Modeling diffraction and imaging of laser beams by the mode-expansion method

$: Modeling diffraction and imaging of laser beams by the mode-expansion method$
556 J. Opt. Soc. Am. B/Vol. 22, No. 3 /March 2005 N. S. Petrovic and A. D. Rakic

Modeling diffraction and imaging of laser beamsby the mode-expansion method

Novak S. Petrovic* and Aleksandar D. Rakic

School of Information Technology and Electrical Engineering, The University of Queensland,St Lucia QLD 4072, Brisbane, Australia

Received July 1, 2004; accepted September 26, 2004

We present a new method of modeling imaging of laser beams in the presence of diffraction. Our method isbased on the concept of first orthogonally expanding the resultant diffraction field (that would have otherwisebeen obtained by the laborious application of the Huygens diffraction principle) and then representing it by aneffective multimodal laser beam with different beam parameters. We show not only that the process of ob-taining the new beam parameters is straightforward but also that it permits a different interpretation of thediffraction-caused focal shift in laser beams. All of the criteria that we have used to determine the minimumnumber of higher-order modes needed to accurately represent the diffraction field show that the mode-expansion method is numerically efficient. Finally, the characteristics of the mode-expansion method are suchthat it allows modeling of a vast array of diffraction problems, regardless of the characteristics of the incidentlaser beam, the diffracting element, or the observation plane. © 2005 Optical Society of America

OCIS codes: 050.1960, 260.2110, 200.4650.

1. INTRODUCTIONBecause of its great theoretical and practical importance,the problem of laser beam diffraction by a circular aper-ture has frequently been studied in the past.1–7 Given aparticular form of incident light, as well as the shape, po-sition, size, and transmission function of the diffractingaperture, the diffraction problem consists of determiningthe form of the perturbed distribution of light behind theaperture. The most common way of determining the la-ser beam diffraction field is through application of themathematically formalized Huygens principle, which re-sults in the formulation of an integral equation for the un-known field distribution. Solving this equation eitheranalytically or numerically, usually with the aid of simpli-fying approximations and assumptions, gives us the de-sired result. Unfortunately, simple and apparent solu-tions of the diffraction integral occur only in abstractcases; in most practical situations the diffraction fieldmust be obtained by employing special procedures andtechniques. For example, one may need to use advancedintegration techniques,8,9 simplify the integral equationby introducing further approximations,10,11 resort to nu-merical algorithms,12,13 solve the diffraction problem by‘‘equivalent representation,’’14,15 or turn to a completelydifferent theory.16,17

On the other hand, the number of areas where lasersare applied is constantly growing. One example is free-space optical interconnects (FSOIs), which could be usedto solve the communication bottleneck that is due to in-adequate performance of the traditional electricalinterconnects.18–20 As schematically shown in Fig. 1, anFSOI would be made out of aligned arrays of vertical-cavity-surface-emitting lasers, microlenses, and photode-tectors. Given a particular required data transfer rate,as well as all other device parameters, the most generalFSOI design goals are to maximize its length L and chan-

0740-3224/2005/030556-11$15.00 ©

nel density D. L is measured as the physical distancefrom the laser array to the photodetector array, while thechannel density indicates the number of FSOI channelsper unit area. Laser beam diffraction, quantified as‘‘optical-channel-crosstalk noise (OCCN),’’ is the main fac-tor that limits both L and D.

Let us consider channel C0 in Fig. 1 as the representa-tive one.21 As the laser beam propagates from its origi-nal beam waist to the transmitter microlens, it spreads asa result of diffraction. Some of its power crosses overinto the neighboring channels (C1 and C2 in Fig. 1) and iseventually imaged onto the photodetectors for which itwas not intended. The remaining portion of the incidentlaser beam is transformed by the transmitter microlensinto a new laser beam with different beam parametervalues,22 namely, beam waist size and position. Fromthe imaged beam waist, the laser beam continues propa-gating to the receiver microlens plane, where more OCCNis introduced in the same way as at the transmitter mi-crolens plane. This process continues, for as many mi-crolens planes as there may be, until the laser beam fi-nally reaches its target photodetector. At the plane ofeach microlens array, a portion of the incident beamcrosses over into the neighboring channels, from which itis always imaged in such a way as not to reach the pho-todetector for which it was intended; it is received asnoise power on other photodetectors instead. The sum ofthe optical power that is lost at each plane is the totalOCCN.

