hankel–bessel laser beams

Hankel–Bessel laser beams

Victor V. Kotlyar,1 Alexey A. Kovalev,1,* and Victor A. Soifer2

1Laser Measurements Laboratory, Image Processing Systems Institute of the Russian Academy of Sciences,151 Molodogvardejskaya Street, Samara 443001, Russia

2Technical Cybernetics Sub-department, S. P. Korolyov Samara State Aerospace University,34 Moskovskoye Shosse, Samara 443086, Russia

*Corresponding author: [email protected]

Received October 26, 2011; accepted December 19, 2011;posted December 23, 2011 (Doc. ID 157152); published April 19, 2012

An analytical solution of the scalar Helmholtz equation to describe the propagation of a laser light beam in thepositive direction of the optical axis is derived. The complex amplitude of such a beam is found to be in directproportion to the product of two linearly independent solutions of Kummer’s differential equation. Relationshipsfor a particular case of such beams—namely, the Hankel–Bessel (HB) beams—are deduced. The focusing of the HBbeams is studied. © 2012 Optical Society of America

OCIS codes: 050.1960, 050.4865.

1. INTRODUCTIONIn optics, there are laser beams whose scalar complex ampli-tudes are described by the exact solutions of the nonparaxialHelmholtz equation. These include well-known planar andspherical waves [1] as well as more recently proposed Besselmodes [2], Mathieu beams [3], and parabolic laser beams [4].There also exist paraxial hypergeometric laser modes [5],hypergeometric (HyG) laser beams [6,7], and their generaliza-tions (circular beams) [8] with their complex amplitude beingdescribed by the Kummer function. The nonparaxial hyper-geometric (nHyG) laser beams [9] were derived as a solutionof the Helmholtz equation by calculating the integral for theangular spectrum of plane waves with the even numbers of thetopological charge of the beam spiral phase. The nHyG beamsare described by a complex amplitude in the form of a productof two Kummer functions 1F1�a; b; x� [10] with different argu-ments x. They represent a superposition of two identical lightwaves propagating in the positive and negative directionsalong the optical axis z. Note that approximate relations forthe complex amplitude describing the propagation of the lightwaves in the positive and negative directions (the HyG laserbeams) were also derived in [9] using the asymptotic decom-position of one of the Kummer functions.

In this paper, extending the results reported in [9], we de-rive an exact solution of the Helmholtz equation for any inte-ger topological charge to describe a nonparaxial laser beamwith the complex amplitude in the form of a product of Kum-mer function 1F1�a; b; x1� and the second solution of Kummerequation U�a; b; x2�. Considering that, under certain param-eters, the U�a; b; x2� function is proportional to the Hankelfunction and Kummer function to the Bessel function, suchbeams have been given the name nonparaxial Hankel–Bessel(HB) beams. It is noteworthy that the solution proposed canbe found neither in the handbooks of mathematical functions[10,11] nor in the well-known work by Miller [12].

The HB beams (n � 0) are generated by a source withinfinite energy density located in the initial plane. As theypropagate along the positive z axis, the divergence of the

beams is proportional to��z

p. For a nonzero topological

charge, n ≠ 0, the source produces no radiation along theoptical axis.

2. SOLUTION OF THE HELMHOLTZEQUATION IN THE PARABOLICCOORDINATESThe Helmholtz equation in the cylindrical coordinates �r;φ; z�takes the form

�∂2

∂r2� 1

r∂

∂r� 1

r2∂2

∂φ2 �∂2

∂z2� k2

�E�r;φ; z� � 0; (1)

where k is the wavenumber.We will seek the solution E�r;φ; z� in the following form:

E�r;φ; z� � E�r; z�rjpj exp�inφ� ikz�; (2)

where n and p are integer numbers (we use jpj to avoidamplitude singularities).

Then, for E�r; z�, Eq. (1) is

∂2E

∂r2� ∂2E

∂z2�

�2jpj � 1

r

�∂E∂r

� 2ik∂E∂z

��p2 − n2

r2

�E � 0: (3)

At n � �jpj, the third term in Eq. (3) is eliminated.Changing to the parabolic coordinates

�u �

��r2 � z2

p� z;

v ��r2 � z2

p− z;

�4�

Eq. (3) takes the form (at n � �jpj)

Kotlyar et al. Vol. 29, No. 5 / May 2012 / J. Opt. Soc. Am. A 741

1084-7529/12/050741-07$15.00/0 © 2012 Optical Society of America

u∂2E

∂u2 � v∂2E

∂v2��n� 1� iku�∂E

∂u��n� 1− ikv�∂E

∂v� 0: (5)

All further equations are valid also for n � −jpj but with nreplaced by jnj.

