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484 J. Opt. Soc. Am. A/Vol. 27, No. 3 /March 2010 Cagniot et al.

Modeling the propagation of apertured high-orderLaguerre–Gaussian beams by a user-friendly

version of the mode 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 Boulevard Maréchal Juin,F-14050 Caen, France

*Corresponding author: Emmanuel.Cagniot@ensicaen.fr

Received September 28, 2009; revised January 8, 2010; accepted January 8, 2010;posted January 11, 2010 (Doc. ID 117880); published February 23, 2010

The mode expansion method (MEM) models the propagation of an apertured beam by expressing the diffractedfield as a finite series of Laguerre–Gaussian or Hermite–Gaussian modes. An optimal expansion parameter set(beam waist of the modes and its location) reduces the number of modes, which is difficult to derive, especiallyfor high-order incident beams. We propose a user-friendly version of the MEM in which the expansion param-eter set and the suitable number of modes are simply deduced from the approximation of the apertured inci-dent beam. © 2010 Optical Society of America

OCIS codes: 050.0050, 050.1220, 050.1380.

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. INTRODUCTIONne way to derive analytical expressions for the Fresnel–irchhoff diffraction integral consists in using orthogonalolynomial families. Laguerre–Gaussian (circular geom-try) and Hermite–Gaussian (rectangular geometry)eam families refer to the mode expansion method (MEM)1–3], in which the diffracted field is synthesized as a su-erposition of expansion modes belonging to the sameamily as the incident beam.

Since the diffracted field is expressed as a finite seriesf modes, both their beam waist and its location directlyffect the truncation error. These parameters are referredo as the expansion parameter set. When the method ispplied to a laser resonator [4], the optimal parameteret, the suitable number of modes, and the resulting trun-ation error can be deduced from the geometric features ofhe cavity (length and mirrors) and the aperture radii [5].he problem is more difficult when the method is appliedo free-space propagation.

The case of a Gaussian beam diffracted through a hard-dge aperture has already been studied [1–3,6]. In thisase, the choice of the optimal parameter set is straight-orward, since it is the set that maximizes the powerransfer from the incident beam to the expansion mode ofhe same order. However, although an optimal parameteret reduces the number of modes, the difficulty is to de-ermine this number as well as the resulting truncationrror.

Few reports provide quantitative results for high-ordereams diffracted through hard-edge apertures [7,8], al-hough forcing a laser to oscillate on a single high-orderransverse mode is a current challenge in today’s researchorld [9,10]. Since, outside the resonator, the selectedode propagates through lenses or other optical systems

1084-7529/10/030484-8/$15.00 © 2

ith finite apertures [11], there is a need to extend theeld of the MEM to high-order beams in order to optimizeuch optical devices numerically.

We propose a user-friendly version of the MEM thatimply addresses the problem of the diffraction of high-rder Laguerre–Gaussian beams through hard-edge aper-ures by exploiting the only available information, i.e., thepertured incident beam. Note that since the incidenteam must be symmetric, this version cannot be appliedo nonsymmetric elegant Laguerre–Gaussian beamsEMpl that possess an orbital angular momentum.This paper is organized as follows. Section 2 presents

he MEM, while Section 3 relates to its restriction toard-edge apertures. Section 4 shows the expansion of anpertured high-order beam onto a specific set ofaguerre–Gaussian modes and the normalized meanquare error that measures its accuracy. Since the coeffi-ients of the expansion and the normalized mean squarerror can be quickly computed by a recurrence formula,n algorithm for determining the suitable number ofodes is derived. Section 5 lists numerical results ob-

ained for various incident beam orders. Finally, the con-lusion is given in Section 6. Appendix A contains the re-urrence formula that is the backbone of this work.