There are three main factors affecting the amount ofOCCN produced in an FSOI: spacing between channels(channel density), spacing between microlens planes(FSOI length), and the manner in which beams are im-aged by microlenses. Small channel spacings and largeinterplanar distances result in more OCCN, as the beamsstray and diffractively expand further. However, the ef-

2005 Optical Society of America

N. S. Petrovic and A. D. Rakic Vol. 22, No. 3 /March 2005/J. Opt. Soc. Am. B 557

fect of the imaging of laser beams on OCCN is consider-ably subtler. Given that a microlens aperture is largeenough, the incident beam is transformed into a beamwith the same functional form as the incident beam, butwith different beam parameter values. The new param-eter values can be obtained by the application of theABCD law.22–24 The reason for using microlens transfor-mations is to periodically refocus the incident laser beamand hence allow it to travel a greater distance. However,when the size of the microlens aperture is decreased, andwhen it starts to ‘‘clip’’ more of the incident laser power,the incident laser beam is not only imaged, but diffractedby the microlens.25,26 Quantitatively, the resultant dif-fraction field, depending on the extent or ‘‘strength’’ of dif-fraction, and relative to the field distribution when no dif-fraction occurs, will have a mathematically different andwider power distribution as well as different beam pa-rameter values. The precise distribution of the diffrac-tion field, and the beam parameter values must be ob-tained by applying the general Huygens diffractionprinciple, as the simple ABCD law no longer applies. Onthe other hand, in an FSOI context a precise descriptionof the diffraction field must be known if the OCCN is to becalculated precisely; OCCN must be calculated preciselyif the performance of an FSOI is to be balanced at an op-timum.

It has recently been shown that the mode-expansionmethod (MEM), first pro-posed by Tanaka et al.,8 is verywell suited for quickly and accurately determining thediffraction field in an FSOI context.21 The reasons are asfollows:

1. The MEM allows the designer to work out the dif-fraction field without having to perform integrationsbased on the traditional Huygens diffraction principle;

2. practical application of the MEM is conceptuallysimple and numerically mollifying;

3. the MEM describes the diffraction field by using themathematical forms already present in the description ofthe incident laser beam; as no new functions are intro-duced it is easier to see the relative effect of diffraction onthe incident optical field distribution;

Fig. 1. Cross-sectional diagram of a portion of a free-space op-tical interconnect. In the design of FSOIs, it is important to beable to determine quickly and accurately the effect of laser beamdiffraction on the overall device performance. The diffractivespreading of laser beams results in optical-channel-crosstalknoise.

4. the MEM has been shown to be more accurate in awider range of practical situations than other methodscommonly used to model diffraction in FSOIs;

5. while primarily examined in an FSOI context, theMEM has the potential to be applied to a greater varietyof diffraction problems.

The essence of the MEM lies in the representation ofthe diffraction field by a weighted sum of the elements ofa suitable set of orthogonal functions; the key character-istic of the method is that the weighting coefficients canbe calculated without any utilization of the Huygens dif-fraction principle. However, the most important issue inthe practical application of the MEM lies in the determi-nation of the number of orthogonal functions that need tobe used to correctly represent the diffraction field. Math-ematically, an infinite number of modes is required forproper representation; practically, and especially in thecases of ‘‘weak’’ diffraction or diffraction at multiple aper-tures, few functions are desirable.

The MEM was originally developed and has been ap-plied in the case of empty-aperture diffraction.27,28 Thisoriginal approach could still be applied in the case of dif-fraction at a microlens or other similar optical element byconsidering the effect of the aperture and the effect of theoptical element separately.29 However, this approach isnot really the best in this more complicated diffractionproblem.

The purpose of this paper is twofold. First is to de-velop authentically the MEM in the case of laser beamdiffraction at a microlens. Second is to propose and in-vestigate several different ways in which the quality ofthe MEM approximation, as well as the minimum num-ber of required expanding modes, can be determined. InSection 2 we present the application and characteristicsof the MEM in the case of laser beam diffraction at a mi-crolens. We describe and illustrate the way in which theMEM is to be applied in a practical situation, and we alsoshow how the parameters of the diffracted beam changecompared with the diffraction-free ABCD values. In Sec-tion 3 we propose and evaluate several differentgoodness-of-fit criteria and show that few modes areneeded to obtain a good fit, especially if the encircledpower rather than the exact distribution of the diffractionfield is sought.

2. MODE-EXPANSION METHODThe schematic diagram of the diffraction problem that weconsider is shown in Fig. 2. The incident laser beamc (r, z) is the fundamental Gaussian mode whose waist,denoted by ws , is located at z 5 zs . The transverse pro-file of this incident beam is given (in cylindrical coordi-nates) by2,8,30

c ~r, z ! 52

wA2pexpS 2

r2

w2D expF j arctanl~z 2 zs!

pws2

2 jk~z 2 zs!G , (1)

function of the number of expanding modes used N. We


where k 5 2p/l is the wave number, l is the wavelengthof incident light, and the beam spot size at any plane z isgiven by

w 5 w~z ! 5 wsH 1 1 Fl~z 2 zs!

pws2 G 2J 1/2

. (2)

While the MEM can in principle handle any type of inci-dent optical field, we chose this relatively simple form soas not to obscure the presentation of the main principlebehind the method. As further shown in Fig. 2, the cir-cular diffracting aperture A of radius a is located at z5 z0 . We shall first consider the case when the aper-ture is empty, i.e., when it is fully transparent over thewhole area of the circle. Mathematical application of theHuygens diffraction principle in this situation would leadto the formulation of an integral equation for the diffrac-tion field U(r, z) due to the laser beam passing through Aat an observation plane parallel to A. With the assump-tions and approximations that are generally acknowl-edged to be appropriate, the very general original equa-tion for the diffraction field simplifies to the Fresnel–Kirchhoff diffraction integral31:

U~r, z ! 5jk exp@2jk~z 2 z0!#

~z 2 z0!E

0

a

c ~r0 , z0!