Equation (5) is solved in separable variables. Assuming inEq. (5) that E�u; v� � P�u�Q�v�, we obtain

�uP00

P��n�1� iku�P

0

P

��−v

Q00

Q− �n�1− ikv�Q

0

Q

��C; (6)

where C is a constant independent of u and v. Then, Eq. (5) isreplaced by two equations:

(u d2P

du2 � �n� 1� iku� dPdu − CP � 0;

v d2Qdv2

� �n� 1 − ikv� dQdv � CQ � 0:�7�

Introducing the designations ξ � −iku, η � ikv, andC � −ikD, both equations in Eq. (7) are transformed intoKummer equations [10, Eq. 13.1.1]:

(ξ d2Pdξ2 � �n� 1 − ξ� dPdξ − DP � 0;

η d2Qdη2 � �n� 1 − η� dQdη − DQ � 0;

�8�

with their solutions described by Kummer functions

1F1�a; b; z� [10]:�P�ξ� � 1F1�D; n� 1; ξ�;Q�η� � 1F1�D; n� 1; η�: �9�

Then, P�u� � 1F1�D; n� 1;−iku� and Q�v� �1F1�D; n� 1; ikv�, and the solution of the original Helmholtzequation Eq. (1) can now be written down as

E�r;φ; z� � A0�kr�n exp�inφ� ikz�1F1�D; n� 1;−iku�1F1�D; n� 1; ikv�; (10)

where A0 is a constant characterizing the beam power.The said solution was offered without derivation in [12]. In

particular, when D � �n� 1�∕2, expressing the Bessel func-tions through Kummer functions [10],

1F1

�v� 1

2;2v� 1;2iz

�� Γ�1� v�exp�iz�

�z2

�−vJv�z�; (11)

Eq. (10) can be rearranged as

E�r;φ; z� � A0Γ2

�1� n

2

�4n exp�inφ�Jn

2

�k2

�z�

��r2 � z2

p ��

× Jn2

�k2

� ��r2 � z2

p− z

��: (12)

Note that Eq. (10) does not contain multiplier exp�ikz�, so itdoes not describe the propagation of laser light in a certaindirection. What it describes is a standing wave resulting from

the interference of two identical waves propagating in thepositive and negative directions along the optical axis z.

Actually, for large distances z ≫ r, using the approximaterelations for the Bessel function for small and large values ofthe argument, the approximate expression of Eq. (12) given by

E�r;φ; z� � A0Γ�1� n

2

�exp�inφ�

��2

πkz

r �2kr2

z

�n2

× cos�kz −

πn4

−π4

�(13)

suggests that the amplitude is proportional to the cosinefunction of the distance z.

Let us separate the light field in Eq. (10) into the forwardand backward waves. In doing so, we make use the relationbetween Kummer function M�a; b; z� � 1F1�a; b; z� and thesecond solution of Kummer equation [10, Eq. 13.1.3]

U�a;b;z�� πsin πb

�M�a;b;z�

Γ�1�a−b�Γ�b�−z1−bM�1�a−b;2−b;z�

Γ�a�Γ�2−b�

�:

(14)

Note that the function Eq. (14) is also defined for the integervalues of the parameter b [10].

The inversion of Eq. (14) yields the expression of

1F1�a; b; z� through U�a; b; z�:

1F1�a; b; z� � AU�a; b; z� � BU�b − a; b;−z�; (15)

where

A � Γ�b�Γ�b − a�T;

B � Γ�b�Γ�a� �−1�

bezT;

T � sin πbsin π�b − a� � �−1�b sin πa : (16)

Making use of Eqs. (15) and (16) and neglecting the con-stant factors, the solution of the Helmholtz equation Eq. (10)takes the form

E�r;φ; z� � A0�kr�n exp�inφ��

exp�ikz�Γ�n� 1 − D�U�D; n

� 1;−iku�1F1�D; n� 1; ikv�

� �−1�n�1 exp�−ikz�Γ�D� U�n� 1 − D; n

� 1; iku�1F1�n� 1 − D; n� 1;−ikv��: (17)

Designating D � �n� β� 1�∕2 and multiplying both sides ofthe equation by Γ�D�Γ�n� 1 − D�, we obtain

742 J. Opt. Soc. Am. A / Vol. 29, No. 5 / May 2012 Kotlyar et al.

E�r;φ; z� � i2n�1A0�kr�n exp�inφ� ×�−iΓ

�n� β� 1

2

�exp�ikz�U

�n� β� 1

2; n� 1;−iku

�1F1

�n� β� 1

2; n� 1; ikv

�

� iΓ�n − β� 1

2

�exp�−ikz�U

�n − β� 1

2; n� 1; iku

�1F1

�n − β� 1

2; n� 1;−ikv

��: (18)

