Adaptive Laguerre-Gaussian variant of the Gaussian beam expansion method

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Cagniot et al. Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. A 2373

Adaptive Laguerre–Gaussian variant of theGaussian beam expansion method

Emmanuel Cagniot,* Michael Fromager, and Kamel Ait-Ameur

Centre de recherche sur les Ions, les Matériaux et la Photonique, Unité Mixte de Recherche 6252,Commissariat à l’Énergie Atomique, Centre National de la Recherche Scientifique, École Nationale Supérieure

d’Ingénieurs de Caen, Université de Caen, 6 Bd Maréchal Juin, F-14050 Caen, France*Corresponding author: [email protected]

Received July 8, 2009; revised September 17, 2009; accepted September 17, 2009;posted September 21, 2009 (Doc. ID 113885); published October 15, 2009

A variant of the Gaussian beam expansion method consists in expanding the Bessel function J0 appearing inthe Fresnel–Kirchhoff integral into a finite sum of complex Gaussian functions to derive an analytical expres-sion for a Laguerre–Gaussian beam diffracted through a hard-edge aperture. However, the validity range ofthe approximation depends on the number of expansion coefficients that are obtained by optimization–computation directly. We propose another solution consisting in expanding J0 onto a set of collimatedLaguerre–Gaussian functions whose waist depends on their number and then, depending on its argument,predicting the suitable number of expansion functions to calculate the integral recursively. © 2009 OpticalSociety of America

OCIS codes: 050.0050, 050.1380, 050.1220, 050.0050.

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. INTRODUCTIONhe study of the diffraction of Gaussian beams throughard-edge apertures has been a workhorse for a long timessentially because forcing lasers to oscillate on a funda-ental mode having a beam quality factor M2 as close to

nity as possible was also a workhorse. In contrast, todayne may prefer to force the laser to oscillate in a singleigh-order mode by inserting adequate amplitude orhase masks into the cavity [1,2] and then, outside thisavity, to reduce the M2 factor of the selected mode by us-ng a diffractive optical element (DOE) [3,4]. As a conse-uence, there is a need for investigating the diffraction ofigh-order beams, as this point has received only a littlettention in the literature.Let us consider a circular DOE located at z=0. It is il-

uminated by a normally incident Laguerre–Gaussianeam whose electric field is written as

�p��,z� =�2

�

1

w�z�exp�−

�2

w�z�2�Lp� 2�2

w�z�2��exp�− j

k�2

2R�z��exp�j�2p + 1��z��

�exp�− jk�z − z0��, �1�

here z0 is the location of the beam waist radius, k is theave number, and Lp is the p-order Laguerre polynomial.he lower-order function is a Gaussian beam character-

zed by its waist w�z�, its radius of curvature R�z�, and theouy phase shift ��z�, which obey the following formulas:

w�z� = w0�1 + � z − z0

zR2�1/2

, �2�

1084-7529/09/112373-10/$15.00 © 2

R�z� = �z − z0��1 + � zR

z − z02� , �3�

��z� = arctan� z − z0

zR , �4�

here zR is the Rayleigh range and w0 is the beam waistadius at z0.

The diffracted field E�r ,z� at a distance z from the DOEs given by the Fresnel–Kirchhoff diffraction integral

E�r,z� = −2� exp�− jkz�

j�zexp�−

j�r2

�z �

0

+�

��p��,0�exp�−jk�2

2z J0�2�

�zr��d�,

�5�

here �� and J0 are, respectively, the transmittanceunction of the DOE and the zero-order Bessel function ofhe first kind. For the sake of clarity, this equation is re-ritten as

E�r,z� = C0

+�

��Lp� 2�2

w�0�2�exp�− � 1

w�0�2

+ jk

2� 1

R�0�+

1

z��2�J0�b��d�, �6�

here

C = −2� exp�− jkz�

j�zexp�−

j�r2

�z �2

�

1

w�0�

�exp�j�2p + 1��0��exp�jkz �, �7�
0
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2374 J. Opt. Soc. Am. A/Vol. 26, No. 11 /November 2009 Cagniot et al.