3 J0S krr0

z 2 z0D expF 2 jk~r2 1 r0

2!

2~z 2 z0!Gr0dr0 ,

(3)

where Jn( • ) is the Bessel function of the first kind andorder n,32 and c (r0 , z0) denotes the incident field distri-bution over A. Zero-subscripted variables are usedthroughout this paper to indicate values at the plane ofthe diffracting aperture.

The starting point in the MEM is to realize that the dif-fraction field obtained by solving Eq. (3) can also be de-scribed as a weighted sum of elements of an orthonormalset of functions $Cn(r, z)%8,33–35:

U~r, z ! 5 (n50

`

Cn~z !Cn~r, z !, (4)

with the weighting coefficients given by the inversion ofthe above formula as

Cn~z ! 5 2pE0

`

U~r, z !Cn* ~r, z !rdr, (5)

where the asterisk denotes complex conjugation. Aswritten above Eq. (4) represents a mere reformulation ofEq. (3). Its practical usefulness is limited by the fact thatwe need to know a priori the form of the resulting diffrac-tion field before we can calculate the weightingcoefficients—the weighting coefficients that are used torepresent alternatively the same diffraction field.

The second cornerstone of the MEM is that the weight-ing coefficients defined by Eq. (5) do not depend on the lo-cation of the observation plane.21 This means that theobservation plane at which the coefficients are worked outcan be taken to be any plane along z, including the planejust to the right of the diffracting aperture itself. By tak-ing the observation plane to be this plane, i.e., by assum-ing that z 5 z0 , the limiting need for the a priori knowl-edge of U(r, z) is eliminated. Assuming that thediffracting aperture is infinitesimally thin, the diffractionfield at z 5 z0

1 is effectively equal to the incident fieldover A, i.e., to the field at z 5 z0 :

Cn 5 2pE0

a

c ~r0 , z0!Cn* ~r0 , z0!r0dr0 . (6)

By substituting Eq. (6) into Eq. (4), we obtain the follow-ing expression for the diffraction field:

U~r, z ! 5 2p(n50

`

Cn~r, z !E0

a

c ~r0 , z0!Cn* ~r0 , z0!r0dr0

' 2p(n50

N

Cn~r, z !E0

a

c ~r0 , z0!Cn* ~r0 , z0!r0dr0 ,

(7)

which indicates that the diffraction field can be obtainedwithout using the Fresnel–Kirchhoff diffraction integralat all, as long as we can find one plane where we cansafely set the diffraction field to a particular startingform. Equation (7) represents the central point of theMEM. Given the distribution of the incident field overthe diffracting aperture, as well as its shape and location,the diffraction field at any plane behind the aperture canbe approximated without any reference to the traditionalFresnel–Kirchhoff diffraction equation. We intentionallywrote ‘‘approximated,’’ as the accuracy of the MEM is a

Fig. 2. Schematic diagram of the diffraction problem. Aperture A is first considered to be empty, and then to contain a thin lens.


note that the diffracting aperture need not be a circle, butcan be of any shape, as the integration in Eq. (7) is overthe whole surface of the diffracting element. Also the in-cident beam formulation need not be the simple Gaussianfunction given by Eq. (1).30,36,37 In the same way that weorthogonally expanded the diffraction field, we can ex-pand an arbitrary incident laser beam. Thus Eq. (7) rep-resents a versatile and useful tool for determining the re-sultant optical field distribution due to another opticalfield passing through a planar aperture of arbitrary shapeand size.

Now let us assume that the diffracting aperture A inFig. 2 contains a thin lens whose phase-shifting actionf(r, z) is given by10

f~r0! 5 expS 2 jkr02

2f D , (8)

where f denotes the focal length of the lens. The tradi-tional method of obtaining the diffraction field in the pres-ence of a thin lens V(r, z) would be to add the phase-shifting action of the lens to the integrand of the empty-aperture, Fresnel–Kirchhoff diffraction Eq. (3):

V~r, z ! 5jk exp@2jk~z 2 z0!#

~z 2 z0!E

0

a

c ~r0 , z0!f~r0!

3 J0S krr0

z 2 z0D expF 2 jk~r2 1 r0

2!

2~z 2 z0!Gr0dr0 ,

(9)

where all the symbols have the same meaning as before.This resulting integral equation would have to be solvedin essentially the same way as the original diffraction Eq.(3); one should not, however, underestimate the addi-tional difficulty that the phase-shifting action brings. Al-ternatively we could reformulate the expression forV(r, z) given by the above Eq. (9) by using the firstpremise of the MEM, i.e., we could write the diffractionfield due to a laser beam passing through a circular aper-ture containing a microlens as a weighted sum of mem-bers of an orthogonal set of functions:

V~r, z ! 5 (n50

`

KnCn~r, z !, (10)

where $Kn% now represent the weighting coefficients andare explicitly given as

Kn~z ! 5 2pE0

`

V~r, z !Cn* ~r, z !rdr. (11)

However, as the lens action does not depend on z (theposition of the observation plane), it will not introduceany new z dependence into the solution of Eq. (9). Hence,since the only difference between $Cn% and $Kn% is in thelens action, Kn are also independent of z, thus allowing usto write

V~r, z ! 5 (n50

`

KnCn~r, z !