The above relationship represents a sum of the forward andbackward HyG beams as indicated by the exponent signsexp��ikz� in Eq. (18). Below, we discuss the forward beamspropagating from the plane z � 0 into the semispace z > 0.The positive beam is expressed as

E��r;φ; z� � �−1�nΓ�n� β� 1

2

�A0�kr�n exp�inφ� ikz�

× U�n� β� 1

2; n� 1;−iku

�

× 1F1

�n� β� 1

2; n� 1; ikv

�: (19)

For n ≠ 0, the field complex amplitude on the optical axis(r � 0) equals zero, whereas at n � 0 the amplitude isnonzero:

E0�r � 0;φ; z� � Γ�β� 1

2

�A0U

�β� 12

; 1;−2ikz�exp�ikz�:

(20)

In particular, for β � 0, Eq. (19) that describes the positivebeam takes the form

E��r;φ; z� � i3n�1 π2n!A0 exp�inφ�H�1�

n2

�k2

�z�

��r2 � z2

p ��

× Jn2

�k2

� ��r2 � z2

p− z

��: (21)

In the initial plane (z � 0), the arguments of both the Hankeland Bessel functions equal kr∕2. The Hankel function is diver-ging as the argument tends to zero. Using the approximate re-lations for the Hankel and Bessel functions for small argumentvalues {see [10], the Relationships (9.1.7) and (9.1.9)}, the fieldamplitude can be shown to have a finite value at n ≠ 0 andr � z � 0:

E��r ≈ 0;φ; z � 0� � �−i�n�n − 1�!A0 exp�inφ�: (22)

From Eq. (21), it is seen that, if β � 0, the complex ampli-tude of the nonparaxial HyG beam is proportional to theproduct of the Hankel and Bessel functions of integer andhalf-integer orders. Thus, to distinguish the beams in Eqs. (18)and (21) from those reported in [9], the former will be referredto as HB beams. From Eq. (21), it can be inferred that the HBbeam is not a free-space mode because the arguments of theHankel and Bessel functions depend in a different way on thevariables r and z, and, therefore, given the same z, thesefunctions’ values remain constant at different r. That the beam

in Eq. (21) propagates in the positive direction of the z axisfollows from the asymptotic formula of the first-order Hankelfunction at large z (kz ≫ 1) [10, Relationship 9.2.3]:

H�1�v �z� ∼

��2πz

rexp

�i�z −

πv2−π4

��: (23)

As distinct from the relationship Eq. (13), the z dependencein Eq. (23) is exponential rather than cosine.

When n � 0, the intensity on the optical axis (r � 0) is

I�r � 0;φ; z� ��i π2A0H

�1�0 �kz�

��2 � π24A20�J2

0�kz� � Y 20�kz��;

(24)

with the intensity peak attained at a point z that satisfies theequation

J0�kz�J1�kz� � Y 0�kz�Y 1�kz� � 0: (25)

The simulation has shown that, while at t > 0, the functionf �t� � J0�t�J1�t� � Y 0�t�Y 1�t� never attains the zero value, thebeam is not focused, and the intensity of the diverging beam isdecreased along the optical axis. Moreover, with the functionin Eq. (24) becoming infinite at z � 0, the field in Eq. (21) canbe assumed to be produced by a source of infinite energy den-sity located at the origin of coordinates. Shown in Fig. 1 arethe squared modulus of the function Eq. (21) for z > 0 in theOrz plane [Fig. 1(a)], the intensity plots on the longitudinal zaxis [Fig. 1(b)], and the transverse r axis [Fig. 1(c)]. FromEq. (21), the HB beams are seen to have infinite energy.

It can be seen from Fig. 1(c) that, despite the infinite valueat r � z � 0, Eq. (21) has a strict zero. From Eq. (21) followsthe zero value at r0 ≈ 0.76λ. Thus, we can assume that the fieldof Eq. (21) is generated by a source of infinite energy densityand radius 0.76λ (n � 0).

Let us consider another particular case when β � 1. Then,the positive beam will be expressed as

E��r;φ; z� � �−i�n�n − 1�! πA0k1−n

4exp�inφ�r

×�Jn−1

2

�kv2

�� iJn�1

2

�kv2

��

×�iH�1�

n−12

�ku2

��H�1�

n�12

�ku2

��. (26)

From Eq. (23), it is also seen that the beam in Eq. (26) alsopropagates in the positive direction of the z axis.