b =2�

�zr. �8�

Numerical integrator routines such as like QAG orAWO [5] represent the most common way to compute

he Eq. (6) integral. They use pairs of quadrature rules:he high-order rule computes the best approximation ofhe integral over a small range, while the difference be-ween the result of the high-order rule and that of theow-order one gives an estimate of the error in the ap-roximation. The integration area is split into subinter-als, and on each iteration the subinterval with the larg-st estimated error is bisected. This rapidly decreases theverall error, while subintervals concentrate around localifficulties in the integrand. Therefore, oscillatory inte-rands like those in Eq. (6) may increase computationimes because of specific beam orders, waists at the DOElane, or transmittance functions. As a consequence, us-ng an adaptive numerical integrator within an optimiza-ion process requiring thousands integral computations isroblematic.Many research works have been undertaken to find

nalytical solutions based on orthogonal polynomial fami-ies [6–10] when diffracting Gaussian beams throughard-edge apertures, fewer with high-order Laguerre–aussian ones [11]. One can roughly classify these fami-

ies into two sets. The first set refers to families that syn-hesize the diffracted beam as a superposition ofxpansion beams belonging to the same family as the in-ut beam. This convenience allows one to handle each ofhe expansion beams separately in the same way as thenput one. Laguerre–Gaussian and Hermite–Gaussianeams are the best-known representatives of this set. Thether set refers to purely mathematical families, of whichhe best-known representative is the Chebyshev one.

Depending on the family set, determining the suitableumber of expansion polynomials as well as the trunca-ion error is more or less difficult. Since expansions canork only in a limited range, the real issue is finding this

ange, typically dictated by the nature of the diffractedeld. Once this natural range is exceeded, the expansionrocedure becomes unstable, as testified by the rapidlyrowing number of terms. For this reason one often di-ides the whole range into two parts and employs sepa-ate complementary expansions, one for the inner regionnd one for the asymptotic region, typically in an invertedariable. Expansions onto Chebyshev polynomials illus-rate this technique [9]. When expanding onto Laguerre–aussian functions one has to choose the waist and the

adius of curvature of the expanding modes (expansionarameter set). This choice is straightforward for Gauss-an beams: it is the set that maximizes the power transferrom the incident beam to the expanding beam of theame order [7,8,10]; but this reasoning is no longer rel-vant when applied to higher-order beams, since the pre-ominant expanding mode depends on the radius of theperture [12]. Anyway, the hardest part is determiningow many coefficients to use and the corresponding trun-ation error.

The Gaussian beam expansion method (GBEM) referso a technique that does not use orthogonal polynomials:t consists in expanding the circ function (the transmit-

ance function of the hard-edge aperture) into an approxi-ate sum of a finite set of complex Gaussian functions

13]. Since complex Gaussian functions do not form a ba-is, expansion coefficients are computed by a numericalptimization process. Such an expansion allows one toransform the Eq. (6) integral into a finite sum of Hankelransforms whose analytical expressions are known. As aonsequence, a significant number of papers propose algo-ithms for computing accurate sets of coefficients [14,15].

For specific reasons it may be preferable to expand theessel function J0 onto complex Gaussian functions. Aariant of the GBEM addresses this problem [16,17] byleverly exploiting the following feature: since the zero-rder Bessel function of the first kind can be calculated bysing the first-order one, expansion coefficients of func-ion J0 can be deduced from that of the circ function.owever, since the validity range of the J0 approximationepends on the number of expansion coefficients that arebtained by optimization–computation directly, this is aimitation.

We propose an alternative to this variant that tries toreserve its simplicity while removing its limitation.ince they are the closest to complex Gaussian functions,he zero-order Bessel function of the first kind is ex-anded onto a specific set of collimated Laguerre–aussian functions whose waist depends on their num-er. On the one hand, this choice allows one to continuexpressing the diffracted field as a superposition of ex-ansion beams belonging to the same family as the inputne. On the other hand, it allows one to derive a simpleule that, depending on the argument of J0, predicts theuitable number of expansion functions. Finally, one justas to apply a recurrence formula to compute rapidly bothhe expansion coefficients and the Fresnel–Kirchhoff dif-raction integral.

This paper is organized as follows. Section 2 presentshe expansion of the zero-order Bessel function of the firstind into a finite sum of collimated Laguerre–Gaussianunctions whose waist depends on their number. A simpleay to predict the suitable number of expansion functionsepending on the argument of J0 is proposed. Section 3erives both the on-axis and off-axis diffracted field ana-ytical expressions for simple binary optics like hard-edgepertures and opaque disks. Section 4 presents the corre-ponding numerical results. Finally, the conclusion isiven in section 5. Appendices A and B present, respec-ively, the variant of the GBEM and the recurrence for-ula standing as the backbone of this work.

