Direct measurement of the spatial modes ofa laser pulse: theory

Eric G. Johnson, Jr.

We make an electric-field measuring apparatus by using optical processing, tapered optical fibers, and a pairof detectors at the end of each optical fiber. Using an appropriate computer-generated hologram (CGH), weshow it is possible to discriminate among a set of orthonormal modes used to represent the spatial features ofthe electric field with a SNR of at least 100 to 1. The tapered fiber is a mode filter that is used in the transformplane of the CGH. This fiber allows precise determination of the strength of each of the orthonormal modesbeing used as the spatial basis of the electric field before the optical processing.

1. Introduction

At NBS1 -4 we have been developing an electric-fieldmeasuring system to measure the temporal and spatialfeatures of a laser pulse with at least 1.0% accuracy. Ifcertain critical elements can be set up, we expect theaccuracy to be 0.1%. We have found a number ofconflicting constraints on the design of the equipment.We address these constraints in this paper. First, weshow why the optical processing is desirable for elec-tric-field (as opposed to intensity) measurements.Second, using simplified analytical models, we illus-trate the technical conflicts that result from usingoptical processing. They are (1) mode mixing due toclipping, various offsets in the optical axis, and variousmode mismatches at several locations; (2) mode mix-ing due to the Fourier transform process; and (3) at-tenuating the undesired modes which result from thecross-correlation process. From these illustrations,we show how to get a resolution of the conflicts.

We get this resolution in three key stages:(1) The first stage uses a computer-generated holo-

gram (CGH) to decompose the electric field of a laserpulse by an orthonormal-mode analysis of the originallaser pulse. The CGH focuses a number of spots intothe transform plane; the power in each spot is propor-tional to the strength of the corresponding mode.However, there are fundamental constraints in theallowed form of the electric field at the Fourier plane.These are a consequence of the cross-correlation pro-cess. This creates undesired electric-field signals atthese selected locations in the Fourier plane.

The author is with U.S. National Bureau of Standards, Electro-magnetic Technology Division, Boulder, Colorado 80303.

Received 17 January 1986.

(2) The second stage uses a graded-index opticalfiber at each selected spatial location in the Fourierplane. Each optical fiber starts with (for example) a300-am core diameter and is adiabatically pinched to asingle-mode core diameter around 10 ,um. The funda-mental mode of the fiber is matched to the focusedspots in the transform plane. After the pinch, thesefiber cores return to the 300-,gm diameter. This stageattenuates all but the fundamental propagating mode,thereby transmitting a power proportional to that ofthe laser mode.

(3) The third stage has a detector which measuresthe power from each fiber. This step is done if there isno need for measuring the polarization or the carrierphase of the laser pulse. If there is a need for thelatter, it is necessary to run the pulse from the opticalfiber through a polarization-separation device and fi-nally to heterodyne at the detectors the resulting sig-nals with a selected reference beam from the same laserpulse.

Section II gives a brief functional diagram of thesuggested measuring unit so the reader has a visualsense of how the apparatus works. The final appara-tus is very likely to be different.

Section III addresses the basic technical ideas indeveloping an electric-field measuring apparatus.Here we use Hermite-Gaussian modes so we can haveanalytical formulas to illustrate the ideas. This sec-tion contains the main body of ideas about the electric-field measuring apparatus. The key ideas are: (1) thebeam matching by a pair of telescopes5; (2) the cross-correlation process by an appropriately constructedCGH6; (3) the Fourier-transform lens, which has avariable focal length; (4) the tapered optical-fiber as-sembly which causes attenuation of all modes of propa-gation except the fundamental mode7-10; (5) estimat-ing and design for minimizing the impact of various

1 September 1986 / Vol. 25, No. 17 / APPLIED OPTICS 2967

alignment and mode-matching steps in the appara-tus11"12; and in addition (6) devising the necessary po-larization13 and detection techniques14'7 for the re-sulting fundamental mode.

Section IV addresses the CGH.18 Here we introducethe complex mathematics' 9 necessary to get a filterwhich will allow optical processing with high precision.

II. Why Know the Details of a Laser Pulse?

The complete specification of how a laser pulseevolves in space requires critical details of the electricfield such as the electric-field profile transverse to thedirection of propagation over the duration of the pulse.With current laser sources, mechanical-adjustmentparts, optical fibers, and electrochemical-depositiontechniques, we are beginning to get to the technicalcapability where it is possible to measure the electricfield of a laser pulse. This paper describes an exampleof how this can be done. The actual apparatus is likelyto be different because the fabrication of each partimposes fabrication errors that may render the exam-ple inappropriate.