. MODE EXPANSION METHODet us consider a circular diffractive optical element

DOE) located at plane z=0. It is illuminated by a nor-ally incident p-order Laguerre–Gaussian beam TEMp0.he diffracted field is calculated at the measurementlane located at a distance zm from the DOE.The electric field associated with the incident TEMp0

eam is

010 Optical Society of America

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Cagniot et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. A 485

�p�r,z� =�2

�

1

w�z�exp�−

r2

w�z�2�Lp� 2r2

w�z�2��exp�− j

kr2

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� and its radius of curvature R�z�.hese quantities as well as the Gouy phase shift ��z� aredependent and obey the following formulas:

w�z� = w0�1 + � z − z0

zR2�1/2

, �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 U�r ,z� can be expressed as aeighted sum of a set of orthogonal functions �̄n:

U�r,z� = n=0

+�

Cn�z��̄n�r,z�. �5�

natural choice for �̄n is a slightly modified form of theaguerre–Gaussian modes [3],

�̄n�r,z� =�2

�

1

w̄�z�exp�−

r2

w̄�z�2�Ln� 2r2

w̄�z�2��exp�− j

kr2

2R̄�z��exp�j�2n + 1���̄�z��

�exp�− jk�z − z̄0��exp�− j��, �6�

here � is a constant depending on the order of the inci-ent TEMp0 beam,

� = k�z̄0 − z0� − �2p + 1���0�, �7�

nd where ��̄�z� is the change in the Gouy phase shiftrom the DOE plane to any plane z,

��̄�z� = �̄�z� − �̄�0�. �8�

The aim of the calculation is to determine the (r- and-independent [2]) expansion coefficients Cn,

Cn =�0

2��0

+�

U�r,z��̄n*�r,z�rdrd�

= 2��0

+�

U�r,z��̄n*�r,z�rdr, �9�

here � is the angular coordinate and where the symbol �

enotes the complex conjugate of the quantity.The diffracted field U�r ,z� is simply related to the inci-

ent TEMp0 beam at the DOE plane by

U�r,z = 0� = �r��p�r,z = 0�, �10�

here �r� is the transmittance function of the DOE.From Eq. (10) it is seen that expansion coefficients Cn

epend on the order of the incident TEMp0 beam, and con-equently they will be written in the following as Cpn.aking into account Eq. (10), we get

Cpn =2

w�0�w̄�0��0

+�

�X�Lp�X�Ln��X�exp�− QX�dX,

�11�

here X=r2 and where the parameters are

=2

w�0�2 , � =2

w̄�0�2,

Q =1

w�0�2 +1

w̄�0�2+

jk

2 � 1

R�0�−

1

R̄�0��. �12�

ote that if an ideal thin lens of focal length f is placeduite close behind the DOE then the expression of param-ter Q becomes

Q =1

w�0�2 +1

w̄�0�2+

jk

2 � 1

R�0�−

1

f−

1

R̄�0��. �13�

As a consequence, the diffracted field can be calculatedn any plane z beyond the DOE by using the truncated se-ies

U�r,z� � n=0

N−1

Cpn�̄n�r,z�, �14�

hose accuracy depends on the number N of modes �̄nsed in the expansion.

. CASE OF HARD-EDGE APERTURESntroducing the transmittance function of the hard-edgeperture

circ� r

a = 1 0 � r � a

0 r a � , �15�

q. (11) becomes

Cpn =2

w�0�w̄�0��0

�

Lp�X�Ln��X�exp�− QX�dX, �16�

here �=a2.The goal is to determine the optimal expansion param-

ter set (w̄0 and z̄0) so that the number of expansionodes is the smallest possible.In [6], the author discusses the coupling of power be-

ween incident high-order TEMmn beams and expansionodes at the plane of an infinitely large aperture (mode

onversion method). The fraction of power coupled be-ween an incident circular TEM beam into a circular

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486 J. Opt. Soc. Am. A/Vol. 27, No. 3 /March 2010 Cagniot et al.