. EXPANDING THE ZERO-ORDER BESSELUNCTION OF THE FIRST KINDhis section presents the most significant result of the pa-er. First, by combining the results of previous work, weerive a novel expansion for the Bessel function J0 differ-nt from that given in [7], since it is based on a set of col-imated Laguerre–Gaussian functions whose waist de-ends on their number. Second, this feature is exploited toerive a simple relationship allowing one to predict theuitable number of expansion functions depending on thergument of J .
0

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. Laguerre–Gaussian Expansionn [18], after recalling that any finite support �0,a� can becaled to fit into the interval [0, 1] by a suitable choice ofhe unit length, authors give the following rule of thumb:

hen a function f�r� with finite support [0, 1] is to be ap-roximated by a LG series truncated to the Nth term, thepot size 0 should equal 1/�N.

As a consequence, once the variable change t=u /x isone, Eq. (A3) is rewritten as

J1�x� = x0

+�

circ�t�J0�tx�tdt, �9�

nd then the circ function with finite support [0, 1] is ex-anded as a truncated sum of Laguerre–Gaussian colli-ated functions,

circ�t� �n=0

N−1

fn�0��n�t;0�, �10�

here

�n�t;0� =�2

�

1

0Ln�2t2

02 exp�−

t2

02 , �11�

0 =1

�N. �12�

Since �n�t ;0� is collimated, we have �n�t ;0��n

*�t ;0�, and then expansion coefficients fn�0�, defineds

fn�0� = 2�0

+�

circ�t��n*�t;0�dt, �13�

re calculated by our recurrence formula [Eq. (B4)–(B6)]

fn�0� =�2�

0I�0,n�, �14�

here =2/02 and Q= /2.

The truncation error is expressed as the normalizedean square error

�̂N�0� = 1 −1

� �n=0

N−1

fn2�0�. �15�

Applying the same technique as in Appendix A, Eq. (10)s introduced into Eq. (9) to produce

J1�x� �2

�

1

0x�

n=0

N−1 �fn�0�0

+�

Ln�2t2

02

�exp�−t2

02J0�xt�tdt� . �16�

Recalling the integral formula

0

+�

exp�− �t2�Ln�t2�J0�tx�tdt =�� − �n

2�n+1

�exp�−x2

4�Ln� x2

4�� − �� , �17�

here Re�� 0, and with the variable changes �=1/02

nd =2�, Eq. (16) becomes

J1�x� = − J0��x� 1

2��2

�

1

0x exp�−

x2

4�

��n=0

N−1

�− 1�nfn�0�Ln� x2

2� . �18�

This equation cannot be directly integrated with re-pect to the variable x. However, by making use of

xJ0��x� + J0��x� + xJ0�x� = 0 �19�

nd then

∀p � 1, xLp��x� = p�Lp�x� − Lp−1�x��, �20�

e can finally derive the expansion of the zero-orderessel function of the first kind:

J0�x� 1

2��2

�

1

0exp�−

x2

4��f0�0��1 + L1� x2

2��

+ �n=1

N−1

�− 1�nfn�0��n + 1�Ln+1� x2

2� + Ln� x2

2�

− nLn−1� x2

2�� . �21�

. Prediction Rulehe problem is to determine the suitable number of ex-ansion functions in such a way that J0�x� and its ap-roximation fit within the range �0,x�. Defining a normal-zed mean square error like Eq. (15) is problematic, sincee should integrate the square of J0 within �0,x�. A less

ormal method consists in performing measurements andhen applying a linear regression in the hope of makingppear a strong correlation between the upper bound xnd the corresponding number of expansion functions N.Let xk=k be a coordinate where k is a natural integer

nd =0.1. Let �=10−2 be the maximum permissible erroretween J0 and its approximation J0

N using N expansionunctions so that

∀xk � xk max, �J0�xk� − J0N�xk�� . �22�

Performing measurements for N� �1,700� and then ap-lying a linear regression, let the following correlation ap-ear:

xk max = 2.04192N − 32.7612, �23�

hile the Bravais–Pearson coefficient equals 0.999948,mplying that measurements xk max and N are fully corre-ated.

The value of xk max calculated by Eq. (23) is eitherarger or smaller than the corresponding measurement.

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o

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E

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o

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a

ip

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t


s a consequence, to ensure that the predicted value fork max is always smaller (i.e., the number of expansionunctions is always sufficient), we propose the formula

x̄k max = xk max − 14 = 2.04192N − 46.7612, �24�

here x̄k max is the predicted value. This equation implieshat the minimum number of expansion functions is 23.oreover, it must be supplemented by an exit conditionhen x̄k max tends toward positive infinity. If the precision

equired is 10−5, then J0�285.1� falls to 1.33479�10−6,nd the maximum number of expansion functions is 163.eyond this limit, J0 is considered null.Now, N and therefore parameters 0 and � appearing

n Eq. (21) can be simply deduced from the argument of0.