The current means for measuring the details of laserpulses are incomplete. They can be designed to mea-sure the flux (irradiance, W/m2) or fluence (J/m2 ) witha time resolution ranging from picoseconds to seconds.The details of the phase front are usually only knownby inference. This means that we have lost at leasthalf of the necessary information. Examples of suchmeasurement devices are (1) single detectors whichscan the laser beam and (2) 2-D detector arrays.20'21

Figure 1 shows a block diagram of a hypotheticalelectric-field measuring apparatus. The numbers inthis figure are references to later figures which showthe details of that block. Briefly, the steps in makingan electric-field measurement are:

A, Sample the laser pulse of interest by using a beamsplitter or a high spatial-frequency grating.

B, Modify this pulse so that it has a beamwidth,phase front, and optical axis that are matched to theCGH.

C, Use a CGH which is coded with the followingfeatures: (1) unique beam steering for each orthonor-mal spatial decomposition of the electric field and (2) aproper intensity weighting for the orthonormal decom-position so that there is a cross-correlation patternwhich has a nearly Gaussian profile of standard widthfor the desired correlation and that is a linear combina-tion of the higher-order Hermite modes for the unde-sired correlations. We use six modes for each dimen-sion in our example.

D, Use an adjustable Fourier-transform lens to causethe pattern exiting from the CGH to form the cross-correlation patterns at the Fourier-transform plane ofthis lens.

E, Mount, for example, four blocks of 3 X 3 arrays ofoptical fibers. Each fiber is located so that its opticalaxis is centered on the Gaussian profile resulting fromthe desired cross-correlation process. The fundamen-tal mode of each graded-index fiber is Gaussian andhence is to be matched in phase front, beamwidth, and

Onti1cal Fourier Foure TaperedSampled Beam Match A Trgndorm Plane Fbers

Pulse I EB 1- B2 )~:~~0~ ~~- /

.. Fig. 2) ... Fla. 4)C .) Fig. 3)5) (a.. Fi. a-)

Fig. 1. Block diagram of the steps in the electric field measuringapparatus.

beam center to the Gaussian profile resulting from thecross-correlation. We have a 36-mode decompositionto represent the beam profile for each polarization.

F, Each optical fiber is polished at the front and backso that there is accurate mode coupling at the begin-ning and a good match to a calcite crystal at the end.The latter results in the separation of the two polariza-tion states of the single propagating mode in the opti-cal fiber.

G, A detector is situated to measure the power ineach polarized mode.

111. Analytical Model Using Hermite-Gaussian Modes toIllustrate the Basic Ideas of an Electric-Field MeasuringApparatus

A. Overview

In this section are seven sections. Each sectiondeals with some aspect of processing the critical elec-tric-field information in the laser pulse. Section III.Baddresses the two beam-matching telescopes. SectionIII.C addresses the optical filter. Section III.D ad-dresses the Fourier-transform lenses. Section III.Eaddresses the optical-fiber units and what is present atthe Fourier-transform plane for the simplified analyti-cal case. Section III.F addresses the errors in match-ing the optical axis of the graded-index fiber to theoptical axis of a particular cross-correlation processor.Also it addresses what the finite size of the optical fiberdoes to the information. At this stage it is appropriateto use these same formulas to estimate the effects ofthe finite-size apertures at other stages of this appara-tus. Section III.G shows a tapered fiber and discusseshow it is used.

B. Beam-Matching Telescopes

Figure 2 shows the configuration of the four lensesused to set the laser pulse to a standard beamwidth andto a flat phase front. There are two telescopes. Oneacts to create a local image and a flat phase front of theincident laser pulse. The second telescope acts as abeam expander to create a standard beamwidth and aflat phase front at the location of the CGH indicated asqlo in Fig. 2. The f,, are the focal lengths of the simplelenses. The distance between each critical opticalstage is indicated by d,. The critical optical stages areindicated by the complex beam parameters for an idealGaussian beam. Thus we can use q, = ibn + zn, whereb = w(7r/X), w, is the beamwidth at the nth location,and -Zn is the relative distance to a point where theGaussian beam has the width wn.3,5 We use the Gauss-ian beam for choice. The actual laser beam need notbe Gaussian.

2968 APPLIED OPTICS / Vol. 25, No. 17 / 1 September 1986

B1 Part B2 Part

Optical f2 _ f

Axis

LaserPulse f2 f4 fe fa CHG

- d- - d3 -d 5 X d7 L dg

q1 q2 q3 q4 q5 qe q7 q8 q9 q10

Fig. 2. Beam-matching telescope, step B in Fig. 1.