EMp̄0 expansion mode is �Cpp̄�2. By expressing this cou-ling power coefficient as a function of �C00�2, it is thenossible to determine its maximum value.In the case of a Gaussian beam diffracted by a hard-

dge aperture, the choice of the optimal parameter set istraightforward: it is the set that maximizes the powerransfer from the incident beam to the expansion mode ofhe same order. This is a consequence of [6]: when an in-nitely large aperture is illuminated by a normally inci-ent Gaussian beam, the power of the diffracted field ishat of the incident beam. The more the aperture radiusecreases, the less power remains in the Gaussian expan-ion mode, and the more power is redistributed amonghe higher-order modes. As a consequence, most of theransmitted power is always associated with the expan-ion mode of the same order as the incident beam. In1,2], the authors determine the optimal parameter set by

aximizing �C00�2 as a function of both w̄0 and z̄0. In [3],aving pointed out that the Gouy phase shift of the ex-ansion modes depends on both w̄0 and z̄0, and therefore arong choice of either can introduce a phase error, the au-

hor maximizes C00 as a function of w̄0 only by letting

1

R̄�0�=

1

R�0�−

1

f�17�

f a thin lens is placed quite close behind the aperture and

1

R̄�0�=

1

R�0��18�

therwise. Note that Eqs. (17) and (18) imply that expan-ion coefficients Cpn are real numbers.

In [8], the author discusses a little about high-orderEMmn beams diffracted by hard-edge apertures. Unlikeaussian beams, an undisturbed transmission requires aeam truncation ratio �=a /w�0� so that � 2. Moreover,ince it depends on the radius of the aperture, the expan-ion mode of the same order as the incident beam is notecessarily predominant. As a consequence, the authoroncludes, “This only confirms overly general statementsbout the behavior of the coefficients should not be made.heir characteristics should be examined on a case-by-ase basis.”

Anyway, once the optimal parameter set is known, oneeeds to determine how many coefficients and the result-

ng truncation error.

. APPROXIMATING THE APERTUREDNCIDENT BEAMhis section presents the most significant result of the pa-er. First, by combining previous results, we expand thepertured incident beam onto a specific set of Laguerre–aussian modes, and the quality of the approximation iseasured by a normalized mean square error. Second,

ince both the coefficients of the expansion and the nor-alized mean square error can be quickly computed bysing a recurrence formula, we derive an algorithm foretermining automatically the suitable number of modesepending on the required accuracy. Hence, since the co-fficients of the expansion are independent of r and z, an

ccurate approximation of the apertured incident beamill ensure an adequate behavior in both the near fieldnd the far field while a coarse one will ensure a properehavior only in the far field.The benefit of our approach is that the expansion pa-

ameter set and the suitable number of expansion modesre simply deduced from the only available information,.e., the apertured incident beam.

. Laguerre–Gaussian Expansiont the aperture plane Eq. (10) is rewritten as

U�r� = circ� r

a�p�r�, �19�

here the axial coordinate z=0 has been omitted for theake of clarity.

In [12], having pointed out that any finite support �0,a�an be scaled to fit into the range [0,1] by using a suitablenit length, the authors state the following rule of thumb:When 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 to 1/�N,” where f is a circularlyymmetric function.

Scaling Eq. (19) requires multiplying all the quantitieswavelength �, waist w�0� and radius of curvature R�0� athe aperture plane, focal length f, and location of the mea-urement plane zm and radial coordinates r in this plane)y a factor to get the suitable unit length so that the ra-ius a of the aperture is exactly 1. In the following, we as-ume that this operation has been performed, and there-ore we restrict our study to the case a=1.

Following [12] (waist of the expansion modes at the ap-rture plane) and [3] (radius of curvature of the expansionodes at the aperture plane), the function U�r� can be ap-

roximated by a �̄n function series UN�r� truncated to theth term so that

U�r� � UN�r� = n=0

N−1

Cpn�̄n�r�, �20�

here w̄�0�=1/�N while R̄�0� obeys either Eq. (17) or Eq.18). As a consequence, expansion coefficients Cpn definedy Eq. (16) are real numbers.Approximation UN�r� minimizes the normalized mean

quare error [12]

�N =�U − UN�2

�U�2 , �21�

here

�U�2 =�0

2��0

+�

�U�r��2rdrd� = 2��0

+�

�U�r��2rdr

= 2��0

1

��p�r��2rdr. �22�

y use of the Bessel equality, Eq. (21) is rewritten as

�N = 1 −1

�U�2 n=0

N−1

Cpn2 . �23�

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Cagniot et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. A 487