. SIMPLE BINARY OPTICSwo simple amplitude binary optics are investigated asxamples: a circular hard-edge aperture and an opaqueisk. For each of them we derive both the on-axis and off-xis diffracted field analytical expressions with respect tour recurrence formula.

. Hard-Edge Aperturentroducing the transmittance function of the hard-edgeperture given by Eq. (A2), Eq. (6) becomes

E�r,z� = C0

a

Lp� 2�2

w�0�2�exp�− � 1

w�0�2

+ jk

2� 1

R�0�+

1

z��2�J0�b��d�. �25�

Since J0�0�=1, the on-axis diffracted field is

E�0,z� = C0

a

Lp��2�exp�− Q�2��d�, �26�

here

� =2

w�0�2 , �27�

Q =1

w�0�2 + jk

2� 1

R�0�+

1

z . �28�

Once the variable change X=�2 is done, we derive then-axis analytical expression of the diffracted field:

E�0,z� =C

2I��p,0�, �29�

here �=a2.Introducing the J0 expansion defined by Eq. (21) into

q. (25) leads to

E�r,z� = C1

2��2

�

1

0�f0�0�

0

a

Lp��2��L0��2�

+ L1��2��exp�− Q�2��d�

+ �n=1

N−1

�− 1�nfn�0��n + 1�0

a

Lp��2�Ln+1��2�

�exp�− Q�2��d� +0

a

Lp��2�Ln��2�

�exp�− Q�2��d� − n0

a

Lp��2�Ln−1��2�

�exp�− Q�2��d�� , �30�

here

� =2

w�0�2 , �31�

=b2

2�, �32�

Q =1

w�0�2 +b2

4�+ j

k

2� 1

R�0�+

1

z . �33�

Once the variable change X=�2 is done, we derive theff-axis analytical expression of the diffracted field,

E�r,z� C

2

1

2��2

�

1

0�f0�0��I��p,0� + I��p,1��

+ �n=1

N−1

�− 1�nfn�0��n + 1�I��p,n + 1� + I��p,n�

− nI��p,n − 1�� , �34�

here �=a2.Note that if an ideal thin lens whose focal length is f

nd whose transmittance function is

�� = −k�2

2f�35�

s placed quite close behind the aperture, then the new ex-ression of parameter Q is simply

Q =1

w�0�2 +b2

4�+ j

k

2� 1

R�0�−

1

f+

1

z . �36�

. Opaque Diskntroducing the transmittance function of the opaque disk

��

a = �1, � a

0, 0 � � � a� , �37�

hen Eq. (6) becomes

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E�r,z� = C�0

+�

Lp� 2�2

w�0�2�exp�− � 1

w�0�2

+ jk

2� 1

R�0�+

1

z��2�J0�b��d�

−0

a

Lp� 2�2

w�0�2�exp�− � 1

w�0�2

+ jk

2� 1

R�0�+

1

z��2�J0�b��d�� . �38�

By combining Eqs. (17) and (29) we derive the on-axisnalytical expression of the diffracted field,

E�0,z� =C

2 � �Q − ��p

Qp+1 − I��p,0�� , �39�

here � and Q are, respectively, defined by Eqs. (27) and28) and where �=a2.

In the same way, by combining Eqs. (17) and (34) weerive the off-axis analytical expression of the diffractedeld,

E�r,z� C

2��Q − ��p

Qp+1 exp�−b2

4QLp� �b2

4Q�� − Q��−

1

2��2

�

1

0�f0�0��I��p,0� + I��p,1��

+ �n=1

N−1

�− 1�nfn�0��n + 1�I��p,n + 1� + I��p,n�

− nI��p,n − 1�� , �40�

here parameters � and Q are, respectively, defined byqs. (27) and (28) and where �=a2.

. NUMERICAL RESULTSomputations have been carried out on a Hewlett Pack-rd HDX X18-1250EF Premium Notebook PC [Intel 64 bitore 2 Duo P8600 2.40 GHz, 4 Gbytes of DDR3 �1 GHz�emory, 3 Mbytes of L2 cache] running Linux version

.6.27.24 (Fedora distribution, “Cambridge”). The appli-ation is written in C�� and makes intensive use of in-eritance, polymorphism, and templates. It requires theNU Scientific Library 1.11 (GSL, numerical integratorsnd mathematical functions) and it is compiled with theNU g�� compiler 4.3.2 and the following optimizations:march=native (instruction set of the host processornly), -funroll-loops (loop unrolling), -msse2 -msse3mssse3 -msse4 (all available streaming instruction setsor double-precision floating-point numbers) and -O3level 3 optimizations).