5 2p(n50

`

Cn~r, z !

3 E0

a

c ~r0 , z0!f~r0!Cn* ~r0 , z0!r0dr0

5 2p(n50

`

Cn~r, z !

3 E0

a

c ~r0 , z0!expS 2 jkr02

2f D3 Cn* ~r0 , z0!r0dr0

' 2p(n50

N

Cn~r, z !E0

a

c ~r0 , z0!

3 expS 2 jkr02

2f DCn* ~r0 , z0!r0dr0 . (12)

Equation (12) is a more general formulation of the MEM,as it allows a phase-altering element to be present in thediffracting aperture as long as the action of the element isnot a function of z. Equation (12) also shows that theproblem of diffraction in the presence of a microlens canbe solved without use of the traditional Fresnel–Kirchhoffdiffraction integral. Furthermore, while the complexityof the Fresnel–Kirchhoff diffraction integral in the pres-ence of a microlens is much higher than the complexity ofthe same diffraction integral in the empty-aperture case,neither the characteristics of the diffracting element northe characteristics of the incident laser beam have anystructural bearing on the evaluation of Eq. (12). How-ever, before either Eq. (7) or Eq. (12) can be applied inpractice, several issues need to be resolved, the first beingthe choice of the expanding beam functions.

Bearing in mind the geometry of the two diffractionproblems considered so far, as well as the nature of laserbeams, we think an intuitive choice for the set of expand-ing functions is the set of modes of the free space, herewritten in terms of the Laguerre–Gaussian functions8,2,30:

Cn~r, z ! 52

wA2pLnS 2r2

w2 D expS 2r2

w2D3 expF j~2n 1 1 !arctan

l~z 2 zs!

pws2

2 jk~z 2 zs!G , (13)

where Ln( • ) is the generalized Laguerre polynomial32

with radial number n. As the nondisturbed laser beamsin the free space are easily represented in terms of theLaguerre–Gaussian functions, we expect that their dif-fraction fields can be represented as a sum of a minimumnumber of the same type of functions, regardless of the


characteristics of the diffraction element, and especially ifthe extent of diffraction is moderate. Hermite–Gaussianor any other orthogonal set30,36,37 could equally well beused without any change in the principle of all of theabove equations.

The incident fundamental mode given by Eq. (1) is in-tentionally the first member of the Laguerre-Gaussian setof functions, i.e., c (r, z) 5 C0(r, z). While both the in-cident laser beam and the expanding set have the samefunctional form, we note that the set of incident beam pa-rameters p 5 $ws , zs% (beam waist size and its position)is generally different from the set of expanding beam pa-rameters, p 5 $ws , zs%. Throughout this paper we willwrite the values associated with the expanding beam inboldface. By substituting Eqs. (1) and (13) into Eq. (12),we find the explicit expressions for the coupling coeffi-cients as32:

Kn 5 2pE0

a

c ~r0 , z0!expS 2 jkr02

2f DCn* ~r0 , z0!r0dr0

54

w0w0expF j arctan

l~z 2 zs!

pws2

2 jk~z 2 zs!G3 expF2j~2n 1 1 !arctan

l~z 2 zs!

pws2

1 jk~z 2 zs!G3 (

q50

n S 22

w02D qS 1

q! D S nn 2 q D s 212q

2g ~q 1 1, a2s!,

(14)

where

s 51

w02

11

w02

1jk

2R02

jk

2R02

jp

lf, (15)

and the beam radius of curvature is given as

R 5 R~z ! 5 ~z 2 zs!H 1 1 F pws2

l~z 2 zs!G2J . (16)

There are two outstanding issues yet to be resolved.First is the way in which p is to be determined; second isthe number of modes N to which the infinite sum is short-ened in practice:

V~r, z ! 5 (n50

`

KnCn~r, z ! ' (n50

N

KnCn~r, z !. (17)

These two issues are closely related, as the minimumnumber of modes depends on the particular choice of p.In reality, any value of p could be chosen, including thecase when p 5 p; however, a more informed choice is notvery difficult to make. Consider the case of thefundamental-mode incidence on an empty aperture.Given that the aperture is large enough, we know that anincident beam c (r, z) will pass through unchanged,hence allowing us to conclude that in this case

U~r, z ! 5 (n50

N

CnCn~r, z ! 5 c ~r, z !