. Algorithmntroducing Eq. (A2), then Eq. (16) becomes

Cpn =2

w�0�w̄�0�I�p,n�. �24�

oreover, since Eq. (22) can be rewritten as

�U�2 = �0

1

Lp�X�Lp�X�exp�− X�dX, �25�

here X=r2 and =2/w�0�2, we have

�U�2 = I�p,p�, �26�

here =�=Q. As a consequence, the coefficients of thexpansion and the normalized mean square error can beuickly computed by using our recurrence formula.ence, it is then possible to derive an algorithm to deter-ine the suitable number of expansion modes N, depend-

ng on the accuracy � required for the approximation ofhe apertured incident beam.

The first part of this algorithm computes the rangea ,b� so that �a����b, implying that N� �a ,b�. At the be-inning, the lower bound a is set to 1, while the upperound b is set to an arbitrary number of expansion modes32 in our case). If ai and bi represent the bounds of theange �a ,b� at the ith iteration, then corrections ai+1=bind bi+1=2bi are applied until �a����b.The second part of the algorithm applies a dichotomic

earch to find the number of expansion modes in theange �a ,b� whose corresponding normalized meanquare error is the closest to �.

. NUMERICAL RESULTShe host system chosen to perform computations is aewlett Packard HDX X18-1250EF Premium Notebookersonal computer [processor Intel 64-bit Core 2 Duo8600 2.40 GHz, 4 Gbytes of DDR3 memory �1 GHz�,Mbytes of L2 cache} running Linux version 2.6.30.5 (Fe-

ora distribution, “Leonidas”). The application, which isritten in C��, makes intensive use of inheritance, poly-orphism, and templates. It requires the GNU Scientificibrary 1.12 (numerical integrators and special math-matical functions), and it is compiled with the GNU g��ompiler 4.4.1 using the following optimizations:funroll-loops (loop unrolling), -march=native (in-truction set of the host system processor only), -msse2msse3 -mssse3 -msse4 (all available streaming in-truction sets for double-precision floating point numbers)nd -03 (level 3 optimizations).We consider a TEMp0 beam that is normally incident

pon a hard-edge aperture whose radius is chosen to be=��w�0�, where � is the beam truncation ratio. Theeasurement plane is located at zm. Since we consider

nly small incident beam orders p, our variant, calledyMEM, uses the common double-precision arithmetic ofhe host system. Its results are compared with those pro-uced by QAG, an optimized implementation of theresnel-Kirchhoff diffraction integral based on the adap-ive numerical integrator QAG [13] using a 61-pointauss–Kronrod rule: its absolute and relative errors are

espectively set to 10−10 and 10−8, while its maximumumber of subintervals is set to 1024. If I1�i� and I2�i� arehe intensities computed, respectively, by QAG and myMEMt the ith radial coordinate r of the measurement plane,hen the approximation error in percent is defined as

err =i

�I1�i� − I2�i��

iI1�i�

� 100. �27�

Intensity distributions are sampled onto large quanti-ies of points so that computational times exceed 1 s.

. Gaussian Beamn this test case, inspired from [2,3], we consider a Gauss-an beam whose wavelength is �=850 nm while its waistnd its radius of curvature at the aperture plane are, re-pectively, w�0�=0.0752 mm and R�0�=0.835 mm. Theeam truncation ratio and the accuracy required for thepproximation at the aperture plane are, respectively, �10% and �=1%. The multiplication factor to get the suit-ble unit length is 133.The radial intensity distribution of the diffracted field

s measured at various coordinates zm in the near-fieldnd the far-field areas that are characterized by theirresnel number

Nf = � 1

R�0�+

1

zm�a2

�. �28�

he near field corresponds to Nf 1, while the far fieldorresponds to Nf�1. Intensity distributions are sampledn 1024�1024 equidistant points in the range0,ub� mm, where ub depends on the Fresnel number todapt to the changing feature scale of the diffracted field.The suitable number of expansion modes chosen byyMEM is N=31, while the corresponding normalizedean square error is 1.03%. Figure 1 plots the incident

eam and its approximation at the aperture plane.Table 1 presents results obtained for this test case. The

olumns represent the Fresnel number Nf and its corre-ponding coordinate zm, the upper bound ub, the compu-

0

20

40

60

80

100

120

140

160

-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)

Apertured Gaussian beamApproximation

ig. 1. Gaussian beam and its approximation using 31 expan-ion modes at the aperture plane. The beam truncation ratio andhe accuracy required for the approximation are, respectively, �10% and �=1%.