In the following we consider a TEMp0 beam that is nor-ally incident upon a DOE whose radius is chosen to be=Y�w�0�, where Y is the beam truncation ratio. Theeasurement plane is located at coordinate z .
m
Since input beam orders are low, our variant does notse quad-double arithmetic. Two versions are proposed:daptive uses the prediction rule defined by Eq. (24),hile Fixed uses a fixed number of expanding functions

hat is equal to the maximum one. Results produced byhese two versions are compared with those produced byAG, the adaptive numerical integrator QAG using a 61-oint Gauss–Kronrod rule whose absolute and relative er-ors are, respectively, set to 10−10 and 10−8, while theaximum number of subintervals is set to 1024. Note

hat the purpose of Fixed is simply to stand as an upperound for Adaptive (accuracy and computation time): itas no other import.Let, respectively, I1�i� and I2�i� be the intensities calcu-

ated by QAG and Adaptive or Fixed at the ith off-axisoordinate of the measurement plane. The approximationrror in percent is defined as

� =�i

�I1�i� − I2�i��

�iI1�i�

� 100. �41�

Since the host system is powerful, intensity distribu-ions are sampled onto disproportionate numbers of equi-paced points so that computation times exceed 1 s.

. Gaussian Beam Diffracted by a Hard-Edge Aperturee first apply our variant to a test case based on [8,10],here a Gaussian beam is diffracted by a hard-edge ap-rture. It has a beam waist w0=0.003 mm that is locatedt z0=−0.8333 mm and a wavelength �=850 nm. Its waistnd its radius of curvature at the aperture plane are, re-pectively, w�0�=0.0752 mm and R�0�=0.835 mm. Theperture radius is chosen to be a=0.1�w�0�, while theeasurement plane is located at various coordinates in

he near-field and the far-field areas that are character-zed by their Fresnel number,

Nf = � 1

R�0�+

1

zma2

�. �42�

he near field extends a distance away from the apertureuch that Nf 1, while the far field corresponds to Nf�1.

The off-axis intensity distributions are sampled on024�1024 equispaced points chosen in a range [0, ub]m whose upper bound ub depends on the Fresnel num-

er to deal with the changing feature scale of the dif-racted field.

Table 1 presents results obtained for this test case. Theolumns stand, respectively, for the Fresnel number Nf,he corresponding location of the diffraction plane zm, thepper bound ub, the computation time required by QAG,nd then both the computation time and the approxima-ion error corresponding to Adaptive and Fixed. Figureplots the intensity distributions corresponding to Nf=8.These results show that the computation time required

y QAG increases with the Fresnel number. As expected,ixed is characterized by both a very good approximationrror and a constant computation time. Since the approxi-ation error does not exceed 6% even in the extreme neareld, these results illustrate the good behavior of

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daptive, whose computation time is nearly constantnd is approximately half that of Fixed.

. Collimated TEM20 Beam Diffracted by a Hard-Edgeperture with a Thin Lense consider a collimated TEM20 beam that is diffracted

y a hard-edge aperture with a thin lens whose focalength is f=50 mm. Its wavelength and its waist at theperture plane are, respectively, �=1.064 �m and w�0�1 mm. The aperture radius is chosen to be a=0.9w�0� to coincide approximately with the first concentric

ing, while the measurement plane is located at the lensocal plane.

The off-axis intensity distributions are sampled on 101024 equispaced points chosen in the range �0,0.1� mm.AG performs the computation in 355 seconds whiledaptive and Fixed both require less than 1 s. The ap-roximation errors are, respectively, �=7.53% for Adap-ive and �=1.05% for Fixed. Figure 2 plots the corre-ponding intensity distributions.