⇔C0~r, z ! 5 c ~r, z !, (18)

from which it follows that we must have p 5 p. As wedecrease the aperture radius a, diffraction becomes moreprominent. However, especially in the early stages, westill expect that most of the power from the incident fun-damental mode will be transferred to the expanding fun-damental mode, i.e., that most of the power will be trans-ferred to the mode of the same type. If diffraction effectsare weak, we know that the shape of the diffraction fieldwill still very closely resemble the shape of the incidentfield. Hence, the natural choice of p is the one that maxi-mizes the power contained in the expanding fundamentalmode, i.e., p that maximizes uC0u2. A very similar argu-ment can be used in the case when the diffracting aper-ture contains a thin lens. Given that the aperture islarge enough, the incident fundamental Gaussian beamwill simply be transformed into another Gaussian beamwith different beam waist size and position pABCD5 $ws,ABCD , zs,ABCD%, from which it follows that the opti-mal choice is p 5 pABCD . As the size of the diffractinglens decreases, it is still reasonable to expect that most ofthe incident fundamental power will emerge as the fun-damental power in the expanding beam. Hence the opti-mal expanding parameter set p is the one that maximizesuK0u2. An explicit expression for uK0u2 can readily befound to be32

uK0~ p, p!u2

5 S 4

w02w0

2D3 F1 2 2 exp~2ja2!cos~za2! 1 exp~22ja2!

j2 1 z2 G ,

(19)

where

j 51

w02

11

w02

, z 5k

2R02

k

2R02

p

lf, (20)

and where all the symbols have the same meaning as be-fore.

The process of practical application of the MEM is il-lustrated in Fig. 3. We defer the problem of finding anappropriate value of N until Section 3 and temporarily as-sume that we have a sufficient number of modes, includ-ing the one whose power-coupling coefficient we optimizedin the process of finding p. In Fig. 3(a) we see the origi-nal incident field with its beam parameter set p, the dif-fracting aperture with the lens, and the observationplane. Figure 3(b) shows that the same diffraction fieldthat would have been obtained from the original beamcan be found by first replacing the original incident laserbeam and the diffracting plane with an effective laserbeam, and then propagating it to the observation


plane. The effective beam consists of a weighted sum ofmodes of the free space as given by Eq. (12), and the set ofbeam parameters of the effective beam is found by maxi-mizing Eq. (19). By applying the MEM we are replacingthe laser beam diffracted and refracted by the optical el-ement by another (equivalent) laser beam; the more thediffraction effects are pronounced, the more higher-orderterms (modes) the equivalent beam will contain to ac-count for the diffraction-caused ripples in the trans-formed field.

Let us now illustrate the application of the MEMthrough some practical examples. We consider the case

Fig. 3. By use of the MEM the original incident beam and thediffracting plane shown in (a) are replaced by the effective beamshown in (b). Both the original and the effective configurationproduce the same diffraction field at any observation plane. Val-ues associated with the effective laser beam are written in bold-face.

Fig. 4. Contour plot of uK0u2 (given by Eq. (19)) versus p5 $ws , zs%, where brightness is proportional to uK0u2. The setof expanding beam parameters p is the one for which uK0u2 (thefundamental-to-fundamental-power-coupling coefficient) reachesa maximum. This graph was produced for k 5 1.33, which re-sults in microlens aperture radius of a 5 100 mm.

of diffraction of the fundamental Gaussian beam by a mi-crolens in the situation illustrated in Fig. 1. The param-eter values, chosen as typical ones in FSOIs, are asfollows21: l 5 850 nm, ws 5 3 mm, f 5 800 mm, , 5 z02 zs 5 f 1 zR ' 833 mm, where zR , the Rayleigh range,is defined as zR 5 pws

2/l. The clipping ratio, the ratioof the aperture radius and the incident beam radius atthe aperture plane and given by14,25

k 5a

w0(21)

was taken to be the measure of the extent of diffraction.We chose the value of k 5 1.33 mainly based on our em-pirically developed perception of the extent of diffraction.The observation plane is located at a distance of 21.7 mmaway from the diffracting plane. According to Ref. 8 ifk . 2.0 diffraction effects can generally be neglected; ac-cording to Ref. 14 diffraction effects can be neglectedwhen k . 2.12. Also according to Ref. 14 diffraction maybe considered ‘‘weak’’ when 1.13 , k , 2.0. Our percep-tion is that the placement of the lower boundary of theweak diffraction region is very subjective, and that it var-ies case-by-case depending on the characteristics of thegiven problem.21 For k values that are smaller than thelower edge of the weak diffraction region, and certainlyfor k , 1.0, diffraction is considered by most authors tobe very ‘‘strong.’’

Figure 4 shows the behavior of uK0( p, p)u2 as a func-tion of p. The first feature of the plot is the existence ofa unique and prominent maximum value of uK0( p, p)u2

when ws 5 50.97 mm, and zs 5 9.54 mm, which weworked out by use of Mathematica’s38 FINDMINIMUM nu-merical optimization routine. Because of the effect of dif-fraction, the values obtained are clearly different from thevalues of zs,ABCD 5 11.25 mm and ws,ABCD 5 51.02 mm ob-tained by the ABCD Law.