C. Computer-Generated Hologram

In this section we address the CGH. The parame-ters are specific to one manufacturing facility. If adifferent facility is used, the parameters can be differ-ent. Figure 3 shows the sequence. First there is aglass plate with width W = 70 mm. Each cell indicat-ed by Cnm has d = 200 ,gm with the range of n and mfrom 1 to 35. The positioning accuracy of each cell hass = 0.15 ym. Within each cell are 400 spots with theircorner locations at the coordinates em1,fnl. Each spotD has a size given by a and b, which are uniquelyspecified for each cell. This implies that 98.0 milliondata values must be specified for a given CGH. Theterms a and b are less than dl, which is 10 ptm.

D. Fourier-Transform Lenses

Figure 4 shows the nominal structure of the pair oflenses used to define the Fourier-transform segment ofthis apparatus. The anodized tube is used to reducethe spurious radiation on the optical fibers in the Fou-rier plane. This tube is only representational. Theproper shielding may be different. We use the samedesign formulas as were used in Sec. III.B. Here weuse two lenses, and we know the beam parameters at q,to be w1 = 10.0 mm and the phase front to be flat. Acritical adjustment of these two lenses will make theeffective focal length at stage q6 show that Xfeff = 1 mm2

is true for the wavelength of interest.

E. Fourier Plane-What is Present There?

The Fourier plane contains the results of the cross-correlation from the CGH and the Fourier-transformlenses acting on the laser pulse. For our example, Fig.5 shows four physical structures in this plane. Eachstructure is made of a 3 X 3 array of tapered opticalfibers mounted in a square pattern indicated by thedimensions in Fig. 5.

We represent the electric field just before the opticalfilter by the following form:

E1(x,y) = E AnmiHn(,/X/W)n,m=O

X Hd( yw) W(X) W(Y)CnCm), (1)

where the following definitions are noted:

all of Dmn spot

en = X,+md

feN = Y + nd

Fig. 3. Computer-generated hologram, step C in Fig. 1 (front view).

f2 f4

OpticalAxis i

/Tube of black anodized Al i

p tia

CHG f2 f4 ~~~~~~~~~~~~OpticalCHG f2 C4 Pilter

i d' - d3 dn

q, . 1q2 q3 q4 q5 CI

Fig. 4. Fourier-transform lenses, step D in Fig. 1.

Fig. 5. Fourier plane, step E in Fig. 1 (front view).

H0 (x/w) = (d exp(-z2 + 2VJ2 X z/w)l,=o,

W(x) = exp[-(x/w) 2 ],

Z(x,z1,w) exp[-z' + 2xz1/w]W(x)l,,=O,

C,, = 2n!7F, (2)

C A J dxHn(,,Gx/w)Hn,(,1 Gx/w)[W(x)] 2,

Hn(-2-x/w) = (-1)nHn(-x/w),

= x or y polarizations.

1 September 1986 / Vol. 25, No. 17 / APPLIED OPTICS 2969

_ PI I I

Here w is the Gaussian width of the beam, representsthe two polarization states, and n and m represent themode label for x and y coordinates of Hermite func-tions. C and Cm are the necessary normalizations forthe orthonormal expansions. Anml are the strengthsof each mode in the electric field. Z represents thegenerator of the Hermitian modes. z is set to zeroafter the differentiation process. The symmetries ofthe Hermitian modes are also indicated.

We assume that the CGH can be represented as

T(x,y) = T(x)T(y). (3)

This means we can separate the two dimensions and dothe analysis with one dimension only. With this as-sumption we can write the electric field after the holo-gram as

2 0,

E,(x,y) = 2 Anm.,En(X)Em(y)/(CnCn),n,m=0

where we have the 1-D field mode as

En(x) = d ) Z(x,zl,w)T(x).

The definition of the hologram function is5

T(x) = E BjHj(x/w)j=o

X cos[u(j)x + V(j)]W(x)/[W1(x)Cj].

(4)

(5)

(6)

Here we require the function to be real and do notworry about the possibility it can be negative. We alsoassume only the first six Hermitian modes are of inter-est. We are implicitly assuming that most of the ener-gy for each dimension of the electric field is containedwithin these first thirty-six modes-six for each di-mension. If that is not true, it is necessary to use adifferent set of orthonormal modes so we can continueto use six modes per dimension. In Eq. (6) we define aweighting function W1 as

Wl(x) = exp[-(x/w) 2 ]. (7)

Its purpose is to allow the cross-correlation process tohave exact orthogonality for the fundamental propa-gating mode at the Fourier-transform plane. We usev(j) to set the phase symmetry of T(-x) =-T(x):

v(j) = r/2 j = 0,2,4= 0 j = 1,3,5.

