe value of t thus calculated does not correspond toone of the discrete points at which Qek is known,either use of the nearest known value or interpolationfrom the neighboring known values of Q0k is necessary.In computed tomography it has been found thatlinear interpolation yields adequate results.7 Thisprocedure is followed for all the points of the grid, andthe whole process has to be repeated once all thevalues of the filtered projection are determined forthe next angle.
Numerical Calculations
In order to obtain the projection P6(t) of the intensitydistribution I(x, y), one uses the following approach.First, a circumference encircling I(x, y) is definedsuch that the values of I(x, y) are negligible over thecircumference, as shown in Fig. 4. The line integralsof I(x, y) are taken along several equally spacedparallel chords of the circumference. If the distanceof a chord to the origin is t and the radius of thecircumference is r, then the length of the chord is s =2(r2 - t2 )1/2, and the coordinates of the starting pointof the line integral for this chord are x0 = t cos 0 +(s/2)sin 0 andyo = t sin 0 - (s/2)cos 0.
For a given displacement As along the chord therecorrespond the increments Ax = -As sin 0 and Ay =As cos 0 of the coordinates. In order to include theeffect of the finite width Aw of the slit at each newposition along the chord, one evaluates the functionI(x, y) on both sides of the chord. Thus if the newposition is given by (xn, yn), then I(x, y) is evaluated at[xn + (Aw/2)cos 0, y,, + (Aw/2)sin 0] and at [x, -(Aw/2)cos 0, y, - (Aw/2)sin 0]. The average of thetwo values thus obtained is used for evaluation of theline integral along the chord according to the trapezoi-dal rule.
Fig. 4. Intensity distribution I(x, y) is totally contained within thecircumference. The line integrals are taken along a number ofequally spaced chords of the circumference.
The values of t are incremented from -0.8r to+0.8r for a discrete projection P 6 k(t) of I(x, y) to becompleted. After filtering of Pok(tn) the reconstruc-tion is made, as explained above, at the points of asquare grid whose diagonal fits inside the range ofvalues of t.
As a particular example, the tomographic tech-nique was applied to the reconstruction of the inten-sity pattern of the TEM12 laser mode, which for a spotsize equal to ,,/ arbitrary units is given by9
I12(x, y) x2(4y2 - 2)2exp[-(x 2 + y2)]. (6)
This intensity distribution is considerably more com-plex than the rotationally symmetric TEMOO Gauss-ian function normally treated with scanning slits andserves to illustrate the general applicability of themethod. The three-dimensional plot of this functionis shown in Fig. 5, and its reconstructed version isshown in Fig. 6. As an aid for comparison, these two
Fig. 6. Reconstructed version of I12 (x, y) obtained with 125 projec-tions and 125 line integrals per projection.
functions were normalized to have peak values equalto 1.
The parameters used to obtain the reconstructedfunction are as follows: The radius r of the circum-ference encircling I12(x, y) is 10. The number of lineintegrals per projection is 125, and the number ofprojection angles is also 125, covering from 0 toalmost 180°. The width of the slit is 1/20 of the spotsize. The backprojection is made on a square gridwith sides equal to 7.5. This grid is divided into 99rows and 99 columns. For the plotting program thegrid also consisted of 99 rows and 99 columns.
Smaller values of r were tried but the reconstructedimage developed larger negative fluctuations andother spurious effects. It was also found that theshape of the reconstructed function is a slowly vary-ing function of the width of the slit as long as itremains much smaller than the spot size.
It is clear that the reconstructed function closelyresembles the original function, thus validating theapplicability of the tomographic treatment to theoptical slit problem. It is expected that this resem-blance will be improved by an increase in the projec-tion and reconstruction data, just as in the case of xrays. In computed tomography the number of projec-tions and the number of line integrals per projectioncan be 200 or more in each case.
Aliasing, Resolution, and Photon Noise
The artificial structures in the form of small ripples atthe corners and in other regions of the reconstructedfunction represent a particular instance of aliasing,which is a reconstruction error caused by undersam-pling. The origin of aliasing is described below.