As shown in Fig. 5 when k is sufficiently large, p be-comes identical to pABCD . As we decrease k, and henceincrease the extent of diffraction, the effective beam waistbecomes smaller and also moves closer to the diffractingaperture; in an FSOI both of these effects would result inmore OCCN. The phenomenon whereby the focus of thediffraction field is produced by an incident field being im-aged and diffracted by a lens has been studied previously,and, depending on the definition of the focus, variousquantifications of the phenomenon exist.39–46 If we havean incident laser beam mode of order n with beam param-eters p imaged and diffracted by a lens as illustrated inFigs. 2 and 3, we could define the focal point zs such thatzs P p and p is such as to maximize the power coefficientuKnu2, where uKnu2 indicates how much of the power in theincident mode of order n is coupled into the effective modeof the same order n. In the same process of finding theoptimal uKnu2 we would find not only the position of thefocal point zs but also the beam waist at that position ws .

Apart from some consideration in the case of diffractionat an empty aperture, this interpretation and evaluationof both the focal point and the diffraction-caused focalshift has not, to the best of the authors’ knowledge, beenconsidered so far. While not in the direct line of our cur-rent inquiry, we believe that this new insight offered by


the MEM is highly beneficial. It essentially allows us towork out the location and the size of the beam waist of adiffracted and imaged laser beam in the same simple wayregardless of the characteristics of the perturbing opticalelements or the incident laser beam. This same propertyof the MEM could also be used to gain new insight intothe definition and evaluation of the actual beam waistsize and the position of laser beams, especially for higher-order modes.

In Fig. 6 we give some indication of how the MEM ap-proximates the intensity of the diffraction field with anincreasing number of expanding modes, again with thesame parameter values as before: l 5 850 nm, ws5 3 mm, f 5 800 mm, , 5 z0 2 zs 5 f 1 zR ' 833 mm,k 5 1.33, and d 5 (z 2 z0) 5 21.7 mm. Our reference,or ‘‘measured,’’ diffraction field Vm(r, z) was obtained bynumerically solving Eq. (3) by the procedure outlined inRef. 9:

Fig. 5. Values of the optimal parameter set of the effective laserbeam ws in (a) and zs in (b) for different values of the clippingratio k. The dashed line in both (a) and (b) represents thediffraction-free values obtained from the ABCD law. As the dif-fraction increases and k decreases, the effective laser beammoves toward the diffracting aperture [as shown in (b)] andeventually becomes smaller [as shown in (a)].

Vm~r, z ! 51

aA2

p

jF

a~1 2 M !

3 expF2jk~z 2 zs! 1 j arctan x

2jFR2

2~1 2 M !G

3 E0

1

R1 expS 2t 2

a2R1

2D J0S FRR1

1 2 M D dR1 ,

(22)

where

x 52~z0 2 zs!

kws2

, R 5r

a, (23)

a 5w0

a, F 5

ka2

f, (24)

M 5f 2 z

f, t2 5 1 1 jx 1

ja2FM

2~1 2 M !.

(25)

The intensity of the measured diffraction field is given as

Fig. 6. Diffraction field produced by the MEM becomes progres-sively better as more expanding modes are used. This figureshows the diffraction field produced by the MEM with (a) 1 mode,(b) 4 modes, (c) 6 modes, and (d) 12 modes. Other parametersare l 5 850 nm, ws 5 3 mm, f 5 800 mm, , 5 z0 2 zs 5 f1 zR ' 833 mm, k 5 1.33, and d 5 (z 2 z0) 5 21.7 mm.


Im~r, z ! 5 uVm~r, z !u2, (26)

whereas the intensity of the diffraction field produced bythe MEM is given by

I~r, z ! 5 uV~r, z !u2. (27)

As shown in Fig. 6, with the fundamental Gaussianmode present in the effective beam only the central lobein the diffraction field is correctly fitted, but none of theother intensity variations are followed. With a sufficientnumber of modes, the MEM approximation converges tothe result given by Eq. (22) in the given observation re-gion. Even extreme diffraction situations, as shown inFig. 7, can be approximated by the MEM. The resultsshown in Figs. 6 and 7 suggest a simple, but somewhatcrude pattern: The more ripples there are in the diffrac-tion field, the more modes are required in the approxi-mate expression. Hence, in the cases of strong diffrac-tion, when the lateral observation distance is large, orwhen the observation plane is close to the diffractionplane, we expect to have to use a larger number of modesN in the effective beam.

Before proceeding to a more formal specification of N,we note that in many applications, as indeed is the case inFSOIs, we could be interested not in the exact intensitydistribution in the diffraction field, but rather in thepower contained in the diffraction field that falls on a par-ticular area V. Assuming that V is a circle of radius v,this encircled power P(V) can also be obtained eitherfrom the ‘‘measured’’ intensity,

Pm~V! 5 2pE0

v

Im~r, z !rdr

5 2pE0

v

uVm~r, z !u2rdr, (28)

or by the MEM,

Fig. 7. MEM can be used to approximate the diffraction field inthe cases of very strong diffraction. In contrast to Fig. 6, herewe have k 5 0.1 and d 5 (z 2 z0) 5 2.71 mm (0.1 of the obser-vation distance used previously).