(8)

This is done because the actual hologram must have adc offset to prevent negative values. This dc offset isimportant only to make sure that a dc signal peak inthe Fourier plane does not match the signal peaks forthe cross-correlation process. We ignore the dc term.The positioning of the cross-correlation processes iscontrolled by the u(j), which is set by

Xf f

x0(j) are specified locations in the Fourier-transformpiane relative to the optical axis. The strength of eachcross-correlation process is controlled by the Bj terms.

The action of the Fourier-transform lenses changesthe 1-D field mode indicated by Eq. (5) to a Fourier-transformed field mode indicated in

S(u) = f J d exp(iux)T(x)H,,G(/x/w)W(x). (10)

Equation (10) has a constant-phase term missingwhich is of no importance in our discussion. If weinsert the form for T(x) we get the transformed fieldmode

5

Sn(U) = E Retexp[iv(j)ISnj(u)ICj,j=o

(11)

where we have the cross-correlation field mode definedin an orthonormal expansion of Hermite functions.Thus

(12)Snj(u) = E' GnjmHm(u/~Ltwf))W2(u)/Cm,m=0

with each mode coefficient given as

j = ()( d) D(Z3 z 2,z1 )

Here we have a fundamental generator function de-fined as

D S2 dudx exp(iux)Z(x,z,,w)

X Z(xz 2,w)Z1(uz3,wf)/W1(x), (14)

where

W2(u) = exp[-(u/w) 21 (15)

and where Z1 is the same as Z with u replacing x and wfreplacing w.

We can integrate analytically over x and u in Eq.(14). First, we make a set of definitions

1 2 1q2 w2 w2

12 1 + q2 , (I

p2 Wf 4

w wb.

The generator D becomes

_Z2 -Z2 2+ 2 p2 Z2 + 2 P 2q2 (Z1 + Z2)2

- w23-Z + i 2 Z3 3 W 22 2+)F~f Wf 2q2 W V) f ii Z3(Z + Z2)

W

This equation form shows us what the cross-correla-tion process is doing. The Z3 = 0 case corresponds tothe fundamental mode in Eq. (12), which correspondsto m = 0. The desired cross-correlation process hasonly correlation between the n = j terms. This canonly happen when the relation

2970 APPLIED OPTICS / Vol. 25, No. 17 / 1 September 1986

LaserBeam

I

.. d -

OpticalAxis ofCross Correlation

- d1- E

OpticalAxis ofGraded-indexFiber

Fig. 6. One-dimensional version of clipping and offset effects dueto the graded-index fiber acting on a cross-correlation structure.

pq =- __

is true. This impliesW = 2w 1 ,

p = wf(l -(WWf)18)112

(18)

(19)

q = w/[CJ(1 -WWf)211)12]

In these conditions D can now be written as

D = exp[2zlz 2 + z3(1-1/b 2 ) + i2z3 (z1 + z2 )/b]. (20)

When b = , we have the ideal cross-correlation, adelta function. Only the Z3 = 0 case has no mixingbetween n 0 j terms. Notice that the mode series isinfinite. This implies a clipping error when the beamis injected into the optical fiber. The value b = 1corresponds to the situation that the m = 0 mode has awidth equal to the Fourier transform of the originalGaussian beam before the optical filter. The best wecan od is to get a nominal minimum in the strength ofthe higher-order modes at the Fourier plane.

F. Effects of Beam Clipping and Optical-Axis Offsetbetween Fiber and Cross-Correlation Beam

Here we look at the mode-mixing effects due toclipping and to a shift in the optical axis of the cross-correlation mode compared to the optical axis of theoptical fiber. Figure 6 illustrates the points. Thevalue d1 is the radius of the optical fiber which accepts

the laser radiation, and E is the amount of relative shiftof the optical axis.

The 1-D electric field for one polarization state canbe written as

(21)E(x + e) = E Am (d Z(x + EZW)Cmm=o

where x is the coordinate perpendicular to the opticalaxis of the optical fiber, E is the offset of the optical axisof the cross-correlation process, w is the width of thefundamental Hermite mode, Am is the strength of themth mode, and Cm and Z are defined in Eq. (2).