When n � 0, we obtain a trivial solution of the Helmholtzequation, E��r;φ; z� � 0, because H�1�

1∕2�u� � iH�1�−1∕2�u�≡ 0.

3. PECULIARITIES OF THE HB BEAMSFor large z, the HB beam is diverging hyperbolically, as evi-dent from the expression for the field amplitude at z ≫ λ:

E��r;ϕ; z� ≈ A0 exp�inϕ�H�1�n∕2�kz�Jn∕2

�kr2

2z

�: (27)

Equation (27) suggests that the field amplitude remains un-changed if the coordinates r and z are linked by the relation:

r ��2γzk

r; (28)

where γ is a constant independent of z. The Hankel function inEq. (21) is diverging when the argument tends to zero, whichis only possible when r and z are simultaneously small. In thiscase, the argument of the Bessel function in Eq. (21) alsotends to zero. Making use of the approximate relations of theHankel and Bessel functions for small argument values, wecan show that

E��r;φ; z� ≈ �−i�n�n − 1�!A0 exp�inφ�tann�ξ2

�; (29)

where tan ξ � r∕z. Thus, the amplitude of a point located nearthe origin of coordinates depends on the direction from theorigin of coordinates toward this point (Fig. 2). Figure 2(a)shows the squared modulus of the function Eq. (21) (atn � 1, A0 � 100) in a 2λ × λ region (white denotes zero, black

denotes maximum). Shown in Figs. 2(b), 2(c), and 2(d) are theradial intensity profiles in the planes z � 0.1λ, z � 0.01λ,and z � 0.001λ.

4. PARTICULAR CASES OF THE HB BEAMSFor odd n, the Bessel functions of half-integer order becomeelementary. Let us consider a particular case of n � 1. Then,

E1�r;φ; z� �−2ikr

A0 exp�iφ� sin�k2

� ��r2 � z2

p− z

��

× exp�ik2

�z�

��r2 � z2

p ��: (30)

At r � 0, there will be zero intensity on the optical axis,despite the fact that r occurs in the denominator of Eq. (30).This can be demonstrated, if we assume r ≪ z:

limr→0

E1�r;φ;z��−2ik

A0exp�iφ�ikz�limr→0

�1rsin

�kr2

4z

��0: (31)

The field in Eq. (30) is formed from the following amplitudedistribution in the initial plane (waist):

E1�r;φ; z � 0� � −2ikr

A0 sin�kr2

�exp

�ikr2

� iφ�: (32)

The beam waist diameter is FWHM � 0.93λ [Fig. 2(d)].Note that being equal to En�1�r ≈ 0;φ; z � 0� � −iA0

exp�iφ� near the original plane center, the amplitude is notdefined at the central point itself due to a spiral phase singu-larity; on the other hand, at r � z � 0, the intensity is notequal to zero, I � jE1�r � 0;φ; z � 0�j2 � jA0j2.

1,0z/

r/

0.0

0.2

0.4

0.6

0.81.0

-1.01.0

0.00.4

0.8

I

0 2 4 6 8 100

25

50

z/

I

0 1 2 3 4 50

25

50

r/

I

(a)

(b) (c)

Fig. 1. (a) Squared modulus of the function Eq. (21) in the Orz planefor n � 0 and A0 � 100; the intensity as a function of (b) the longitu-dinal axis at r � 0 and (c) the radial axis at z � 0.

r/

z/

-1 -0.5 0

(a)

(b) (d)(c)

0.5 10

0.5

1

-1 -0.5 0 0.5 10

50 I

r/

-1 -0.5 0 0.5 10

100

r/

I

-1 -0.5 0 0.5 10

100

r/

I

FWHM = 0.93

Fig. 2. (a) Intensity in the Orz plane at n � 1, A0 � 100 (−λ ≤ x ≤ λ,0≤z ≤ λ; white denotes zero, black denotes maximum intensity).(b) Radial sections of the squared modulus of the function Eq. (21)in different planes: (b) z � 0.1λ, (c) z � 0.01λ, and (d) z � 0.001λ.


At large distances z ≫ r, Eq. (30) takes the form

E1�r;φ; z ≫ r� � −2ikr

A0 sin�kr2

4z

�exp�iφ� ikz�: (33)

5. SIMULATION RESULTSFigure 3 presents the simulation results for the field of Eq. (30)for wavelength λ � 633 nm: the intensity and phase are shownin the transverse plane z � 2λ.

Figure 4 depicts the radial intensity profiles in the planesz � 2λ, z � 4λ, and z � 6λ.