P~V! 5 2pE0

v

I~r, z !rdr

5 2pE0

v

uV~r, z !u2rdr, (29)

where all the symbols have the same meaning as before.In Section 3 we look more closely into the way in whichthe MEM approximates the encircled power.

3. DETERMINATION OF THE REQUIREDNUMBER OF EXPANDING MODESThe number of expanding modes that are required in theeffective beam depends on the way in which we define thegoodness-of-fit criterion. In this paper we will primarilybe concerned with approximating the intensity of the dif-fraction field and the encircled power, as they are mostrelevant in the study of FSOIs. However, the same rea-soning could be applied to approximating other quanti-ties, such as the optical field amplitude and phase. Asimple but very stringent criterion would be simply tocompare the two values (one obtained by numerical inte-gration or experimental measurement, the other by theMEM) at each point on the observation plane and calcu-late the total difference as an average percent differenceper point:

« int~N ! 5Dr

~rmax 2 rmin!(

r5rmin

rmax uIm~r, z ! 2 I~r, z; N !u

Im~r, z !

3 100%, (30)

where Im(r, z) represents the intensity of the ‘‘measured’’diffraction field, I(r, z) is the intensity of the diffractionfield obtained by the MEM, z remains the position of theobservation plane, and the interval @rmin , rmax# representsthe region of interest in the observation plane throughwhich r is swept in steps of Dr:

r 5 rmin ,rmin 1 Dr,rmin 1 2Dr,rmin 1 3Dr,...,rmax .(31)

As before N is the number of modes in the MEM. Thenumber of test points (rmax 2 rmin)/Dr is increased if thestep is made finer. As indicated by Eq. (30), at each testpoint we calculate the relative difference between the ap-proximate diffraction intensity and the ‘‘measured’’ inten-sity, ignoring the sign of the difference as irrelevant. Wethen add up all those percent differences and divide thesum by the total number of points considered to obtain anaverage difference per point. If we select any observa-tion point in the interval @rmin , rmax#, a particular numeri-cal value of « int tells us what is the most probable differ-ence between the measured and approximate diffractionfields at that particular observation point.

The main problem with « int(N) as defined by Eq. (30) isthe need for a priori knowledge of Im(r, z). In the case offundamental-mode incidence we have the luxury of know-ing the exact distribution—given by Eq. (22)—but in thecase where other modes are present in the incident beam,or where a different element configuration is considered,we may not have the same information at our disposal.Hence to determine when to stop adding modes, we can


only use the information provided by the MEM. So anadaptive criterion may be more suitable, such as

Cint~N 1 DN !

5Dr 3 100%

~rmax 2 rmin!

3 (r5rmin

rmax uI~r, z; N ! 2 I~r, z; N 1 DN !u

I~r, z; N !. (32)

In Eq. (32) we are again determining the average relativedifference per point as in Eq. (30), but now it is betweenthe approximate intensities obtained by the MEM with Nand N 1 DN modes, where the integer DN denotes theadditional number of modes used. By using Eq. (32) wedetermine the difference caused by adding DN modes tothe already-existing approximation. If we assume thateach new mode contributes to the approximation in thebest possible way (which it does given a proper choice ofp), then a small value of Cint(N 1 DN) indicates that thefit with N modes is already good, and that adding an ad-ditional DN modes does not improve the situation consid-erably: Hence, the approximation could safely stop at Nmodes.

Figure 8 shows « int(N) and Cint(N 1 DN) in the samediffraction situation as considered previously. With lessthan '12 modes in the effective beam the average error ismore than 50% at each point. Increasing the number ofmodes from 20 to 50 results in the error dropping from 3%to 1%, while the minimum of 0.3% (with no more than 100modes ever used) is reached at about 80 modes in the ef-fective beam. On the other hand, as also shown in Fig. 8,the local minima of Cint(N, N 1 DN) seem to coincidewith the local minima of « int(N), hence reinforcing the va-lidity of the implicit assumption underlying Eq. (32):Namely, when the approximation is relatively good [indi-cated by a local minimum of « int(N)], the change in theapproximation [indicated by a local minimum in Cint(N1 DN)] is also minimal. Hence we can determine thenumber of required modes in any situation by looking forthe local minima of Cint(N 1 DN) rather than by looking

Fig. 8. First, simple, and stringent criterion used to determinethe minimum number of required expanding modes « int(N) givenby Eq. (33) is plotted with the label ‘‘Error’’ (open squares). Thecorresponding adaptive criterion Cint(N 1 DN) given by Eq. (35)is plotted and denoted by ‘‘Change’’ (filled circles).

for the local minima of « int(N). Depending on the re-quired accuracy, we could use any of the local minima.The benefit of using Cint rather than « int lies in the MEM’sitself providing all the necessary information needed tocalculate Cint , which is not the case for « int .

Finally we note in relation to Fig. 8 that the approxi-mation error «(N) decreases in a very characteristic spi-ral fashion. At the beginning, as we add more modes, thedecrease in the error is very sharp and rapid. However,we soon reach a point where the approximation is goodexcept for a particularly stubborn peak or valley—probably far away from the propagation axis—for whichmodes of much higher order are required. Once thosemodes are incorporated into the effective beam, the prob-lematic region is fixed. However, all the other approxi-mations, previously correct, are now disturbed, and theoverall error increases. The ensuing disturbances arefixed by adding even more modes, until another problem-atic fold is reached, at which point the process is repeatedagain, but at a much lower error scale.