Once the radiation is launched within the opticalfiber, we have the following form for the electric field:

E,(x) = Z Bm (d ) Z(xlw)/Cm.m=O

(22)

The only change between Eqs. (21) and (22) is thestrength coefficients. We are particularly interestedin the Bo term. It provides all information about thecross-correlation process and about the particular or-thonormal mode. The ideal situation is no offset errorand no error due to clipping. Bo is given as

Bo Am () D2 (ZOdiE)/Cm,,m1=0

where D2 is defined as

D2(z,z1,d,e) = I dxZ(x + e,z,W)Z(xz1 ,W).

If we have the definitions

a e/w,b = CWd1/w,u = Cxlw,

c z - a/2,

D 2 is given as

D 2(z,0,d1 ,e) = exp(-a 2/4 - az)D3 (b,c),

where D 3 is defined as

D 3(b,c) = J exp(-u 2 )du.-b-c

(23)

(24)

(25)

(26)

(27)

If we make the approximation that b is very large and ais small, the relationship between B and A is

B o AO - aA1 /2. (28)

The key fact to notice here is that an error in the opticalaxis can be significant because the strength of A iscomparable with Ao. First, we note that A1 is 90° outof phase with Ao. This means the magnitude of Bo isproportional to a. If a = 0.03, we have a 0.1% errordue to alignment. This implies a positioning accuracyof 0.5 ,um. If we make the approximation that a is zeroand b is large, the relationship between B and A is

Bo Ao[1 - exp(-b 2 /b]

- exp(-b2 ) E Am+[H(b) - Hm(-b)]/Cm+i. (29)m0

Here we simply notice that b needs to be chosen so thatthe highest Am modes are <1% of the Ao mode. The

1 September 1986 / Vol. 25, No. 17 / APPLIED OPTICS 2971

I

-- di + 4E

choice of a graded-index optical fiber of 150-,tm radiusand w = 26.68 /im allows b = 5.62. As long as m is lessthan thirty modes we have no significant clipping ef-fect. There can also be a mismatch in the width of thefundamental propagation mode in the fiber example tothe width of the fundamental cross-correlation pat-tern. This error shows a relationship between B and Aas Bo A + 0.25A2 (w/w). If Aw/w is 4%, thecoupling is 1%.

G. Mode Attenuator-the Tapered Optical Fiber

Figure 7 shows the design example of the taperedoptical fiber. The F1 side needs to be positioned to+0.1-/im accuracy, and the F2 side has a birefringentmaterial to get both polarization states of the funda-mental mode. The taper rate is 5 or less to avoidmode mixing. The resulting diameter at the neckallows only single-mode propagation. It is near 6-10gim in diameter. After the birefringence material,there are two detectors to measure the respective ener-gy in each polarization. The tapered fiber needs to bea grade-index fiber with as low a A as possible and aslarge a diameter as possible to allow accurate modematching. I suggest A <0.003 and a diameter of >300/um.

IV. Realization of CGH Using Abstract Mathematics

A. Overview

In this section we develop a summary of mathemati-cal expressions necessary for the computer programs.(The details are in Ref. 22.) These programs are usedto generate the CGH. Section IV.B generates theexact equation and indicates the subsequent errorsdue to the fabrication process. Section IV.C examinesthe antisymmetric part that is nonlinear.

A,C - Glass support for fiberB - Glass support for tapered position of fiber

- Electroform of nickel with holes to all fibers to be held

E - Electroform with holes to shield out extraneous radiationFt F2 - Polished ends of optical fiber

Fig. 7. Diagram showing a design example of the taper assembly.

+(L/2). This is due to the finite aperture of the opticalfilter.

Because we are interested only in the fundamentalfiber mode, we multiply the expression in Eq. (31) bythat in Eq. (32), namely,

2 exp(-I[u - u(r)/wm}2),TtWm

(32)

and integrate over u to get the strength of each radia-tion pattern in the fundamental mode. There is anapproximation in this expression which is <0.2%. Weget

Sl(n,k,r) = E Hn(y1 )B(zj)

X exp[iu(k)yl - (wnyj/2) 2 ]. (33)

B. Exact Equation Incorporating the Constraints ofFabrication

We assume that the separability into 1-D structuresis sufficient. In these circumstances the resulting pat-tern of a single mode at the Fourier-transform plane isgiven as

4

S(n,u) = S(n,u,r). (30)r=1

Here1 sL/2

S(n,ur) = 2 dxHn(x)T[x,a(r),b(r)] exp(iux). (31)f J-L/2

We have formed four types of radiation pattern due tothe structure of the holes in the optical filter.T[x,a(r),b(r)] gives the pattern for each type. Equa-tion (33) and others define this function. For now,notice the terms Hn(x), which represent the nth modeof the original orthonormal decomposition of the elec-tric field at the optical filter. This nth mode is like theHermite-Gaussian mode but does not have to be exact-ly the same. We assume that n ranges from 1 to 6.Also notice that the range in integration is -(L/2) to