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Fma=

f

Fm

488 J. Opt. Soc. Am. A/Vol. 27, No. 3 /March 2010 Cagniot et al.

ational times required by QAG and myMEM, and, finally,he approximation error err at zm.

These results confirm our basic idea: an accurate ap-roximation of the apertured incident beam ensures ad-quate behavior in the near field. Moreover, one can seehat an accurate approximation ��=1%� does not neces-arily imply a large number of expansion modes �N=31�.s a consequence, myMEM is characterized by both a smallpproximation error (up to Nf=7, where err=4.08%) andconstant and low computational time versus QAG, whose

omputational time increases with the Fresnel number.his result is satisfactory, all the more so because myMEMeeds to determine the suitable number of expansionodes before computing the radial intensity distribution

f the diffracted field, while QAG needs only to computehis last one. Figure 2 plots the radial intensity distribu-ions computed by QAG and myMEM for Nf=7.

If the accuracy required for the approximation at theperture plane is reset to �=0.1%, then the suitable num-er of expansion modes chosen by myMem is N=315, while

Table 1. Radial Intensity DistributionsCorresponding to an Apertured Gaussian Beam

Expanded onto 31 Modesa

QAG myMEM

Nf zm ��m� ub (mm) t (s) t (s) err (%)

11 6.09 0.01 141 2 11.610 6.71 0.01 126 2 9.039 7.46 0.01 107 3 9.098 8.4 0.01 85 2 6.427 9.61 0.01 61 3 4.086 11.2 0.01 61 2 3.285 13.5 0.01 59 2 1.524 17 0.015 61 2 1.213 22.8 0.015 57 3 0.8122 34.6 0.015 55 3 0.3161 72.3 0.02 48 2 0.1540.5 158 0.02 36 2 0.2040.1 3270 0.5 37 2 0.253

aComputed by QAG and myMEM at various coordinates in the near-field and thear-field areas.

0

50

100

150

200

250

300

350

400

450

-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)

QAGmyMEM

ig. 2. Radial intensity distributions computed by QAG andyMEM at N =7 for an incident Gaussian beam.

f

he corresponding normalized mean square error is ex-ctly �. The approximation error falls to err=0.0516% forf=11, while the computational time is 24 s.

. Collimated TEM10 Beame consider a collimated TEM10 beam whose wavelength

s �=1064 �m while its waist at the aperture plane is�0�=0.4 mm. The beam truncation ratio is chosen to be=120% so that the radius of the aperture coincides ap-roximately with the bright concentric ring of the inci-ent beam. The measurement plane is located at variousoordinates in the near-field and the far-field areas. Theccuracy required for the approximation at the aperturelane is �=1%, while the multiplication factor to get theuitable unit length is 2.08. Radial intensity distributionsre sampled on 256�1024 equidistant points.The suitable number of expansion modes chosen byyMEM is N=37, while the corresponding normalizedean square error is �=1.03%. Figure 3 plots the incident

eam and its approximation at the aperture plane.Table 2 presents results obtained for this test case. One

an see the same behavior as for the previous test case: anccurate approximation of the apertured incident beam�=1%� requiring a small number of expansion modesN=37� ensures proper behavior in the near field (up to

f=9, where err=3.19%). Figure 4 plots the intensity dis-ributions corresponding to Nf=9.