As for any other sampled function, the Fouriertransform of a discrete projection POk(tn) consists ofperiodic replicas of the Fourier transform SO(w) of thecontinuous function Po(t), separated by the reciprocalof the sampling interval At. Since in general SO(w) isnot band limited, the information within the measure-ment band becomes contaminated by the tails of thehigher and lower replications of SO(w). This contami-
nated information constitutes the spectrum of thealiasing error. When the image is reconstructed,aliasing appears as streaks, moir6 patterns, and otherartifacts. In general, aliasing effects decrease with adecrease in the sampling interval and with an in-crease in the number of projections.6 7
Another important reduction of aliasing is expectedbecause of the finite width Aw of the slit. In fact, theprojection P0 (t) results from the convolution of therectangular aperture function of the slit with theideal value of the line integral of I(x, y) parallel to theslit. Thus the spectrum of the projection data be-comes highly restricted to the bandwidth (- 1/Aw,1/Aw), where 1/Aw is the first zero of the Fouriertransform of the slit aperture function. In order tocompletely specify a filtered function I(x,y) bandlimited to the above spectral region, one shouldensure that the sampling displacement At of the slitbe less than or equal to half of the slit width. Thisgives the approximate size of the smallest resolvabledetail of I(x, y) by this technique. These consider-ations are made in analogy to the x-ray case in whichthe detector-source line of sight should be steppedwith intervals of half the detector width or less inorder to achieve maximum resolution and in whichthe detector width plays the same role as the slitwidth. 6 7
Because of the large value of the photon energies hvin x-ray CT and because of the necessity of limitingthe radiation dose delivered to the living biologicaltissues, the final image is based on a relatively smallnumber of detected photons. As a consequence, thephoton noise is usually the dominant source of errorin the reconstructed image.6 In contrast, in theoptical slit case, with photon energies typically 4orders of magnitude smaller than those in x-ray andwith the availability and freedom for the use of largenumbers of photons, it is expected that the photonnoise will play an insignificant role compared with
other systematic reconstruction errors, such as theresidual aliasing effect.
Commentary
Although the tomographic slit technique is beingintroduced here for the measurement of intensitydistributions of laser beams, its applicability is notlimited to this case. It could also be useful forreconstructing images in optical systems if the slit isdisplaced and rotated in the image plane of thesystem. This could be particularly valuable at infra-red and far-infrared wavelengths at which photo-graphic films are insensitive and for which detectorarrays are scarce or do not exist at all. However, inthose spectral regions in which detector array areoperational, this new technique does not offer anyadvantage because it is slower and cannot be appliedto fast optical events unless they are repetitive.
References
1. P. J. Shayler, "Laser beam distribution in the focal region,"Appl. Opt. 17, 2673-2674 (1978).
2. R. L. McCally, "Measurement of Gaussian beam parameters,"Appl. Opt. 23, 2227 (1984).
3. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, D. de laClaviere, E. A. Franke, and J. M. Franke, "Technique for fastmeasurement of Gaussian laser beam parameters," Appl. Opt.10, 2775-2776 (1971).
4. M. Mauck, "Knife-edge profiling of Q-switched Nd:YAG laserbeam and waist," Appl. Opt. 18, 599-600 (1979).
5. H. H. Barrett, "The Radon transform and its applications,"Prog. Opt. 21, 217-286 (1984).
6. H. H. Barrett and W. Swindell, Radiological Imaging (Aca-demic, New York, 1981), Vol. 2, Chap. 7, pp. 375-422.
7. A. C. Kak and M. Slaney, Principles of Computerized Tomo-graphic Imaging (Institute of Electrical and Electronics Engi-neers, New York, 1988), Chap. 3, pp. 49-75.
8. A. Rosenfeld and A. C. Kak, Digital Picture Processing, 2nd ed.(Academic, New York, 1982), Vol. 1, Chap. 8, pp. 353-430.
9. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 329.