We have a large number of definitions to note:First, we give the form of T:

NT[x,a(r),b(r)] -- 10[x - x - al(r)] - O[x - x - b(r)] ,

I=-N+1

(34)

where we have below the step function, the centroidcoordinate, the halfwidth of each hole in the opticalfilter, and the nominal stepping interval between theholes:

O(x) I x>0 O(x)0= x<O,

x1Xi + [al(r) + b1 (r)]/2,

z - [bl(r) - al(r)]12, (35)

xl-= D4(21 -1).

The width of the fiber in u coordinates is related to theactual fiber mode width in position coordinates of theFourier plane as

wm = 2rWfiber/Xf = 0.168 (mm)-'. (36)

Some values that are specified by the conditions ofdesign are

2972 APPLIED OPTICS / Vol. 25, No. 17 / 1 September 1986

Table 1. Values of a(n) etc. vs Strength

TYPE r al(r) bl(r) zI yl - Xi

DC 1 -D4 0 D4/2 -D4/2AC 2 0 g1 g1/2 g1/2El 3 El - D4 -D4 -el/2 eJ/2 - D4E2 4 g1 gl + El El/2 g + l/2

u(k) = N(k)Au,

Xf = 1.0 (mm)2 ,

Au = 27rAx/Xf = 0.0898 (mm)-',

Aucell = 27rAxcell/Xf = 31.42 (mm)-',

Ax = Xf/L = Af/(4ND4) = (1/70) mm,

AXceti = Xf/AXceij = 5 mm,

(37)

AXcell = 200 m.

The cell set is defined by the 2 00-,um cell size. Thisimplies a spacing in the Fourier plane of 5 mm. Thesecond spacing is defined by the finite aperture.

Some further definitions or consequences of con-straints in the design are

D4 = D0/2 = 5m,

N = 3500, (38)

Wfiber = 26.68 for A = 1.0 gim.

DO is the fundamental maximum spot size of the CGH,and Nis the total number of spots available in a 70-mmwidth plate. Wfiber will change as the wavelength ischanged. For maximum accuracy, each new wave-length will require a new CGH. B(zj) is given andapproximated to

B(z,) = 2 sin[u(k)z 1 ]/u(k)

2zl[1 - [u(k)zI2/6]. (39)

The term that is cubic in zj can be a 6% correction to thefilter. Thus it must be retained.

The individual strengths are summed to produce thenet strength of the fundamental mode,

Sl(n,k) = 7 Sl(n,k,r).r=

(40)

Table II. Remaining symmetries vs Strength

r Z-1+1 Y-i+l

1 D4/2 -yl - D42 -gl/2 -yl3 -c-1+1/2 -yl - 2D4 + (l + (1+1)/24 E£4+1/2 -yl + (l + e_1+D/2

The above also shows the restrictions in the size of theerror E and the range of the desired signal.

The beam turning is specified by the N(k). It hasthe following values and properties. Here k rangesfrom -6 to +6.

N(-k) = -N(k),

(42)N(0) = 0,

N(k) = 440, 615, 790, 965, 1140, and 1315,

for k = 1-6. These values are chosen to allow thenecessary physical spacing of the optical fibers andtheir associated equipment as well as avoiding reso-nances due to the cell placement errors. They occur atmultiples of 350. The first-mode correlation N(1) isplaced far enough that the signal on the optical axis hasno significant contribution to that correlation. N(2) isspaced far enough from N(1) so that there is <1%mixing between the N(2) and N(1) terms.

The remaining symmetry relationships are shown inTable II.

This completes the general specification of themathematics for the expected field strengths due tothe four possible radiation patterns. Only the anti-symmetric part, namely, r = 2, is of interest. Theother three imply errors of <0.1% at the points ofinterest (see Ref. 22 for details).

C. Evaluation of the Antisymmetric Nonlinear Part

The strength that has the desired signal is the r = 2case of Table I, and it has the form given by Eq. (43):

C2NSl(n,k,2) = r Hn(y1) exp[iu(k)yl]

1=1

- (-1)n~l exp[-iu(k)yfll

At this point, we specify the a(r), b(r), z, and yl.We use Table I to specify these quantities. The TYPElabel notes the following: DC is the necessary offset toprevent the signal at the hole from becoming negative;AC is the desired signal which can be negative; El isdue to the error in cell placement affecting the offset;and E2 is due to the error in cell placement affectingthe desired signal.