In the same way as we defined « int(N) and Cint(N1 DN) for intensity, we can define them for the purposeof evaluating the encircled power approximation, butwithout the need for the ‘‘per point’’ refinement:

«ep~N ! 5uPm~r, z ! 2 P~r, z; N !u

Pm~r, z !3 100%, (33)

Cep~N 1 DN ! 5uP~r, z; N ! 2 P~r, z; N 1 DN !u

P~r, z; N !

3 100%. (34)

The results are shown in Fig. 9. While the previously ex-hibited trend of the minima of « following the minima of Cis no longer present, we see that even with only one or twomodes in the effective beam, the error is less than 10%.It seems reasonable to say that as soon as the adaptivechange drops below 1%, the number of modes is sufficient,as the overall error is also smaller than 1%. By addingno more than 20 or 30 modes, the error can be decreasedeven to below 1%.

Fig. 9. Plots of «ep(N) and Cep(N 1 DN) in the manner of Fig. 8but now with the encircled power as the approximation objective(resulting in a considerably smaller error scale). We took theencircling radius over which the power is calculated as equal tothe radius of the diffracting aperture.


In our original goodness-of-fit criteria given by Eqs.(30)–(34) we assumed that the difference in the MEM ap-proximation at each point in the observation plane wasequally important. In frequent cases, such as in the de-sign of FSOIs, this may prove to be too strict, as we aregenerally more concerned with fitting the portions of thediffraction field that carry greater amounts of power. Wemay then choose to weight the contribution of each testpoint to the total error, which results in the following re-formulations of the original goodness-of-fit criteria:

« int,w~N ! 5Dr 3 100%

~rmax 2 rmin!

3 (r5rmin

rmax uIm~r, z ! 2 I~r, z; N !u

Im~r, z !expS 2r2

w2 D ,

(35)

and

Cint,w~N 1 DN !

5Dr 3 100%

~rmax 2 rmin!

3 (r5rmin

rmax uI~r, z; N ! 2 I~r, z; N 1 DN !u

I~r, z; N !

3 expS 2r2

w2 D , (36)

where w 5 w(z) represents the spot size of the effectivebeam at the observation plane, and the additional sub-script ‘‘w’’ was added to denote that these are weightedgoodness-of-fit criteria. We choose the Gaussian weight-ing function since it emphasizes the error close to the axisof propagation at the expense of the laterally removedpoints. This is in accordance with our requirements forthe quality of the approximation of the diffraction fieldmainly from the point of view of FSOI design. In othersituations other weighting functions may be chosen.From the results shown in Fig. 10 we see that use of a

Fig. 10. Plots of « int,w(N) and Cint,w(N 1 DN) in the manner ofFig. 8 but with a Gaussian weighting function now included inthe criterion. Considerably fewer higher-order expandingmodes are needed when it is more important to approximate cor-rectly the diffraction field closer to the propagation axis.

weighted fitting function results in the same overall be-havior of the approximation process but with an intrinsi-cally smaller scale of approximation errors.

4. CONCLUSIONWe have found that the mode-expansion method is a veryuseful tool for modeling the imaging and diffraction of la-ser beams for several reasons. First, the extension of theMEM formalism from the empty-aperture-diffraction sce-nario to the more general scenario of diffraction at aphase-shifting optical element was found to be straight-forward. Second, the main formulation of the MEM asgiven by Eqs. (12) and (14) was found to be applicable re-gardless of the placement of the observation plane or pa-rameters of the incident laser beam, and applicable notonly to circular microlenses but to optical elements ofother shapes and characteristics. Third, the process ofdetermination of the parameters of the expanding beamset, based as it is on maximizing the power coupled fromthe incident to the expanding mode of the same type[which translates to maximizing Eq. (19) for the situationdiscussed in this paper], provides another way of inter-preting the focal shift of Gaussian beams imaged in thepresence of diffraction. Finally, all of the three ways ofdetermining the minimum number of required expandingmodes examined in this paper have shown that the MEMis not numerically intensive.

We have found that using an appropriately weightedcomparison criterion results in having to use only '20 ex-panding modes to achieve an approximation error of lessthan 0.1%. If the encircled power rather than the inten-sity of the diffraction field is approximated, then using 20expanding modes results in an approximation error ofless than 0.01%. If a nonweighted criterion for approxi-mating the intensity is used, '80 modes are required foran error of less than 1%.

The best quality of the MEM formalism presented inthis paper is the power to enable one to undertake thestudy of laser beam imaging and diffraction in more com-plicated configurations of practical interest.

*Present address, Filtronic Comtek Pty Ltd, MetroplexAvenue, Murarrie QLD 4172, Brisbane, Australia; e-mailaddress is [email protected].

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Modeling diffraction and imaging of laser beams by the mode-expansion method

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