This test case illustrates how a specific combination ofn input beam order (not necessarily high), a beam trun-ation ratio (not necessarily low), and a common idealhin lens can put QAG in difficulty (355 s for a low number

Table 1. Gaussian Beam Diff

Nf zm ��m� ub (mm)

QAG

t (s)

11 6.09 0.01 14610 6.71 0.01 1329 7.46 0.01 1118 8.4 0.01 887 9.61 0.01 636 11.2 0.01 635 13.5 0.01 644 17 0.015 633 22.8 0.015 612 34.6 0.015 551 72.3 0.02 49

0.5 158 0.02 370.1 3270 0.5 40

0

50

100

150

200

250

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01

Inte

nsity

(a.u

.)

r (mm)

QAGAdaptive

Fixed

ig. 1. Off-axis intensity distributions of a Gaussian beam dif-racted by a hard-edge aperture and measured at N =8.
f
f points on a powerful host system, i.e., an eternity…),hich is problematic within an optimization process re-uiring thousands of integral computations. As in the pre-ious test case, our variant achieves satisfying results.

. Collimated TEM10 Beam Diffracted by an Opaqueiske consider a test case presented in [19] in which a colli-ated TEM10 beam is diffracted by an opaque disk. Itsavelength and its waist at the disk plane are, respec-

ively, �=1.064 �m and w�0�=1 mm. The radius of theisk is chosen to be a=0.5��2 mm to coincide exactlyith the dark ring radius of the input beam, while theeasurement plane is located at various coordinates in

he near-field and the far-field areas.The off-axis intensity distributions are sampled upon

4�1024 equispaced points.Table 2 presents results obtained for this test case,

hile Fig. 3 plots the intensity distributions correspond-ng to Nf=7.

From these results we can see that, on the one hand,pproximation errors achieved by Adaptive are more de-

d by a Hard-Edge Aperture

Adaptive Fixed

t (s) � (%) t (s) � (%)

6 5.8 10 1.465 5.61 10 1.36 5.3 10 1.135 4.85 10 1.175 4.56 10 1.115 4.71 10 1.074 4.26 10 0.7945 5.02 10 0.8924 4.32 11 0.794 4.13 10 0.7854 3.49 10 0.4464 3.79 10 0.434 4.11 10 0.469

0

5

10

15

20

25

30

35

40

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

Inte

nsity

(a.u

.)

r (mm)

QAGAdaptive

Fixed

ig. 2. Off-axis intensity distributions of a collimated TEM20eam diffracted by a hard-edge aperture with a thin lens andeasured at the focal plane.

racte

catQ

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a=stittt

�

wi

tegsm

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Fb


eiving than for hard-edge apertures because impuritiesppear at the ring center. On the other hand, results ob-ained by Fixed are also always good. For both, unlikeAG, computation times do not exceed 1 s.

. Collimated Gaussian Beam Diffracted by an Opaqueiskesults achieved by Adaptive in the previous test casehere a collimated TEM10 beam was diffracted by an

paque disk seem to reveal a weakness. To be sure, thisime we consider a test case in which a collimated Gauss-an beam is diffracted by an opaque disk. Its wavelength

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-3 -2 -1 0 1 2 3

Inte

nsity

(a.u

.)

r (mm)

QAGAdaptive

Fixed

ig. 3. Off-axis intensity distributions of a collimated TEM10eam diffracted by an opaque disk and measured at Nf=7.

Table 2. Collimated TEM10 Be

Nf zm (mm) ub (mm)

QAG

t (s)

7 67.1 3 2566 78.3 3 2295 94 3 2024 117 3 1593 157 3 1242 235 3 861 470 3 44

0.1 4700 5 40.05 9400 7 4

Table 3. Collimated Gaussian B

Nf zm (mm) ub (mm)

QAG

t (s)

7 134 2 1046 157 2 925 188 2 724 235 2 593 313 2 502 470 2 331 940 3 16

0.1 9400 8 4

nd its waist at the disk plane are, respectively, �1.064 �m and w�0�=1 mm. The radius of the disk is cho-en to be a=1 mm to occult most of the input beam, whilehe measurement plane is located at various coordinatesn the near-field and the far-field areas. Since such an op-ical layout is known to produce Bessel beams with mul-iple wavelets [20], this test case must be considered ex-reme.

The off-axis intensity distributions are sampled on 641024 equispaced points.Table 3 presents results obtained for this test case,

hile Fig. 4 plots the intensity distributions correspond-ng to Nf=7.