There are some symmetries to note relative to and-1 + 1, namely,

X-+1 =-Xl,

9-l+l -91s

(41)fc • 0.15 Am = 0.03 D4,

jg1 -<D4.

* B(zl) exp[-(w,..yl/2)2]. (43)

This expression contains all the details for the CGH.It is useful to rewrite this expression to get it into areal-function form and to establish the form for the gj.

First we define

S1(n,k,2) = 2\/2DO A(nk) (i)r(k),v Xf

usingWA(y) = exp[-(wnyj/2)2],

yl = xl + ZI,

z = [1/2

g = [lD4h,],q

(44)

1 September 1986 / Vol. 25, No, 17 / APPLIED OPTICS u

(45)

2973

Here [ ] represents the fact that the spot size adjust-ments are discrete rather than continuous. We notethe constraints

0 < a < 1, IhA < 1. (46)

Because of the way a spot is generated we have certainconditions on the values of a.

With the above definitions A(n,k) in Eq. (44) be-comes

NA(nk) = E Hn(y,)B(z,) WA(y,) [u(k)y1]/DO (47)

Cos

with n(k) in Eq. (47) using

r(k) = 1 for k = 1,3,5 use sin in Eq. (47),

= 0 for k = 2,4,6 use cos in Eq. (47), (48)

and with

W(yl) = 1/WA(yi). (49)

Notice that A(n,-k) = -A(n,k) for k odd and A(n,-k)= A(n,k) for k even. This means it is necessary tocalculate only the positive k values. We define fourfunctions for convenience. Namely,

Gl(s,kl,l) W(y 1)H(y 1) cos[u(s)yl] sin[klAuyl],

G2(s,kl,l) W(y 1)H(y 1 ) cos[u(s)yl] cos[klAuyl],

G3(s,kl,l) -W(y1)Hs(y1) sin[u(s)yl] sin[kl/\uyll, (50)

G4(s,kl,1) -W(y,)H,(y1) sin[u(s)yl] cos[klAuyll.

Using these definitions and noting the symmetry of gl,we can further define in terms of some coefficientsB(s,kl), C(s,kl) the composite functions. For s =1,3,5,

N2

U(s,yi) >3 [B(s,kl)G4(s,kl,l)kl=O

+ C(s,kl)G1(s,kl,l)], (51)

and for s = 2,4,6,

N2

U(s,yi) >3 [B(s,kl)G2(s,kl,1)kl=O

+ C(s,kl)G3(s,kl,1)], (52)

(note N2 S 87).hl is finally defined as

6

hi = Pa>3 U(s,yl), (53)s=1

where P is adjusted to make the constraint in Eq. (49)true.

A look at these equations shows that the relation-ships are nonlinear in B(s,kl) and C(s,kl). Thismeans that determining the values of B and C requiresan iteration process to finally get the proper values of Band C. The sequence is to get the value of A(n,k) to beas small as possible for the cases n $ k and to get thevalues of A(n,n) to be approximately the same and as

large as possible consistent with the constraints of theq-

The computation sequence is to start with a selectedset of B(s,kl) and C(s,kl) and compute g, with thechoice of constraints in a and the q operation in aniterative manner until the values of gi are unchanged.For the range of values of gl, this takes no more thanthree iterations. Oncegl is determined, it is possible tocompute A(n,s). The computation sequence de-scribed here is straightforward. The hard part is de-veloping a means for improving the selected set ofB(s,kl) and C(s,kl) to get the best solution in as effi-cient way as possible (see Ref. 22 for those details).

V. Conclusion

We finish this paper with three comments:(1) Once the apparatus is constructed, it will be

possible to do a calibration process with a single well-characterized cw beam to evaluate and calibrate theunit. We then can use some form of matrix analysis toreduce substantially some of the remaining systematicerrors.

(2) We have emphasized Hermite-Gaussian modesin this discussion. To have good separation betweenthe cross-correlation structures in the Fourier plane, itis necessary to stay away from sharp cutoff structuresfor representing the orthonormal set.23

(3) This paper has removed construction detailswhich may be found in Ref. 22. The final apparatuswill be appropriately modified as the construction de-tails are worked out.

In a complex work of this sort there are a largenumber of people and groups who helped at variousstages in the work. In particular, I acknowledge manyuseful discussions with Matt Young and Aaron Sand-ers. The Calibration Coordination Group (CCG) ofthe Department of Defense partially funded the back-ground development necessary to get this work whereit is today.