. Collimated TEM30 Beam with a Thin Lense consider a collimated TEM30 beam diffracted by a

ard-edge aperture set against a thin lens. Its wavelengthnd waist at the aperture plane are, respectively, �1064 �m and w�0�=0.4 mm, while the lens focal length

s f=50 mm. The beam truncation ratio is chosen to be �140% so that the radius of the aperture coincides ap-roximately with the second bright concentric ring of thencident beam. The measurement plane is located at theens focal plane. The accuracy required for the approxi-

ation at the aperture plane is �=1%, while the multipli-ation factor to get the suitable unit length is 1.79. Radialntensity distributions are sampled on 256�1024 equidis-ant points in the range �0,0.4� mm.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Inte

nsity

(a.u

.)

r (mm)

Apertured TEM10 beamApproximation

ig. 3. TEM10 beam and its approximation using 37 expansionodes at the aperture plane. The beam truncation ratio and the

ccuracy required for the approximation are, respectively �120%, and �=1%.

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Cagniot et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. A 489

The suitable number of expansion modes chosen byyMEM is N=35, while the corresponding normalizedean square error is �=1.02%. Figure 5 plots the incident

eam and its approximation at the aperture plane.QAG performs the computation in 25 s, while myMEM re-

uires 1 s. The approximation error is err=0.826%. Fig-re 6 plots the corresponding intensity distributions.rom this result, one can see the same behavior as for therevious test cases.

. CONCLUSIONhe mode expansion method (MEM) is a powerful math-matical tool for modeling the propagation of an aper-ured beam by expressing the diffracted field as a super-osition of expansion modes belonging to the same familys the incident beam. Since the diffracted field is ex-anded as a finite series of modes, their waist and their

Table 2. Radial Intensity DistributionsCorresponding to an Apertured TEM10 Beam

Expanded onto 37 Modesa

QAG myMEM

Nf zm (mm) ub (mm) t (s) t (s) err (%)

11 19.7 0.7 42 1 7.0510 21.7 0.7 40 1 5.49 24.1 0.7 35 1 3.198 27.1 0.7 31 1 2.587 30.9 0.7 25 0 2.146 36.1 0.7 18 1 1.055 43.3 0.7 15 1 0.7524 54.1 0.8 15 1 0.5693 72.2 0.8 14 1 0.2462 108 0.8 13 1 0.09771 217 1.2 12 1 0.1250.5 433 3 12 1 0.1050.1 2170 12 11 1 0.112

aComputed by QAG and myMEM at various coordinates in the near field and the fareld areas.

0

1

2

3

4

5

6

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Inte

nsity

(a.u

.)

r (mm)

QAGmyMEM

ig. 4. Radial intensity distributions computed by QAG andyMEM at N =9 for an incident TEM beam.

f 10

adius of curvature at the aperture plane (parameters re-erred to as the expansion parameter set) directly affecthe truncation error.

When modeling a laser resonator, the optimal set, theuitable number of modes and the resulting truncation er-or can be deduced from the features of the cavity (length,irrors and aperture radii). The case of the free-space

ropagation is more difficult to handle. When the incidenteam is Gaussian, the optimal set is the one which maxi-izes the power transfer from the incident beam to the

xpansion mode of the same order. However, this reason-ng is no longer valid when applied to higher order beamsince the expansion mode of the same order as the inci-ent beam is not necessarily predominant. Once the opti-al set is known, the most difficult task consists in deter-ining the suitable number of modes and the

orresponding truncation error.Today, because one prefers to force a laser to oscillate in

single high-order mode that propagates, outside the cav-ty, through lenses or other optical systems with finite ap-rtures, there is a need to extend the field of the MEM toigh-order beams.

0

0.5

1

1.5

2

2.5

3

3.5

4

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Inte

nsity

(a.u

.)

r (mm)

Apertured TEM30 beamApproximation

ig. 5. TEM30 beam and its approximation using 35 expansionodes at the aperture plane. The beam truncation ratio and the

ccuracy required for the approximation are, respectively, �140% and �=1%.

0

2

4

6

8

10

12

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Inte

nsity

(a.u

.)

r (mm)

QAGmyMEM

ig. 6. Radial intensity distributions computed by QAG andyMEM at the focal plane of a lens for an incident TEM beam.