These results confirm our suspicions because Adap-ive completely fails to rebuild the diffracted field. How-ver, approximation errors achieved by Fixed remainood despite significant numbers of wavelets to recon-truct. As a consequence, a conclusion is essential: theinimum number of expanding functions is too weak.Noticing that in the case of an opaque disk the upper

ound of the integral in Eq. (6) is positive infinity, let usefine the following rule of thumb: “the maximum num-er of expansion functions for hard-edge apertures is theinimum one for opaque disks.” If min and max stand,

espectively, for the minimum and the maximum numberf expansion functions for hard-edge apertures, then ap-lying this rule leads simply to add max–min expansionunctions to the result given by the prediction rule. There-ore, in our case, the minimum number of expansion func-ions is 163, while the maximum one is 303. Now, rerun-

iffracted by an Opaque Disk

Adaptive Fixed

t (s) � (%) t (s) � (%)

0 4.18 1 0.5320 3.71 1 0.5111 3.27 1 0.5081 3.01 1 0.520 2.75 1 0.5280 2.88 1 0.5260 3.07 1 0.4980 0.635 1 0.06950 0.963 1 0.104

Diffracted by an Opaque Disk

Adaptive Fixed

t (s) � (%) t (s) � (%)

1 80.8 1 151 71.3 1 130 60.2 1 11.21 51.5 1 8.470 43.2 1 70 28.3 1 4.550 21 1 2.731 9.41 1 1.09

am D

eam

n�sa

5Steilr

HkmptcppHdcG

viGfiLtcoisdptHsK

crtthpt

ossfatvherem

AGT[

fi

E

t

wto

w

Fb


ing Adaptive and Fixed for Nf=7 leads, respectively, to=13.4% and �=7.79%, while computation times are, re-pectively, 1 and 2 s. As a consequence Adaptive is againttractive for opaque disks.

. CONCLUSIONince computation times required by numerical integra-ors such as QAG are case dependent, deriving analyticalxpressions for the Fresnel–Kirchhoff diffraction integrals of the utmost importance to tackle optimization prob-ems involving high-order Laguerre–Gaussian beams andequiring thousands of integral computations.

To transform the diffraction integral into a finite sum ofankel transforms whose analytical expressions arenown, the GBEM expands the circ function, the trans-ittance function of the hard-edge aperture, into an ap-

roximate sum of a finite set of complex Gaussian func-ions whose coefficients are obtained by optimization–omputation directly. For specific reasons it may bereferable to expand the Bessel function J0. Such an ex-ansion is simply deduced from that of the circ function.owever, since the validity range of the J0 approximationepends on the number of expansion coefficients that arealculated by an optimization process, this variant of theBEM suffers from a limitation.In this paper we have proposed an alternative to this

ariant that tries to preserve its simplicity while remov-ng its limitation. Since they are the closest to complexaussian functions, the zero-order Bessel function of therst kind is expanded onto a specific set of collimatedaguerre–Gaussian functions whose waist depends onheir number. On the one hand, this choice allows one toontinue expressing the diffracted field as a superpositionf expansion beams belonging to the same family as thenput one. On the other hand, it allows one to derive aimple rule that, depending on the argument of J0, pre-icts the suitable number of expansion functions. Com-ared with an expansion onto complex Gaussian func-ions, the number of expansion functions is greater.owever, since Laguerre–Gaussian functions form a ba-

is, the expansion coefficients as well as the Fresnel–irchhoff diffraction integral are computed by using a re-

0

0.02

0.04

0.06

0.08

0.1

0.12

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Inte

nsity

(a.u

.)

r (mm)

QAGAdaptive

Fixed

ig. 4. Off-axis intensity distributions of a collimated Gaussianeam diffracted by an opaque disk and measured at Nf=7.

urrence formula. The amount of processor instructionsequired by the recurrence formula is broadly superior tohat of an adaptive numerical integrator. However, sincehese are elementary operations, they are efficientlyandled by dedicated pipeline operators on all modernrocessors. As a consequence, computation times are ex-remely low.

Two versions of our variant have been tested. The firstne, Adaptive, predicts the suitable number of expan-ion functions depending on the argument of J0, while theecond one, Fixed, uses a maximum number of expansionunctions. Note that the only import of Fixed is to stands an upper bound for Adaptive (accuracy and computa-ion time). Approximation errors achieved by Fixed areery good in all cases, while Adaptive succeeds withard-edge apertures but fails with opaque disks. How-ver, the problem was solved by applying the followingule of thumb for opaque disks: “the maximum number ofxpansion functions for hard-edge apertures is the mini-um one for opaque disks.”

PPENDIX A: VARIANT OF THEAUSSIAN BEAM EXPANSION METHOD

his variant was proposed first for the field of acoustics16] and then applied to that of optics [17].