References1. E. G. Johnson, Jr., "Laser Beam Profile Measurements using

Spatial Sampling, Fourier Optics, and Holography," Natl. Bur.Stand. U.S. Tech. Note 1009 (Jan. 1979).

2. E. G. Johnson, Jr., "Design of a Reflection Apparatus for LaserBeam Profile Measurements," Natl. Bur. Stand. U.S. Tech.Note 1015 (July 1979).

3. E. G. Johnson, Jr., "Beam-Profile Measurements of LaserPulses Using a Spatial Filter to Sample the Hermite Modes for aString of Pulses," Natl. Bur. Stand. U.S. Tech. Note 1057 (Sept.1982).

4. E. G. Johnson, Jr., "Using Optical Processing to Find the BeamProfile of a Laser Pulse (Theory)," Proc. Soc. Photo-Opt. In-strum. Eng. 499, 75 (1985).

5. H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc.IEEE 54,1312 (1966).

6. W. T. Cathey, Optical Information Processing and Holography(Wiley, New York, 1974), pp. 1-200.

7. W. R. Allan, Fiber Optics, Theory and Practice (Plenum, NewYork, 1973), Chap. 9, pp. 177-197.

8. L. Lindgren and K. Vilhelmsson, "Mode Excitation in Graded-Index Optical Fibers," IEEE/OSA J. Lightwave Technol. LT-2,559 (1984).

2974 APPLIED OPTICS / Vol. 25, No. 17 / 1 September 1986

9. T. Ozeki, T. Ito, and T. Tamura, "Tapered Section of MultimodeCladded Fibers as Mode Filter and Mode Analyzers," Appl.Phys. Lett. 26, 386 (1975).

10. F. Montalti, W. Valli, and R. Vannucci, "Optical Fiber Technol-ogy-Analysis of Tapered Graded-Index Fibers," Electon.Commun. 56, 431 (1981).

11. Modern examples of piezoelectric coefficients are shown in re-port PD-9247, Modern Piezoelectric Ceramics, Vernitron Corp.Certain trade names are identified in this paper to specify ade-quately the experimental procedure. Such identification doesnot imply recommendation or endorsement by the NationalBureau of Standards or that the materials or equipment identi-fied are necessarily the best available for the purpose.

12. J. P. Vidosic, in Standard Handbook for Mechanical Engineers,T. Baumeister, E. A. Avalline, and T. Baumeister III, Eds.(McGraw-Hill, New York, 1978), Chap. 5, pp. 5-38.

13. E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, MA,1979), p. 234.

14. M. K. Barnoski, Introduction to Integrated Optics (Plenum,New York, 1974).

15. M. C. Teich, "Infrared Heterodyne Detection," Proc. IEEE 56,37 (1968).

16. S. M. Sze, Physics of Semiconductor Devices (Wiley-Inters-cience, New York, 1969), pp. 625-686.

17. G. W. Day, C. A. Hamilton, P. M. Gruzensky, and R. J. Phelan,Jr., "Performance and Characteristics of Polyvinylidene Fluo-ride Pyroelectric Detectors," Ferroelectics 10, 99 (1976).

18. National Computer Holography Facility at Aerodyne Research,Inc., see discussion in Laser Focus, 42 (Nov. 1982). The designspecifications were chosen since they have been well specified Ufor use by the general public.

19. J. C. Nash, Compact Numerical Methods for Computers: Lin-ear Algebra and Function Minimisation (Wiley, New York,1979). Of particular interest here is the discussion in Chap. 3 onthe singular value decomposition procedure.

20. S. D. Wilson and T. Reed, "Beam and Spot Profiles Measure-ment Methods for Optical Storage Systems," Proc. Soc. Photo-Opt. Instrum. Eng. 499, 65 (1984).

21. J. D. Frank, "Beam Profile Measurement for Target Designa-tors," Proc. Soc. Photo-Opt. Instrum. Eng. 499, 56 (1984).

22. Eric G. Johnson, Jr., Direct Measurement of the Electric Fieldin a Laser Pulse-Theory, Nat. Bur. Stand. (U.S.) Tech. Note1084 (1985).

23. M. B. Priestley, Probability and Mathematical Statistics,Spectral Analysis and Time Series, Vol. 1, Univariate Series,(Academic Press, 1981). Here are some examples of cutofftechniques to avoid ringing.

P. F. Liao of Bell Communications Research-Red Bank, photo-graphed by W. J. Tomlinson of Bell Communications Research-Red Bank, at the 1985 OSA Annual Meeting in Washington, DC.

1 September 1986 / Vol. 25, No. 17 / APPLIED OPTICS 2975