30

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ATbvss[

w

w

w

g

wt

w

tindnMsbMObrt

490 J. Opt. Soc. Am. A/Vol. 27, No. 3 /March 2010 Cagniot et al.

In this paper we have proposed a user-friendly versionf the MEM that simply addresses the challenge of theiffraction of high-order Laguerre–Gaussian beamshrough hard-edge apertures. In this one, the aperturedncident beam is expanded onto a specific set of Laguerre–aussian modes with the waist at the aperture plane de-ending on the number of modes, while the radius of cur-ature is that of the incident beam. The accuracy of thepproximation is measured by a normalized mean squarerror. Both the expansion coefficients and the normalizedean square error are quickly computed by using a recur-

ence formula. These quick computations allow derivingn algorithm for automatically determining the suitableumber of expansion modes depending on the requiredccuracy. As a consequence, since the coefficients of ex-ansion are independent of both radial and axial coordi-ates, an accurate approximation of the apertured beamnsures adequate behavior in both the near field and thear field, while a coarse one ensures proper behavior onlyn the far field. Numerical results obtained for various in-ident beam orders confirm this assumption. The mainenefit of our approach is that we exploit the only avail-ble information, i.e., the apertured incident beam. Notehat since the incident beam must be symmetric, our ap-roach cannot be extended to nonsymmetric elegantaguerre–Gaussian beams TEMpl that possess an orbitalngular momentum.

PPENDIX A: RECURRENCE FORMULAhis recurrence formula was originally proposed as aackbone for an alternative model based on the mode con-ersion method for diffraction through apertured ABCDystems [14]. It was then applied to optimize a specific la-er resonator by using the simulated annealing algorithm15].

Let us consider the finite integral

I��p,q� =�0

�

Lp��x�Lq��x�exp�− Rx�dx, �A1�

here � 0, � 0, and Re�R� 0.Once the variable change y=x /� is done, Eq. (A1) is re-

ritten as

I��p,q� = ��0

1

Lp�y�Lq��y�exp�− Qy�dy = �I�p,q�,

�A2�

here =��, �=��, and Q=R�.By making use of the recursive definition of the La-

uerre polynomials

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

�A3�

e derive the following recurrence formula to calculatehe integral I�p ,q�:

I�0,0� =1

Q�1 − exp�− Q��, �A4�

I�0,1� = �1 −�

QI�0,0� +�

Qexp�− Q�, �A5�

∀q � 2, I�0,q� = �2q − 1

q−

�

QI�0,q − 1�

+q − 1

q � �

Q− 1I�0,q − 2� +

�

qQexp�− Q�Lq−1���,

�A6�

I�1,0� = �1 −

QI�0,0� +

Qexp�− Q�, �A7�

∀q � 1, I�1,q� = �1 −�q + 1�

Q �I�0,q� +q

QI�0,q − 1�

+

QLq���exp�− Q�, �A8�

∀p � 2, I�p,0� = �2p − 1

p−

QI�p − 1,0�

+p − 1

p �

Q− 1I�p − 2,0� +

pQexp�− Q�Lp−1��,

�A9�

∀p � 2, ∀ q � 1, I�p,q� =1

p�T1�p,q� + T2�p,q�

+ T3�p,q� +

Qexp�− Q�Lp−1��Lq���� , �A10�

here

T1�p,q� = �2p − 1 −�p + q�

Q �I�p − 1,q�, �A11�

T2�p,q� = �p − 1��

Q− 1I�p − 2,q�, �A12�

T3�p,q� =q

QI�p − 1,q − 1�. �A13�

A recurrence formula is not unstable, but its implemen-ation onto a host system can be associated with instabil-ty. In our case, high values of both p and q may induceumerical instability due to the standard IEEE 754 thatefines the internal representation of a floating-pointumber as −1S�M�2E, where S is the sign (+1 or −1),

is the mantissa, and E is the exponent. Of the threepecified formats, the best known is double precision (64its � 1 bit sign � 11 bit exponent � 52 bit mantissa).any solutions make it possible to solve this problem.ne of them consists in using a quad-double arithmetic li-rary such as QD [16] allowing the accuracy (not theange) of the double-precision format to be extended fourimes. This solution is applied in [15].

R

1

1

1

1

1

1

1

Cagniot et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. A 491

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