Let us consider the first-order Bessel function of therst kind,

xJ1�x� =0

x

uJ0�u�du. �A1�

On introducing the circ function,

circ� �

a = �1 0 � � � a

0 � a � , �A2�

q. (A1) becomes

xJ1�x� =0

+�

circ�u

xuJ0�u�du. �A3�

The circ function can be expanded as a linear combina-ion of complex Gaussian functions,

circ�u

x �n=0

N−1

An exp�−Bn

x2 u2 , �A4�

here complex constants An and Bn denote, respectively,he expansion and Gaussian coefficients obtained byptimization–computation directly [13].

Introducing this expansion into Eq. (A3) leads to

xJ1�x� �n=0

N−1 �An0

+�

exp�−Bn

x2 u2uJ0�u�du� . �A5�

Recalling the integral formula

0

+�

exp�− pt2�J0�bt�tdt =1

2pexp�−

b2

4p , �A6�

here Re�p� 0, we can rewrite Eq. (A5) as

(

czzt

wq

ATfan

wc

w

im

∀

w

bsn�twpmitc

R


J1�x� = − J0��x� �n=0

N−1 An

2Bnx exp�−

x2

4Bn . �A7�

Finally, we get the expansion of J0 by integrating Eq.A7) with respect to the variable x:

J0�x� �n=0

N−1

An exp�−x2

4Bn . �A8�

If complex coefficients An and Bn appear in complexonjugate pairs, then the imaginary part of Eq. (A8) isero. Otherwise, since this imaginary part is very close toero, it is canceled by taking only the real part of the al-ernative expansion,

J0�x� 1

2 �n=0

N−1 �An exp�−x2

4Bn + An

* exp�−x2

4Bn*� ,

�A9�

here the symbol � denotes the complex conjugate of theuantity.

PPENDIX B: RECURRENCE FORMULAhis recurrence formula was first proposed to model dif-

raction through apertured ABCD systems [21] and thenpplied to optimize a plano-concave laser cavity with twoonequivalent apertures inside [22].Let us consider the finite integral

I��p,q� =0

�

Lp��x�Lq�x�exp�− Rx�dx, �B1�

here � 0, 0 and Re�R� 0. Once the variablehange y=x /� is done, Eq. (B1) can be rewritten as

I��p,q� = �0

1

Lp��y�Lq�y�exp�− Qy�dy = �I�p,q�,

�B2�

here �=��, =�, and Q=R�.Since Laguerre polynomials are defined recursively,

∀p � 2, pLp�x� = �2p − 1 − x�Lp−1�x� − �p − 1�Lp−2�x�,

�B3�

ntegral I�p ,q� can be calculated by the recurrence for-ula

I�0,0� =1

Q�1 − exp�− Q��, �B4�

I�0,1� = �1 −

QI�0,0� +

Qexp�− Q�, �B5�

∀q � 2, I�0,q� = �2q − 1

q−

QI�0,q − 1�

+q − 1

q �

Q− 1I�0,q − 2� +

qQexp�− Q�Lq−1��,

�B6�

I�1,0� = �1 −�

QI�0,0� +�

Qexp�− Q�, �B7�

∀q � 1, I�1,q� = �1 −��q + 1�

Q �I�0,q� +�q

QI�0,q − 1�

+�

QLq��exp�− Q�, �B8�

∀p � 2, I�p,0� = �2p − 1

p−

�

QI�p − 1,0�

+p − 1

p � �

Q− 1I�p − 2,0� +

�

pQexp�− Q�Lp−1��,

�B9�

p � 2, ∀ q � 1, I�p,q� =1

p�T1�p,q� + T2�p,q� + T3�p,q�

+�

Qexp�− Q�Lp−1��Lq�� , �B10�

here

T1�p,q� = �2p − 1 −��p + q�

Q �I�p − 1,q�, �B11�

T2�p,q� = �p − 1��

Q− 1I�p − 2,q�, �B12�

T3�p,q� =�q

QI�p − 1,q − 1�. �B13�

Implementations of this formula on a host system mayecome unstable for high values of both p and q. This re-ults from the standard IEEE 754 that defines the inter-al representation of a floating-point number as −1S

M�2E where S is the sign (+1 or −1), M, is the man-issa and E is the exponent. Three formats are specified inhich the best-known representative is the double-recision one (64 bits=1 bit sign+11 bit exponent+52 bitantissa). One solution to solve this problem [22] consists

n using a quad-double arithmetic library such as [23]hat allows one to extend the accuracy of the double pre-ision format four times (not its range).

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Adaptive Laguerre-Gaussian variant of the Gaussian beam expansion method

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