For comparison, the simulation using the finite-differencetime-domain (FDTD) method (Fig. 5) was also conducted.The simulation parameters were as follows: the computationregion dimension, �−8λ; 8λ� × �−8λ; 8λ� × �0; 8λ�; spatial discreti-zation, λ∕16 (for all coordinates); the simulation time, 20 per-iods (i.e., 20λ∕c, where c is the speed of light in vacuum); thetemporal discretization, λ∕�32c�. The time-averaged intensityin the plane z � 2λ is shown in Fig. 5(a), and its radial profileis depicted in Figs. 5(b) and 5(c). The central intensity is seennot to fall toward zero, which may be due to the presence ofthe longitudinal (axial) component Ez (because n � 1). It canalso be seen that the intensities shown in Figs. 4(a) and 5(c)are similar in structure, whereas the intensity in Fig. 5(b) isdifferent, not dropping to zero between the bright rings. Thismay be due to the contribution of the axial component and thelinear polarization, which is regarded when simulating in theFDTD method and disregarded in the scalar theory.

6. FOCUSING OF THE HB BEAMSBy substitution of variables in Eq. (21), z → f − z (f is the focallength) we obtain

E��r;φ; z� � i3n�1 π2n!A0 exp�inφ�

×H�1�n2

�k2

�f − z�

��r2 � �z − f �2

q ��

× Jn2

�k2

� ��r2 � �z − f �2

q− f � z

��: (34)

When n � 0,

x/

y/

-8 -4 0

(a) (b)4 8

-8

-4

0

4

8

-8 -4 0 4 8-8

-4

0

4

8

x/

y/

Fig. 3. Simulation results for the beam Eq. (30) for wavelengthλ � 633 nm: (a) intensity and (b) phase in the transverse plane z � 2λ.

-8 -6 -4 -2 0

(a) (b) (c)2 4 6 8

0

5

r/

I

r/

I

-8 -6 -4 -2 0 2 4 6 80

5

-8 -6 -4 -2 0 2 4 6 80

5

r/

I

Fig. 4. Intensity profiles in the planes z � 2λ, z � 4λ, and z � 6λderived using Eq. (21) at n � 1, A0 � 100.

4r/

I

-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5

-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5

r/

I

(a) (b) (c)

Fig. 5. Simulation of the propagation of the HB beam for n � 1 usingthe FDTD method (TE polarization, Ex ≠ 0): (a) time-averaged inten-sity in the plane z � 2λ, (b) its horizontal and (c) vertical profiles.

x/

z/

-8 -4 0 4 8

0

2

4

6

8

-8 -4 0 4 80

2

4

6

8

x/

z/

x/

y/

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

(a) (b)

(c)

Fig. 6. FDTD-method-based simulation of focusing the HB beam(n � 0, f � 4λ): (a) the modulus of the amplitude Ex at t � 20λ∕c,(b) intensity I � jExj2 � jEyj2 � jEzj2 in the Oxz plane, (c) time-averaged intensity in the plane z � 4λ.

x/

z/

-8 -4 0 4 8

0

2

4

6

8

Fig. 7. FDTD-method-based simulation of focusing the HB beam(n � 3, f � 4λ): the modulus of the amplitude Ex at t � 20λ∕c.


E0�r;φ; z� �iπA0

2H�1�

0

�k2

�f − z�

��r2 � �z − f �2

q ��

× J0

�k2

� ��r2 � �z − f �2

q− f � z

��: (35)

When z > f and r � 0, the argument of the Hankel functionturns to zero, and the intensity becomes infinite. In the initialplane z � 0, such a field is given by

E0�r;φ; z � 0� � iπA0

2H�1�

0

�k2

�f �

��r2 � f 2

q ��

× J0

�k2

� ��r2 � f 2

q− f

��: (36)

When solved using the FullWAVE software, this equationdescribes the propagation of a field with the modulus ofthe amplitude Ex as shown in Fig. 6(a). Figure 6(b) depicts theintensity in the Oxz plane. The simulation was conducted forthe wavelength λ � 633 nm and the focal length f � 4λ �2.53 μm. The computation region dimension was �−8λ; 8λ� ×�−8λ; 8λ� × �0; 8λ�. For all coordinates, the sampling was λ∕16.The simulation was conducted over 20 periods. The temporaldiscretization was λ∕�32c�.

From Fig. 6(c), the beam is seen to be focused into an el-liptical spot with the minimal diameter at half-maximum ofabout FWHM � 0.65λ. Alongside the focal spot, there is abright ring (side lobe) with the 2.5% peak intensity of thatin the spot, which is considerably lower than in the zero-orderBessel beam for which the first-ring intensity is 16% of that inthe diffraction pattern center.

The focusing into the axial line r � 0 (z > f ) does notoccur, as it might have appeared from Eq. (35). This maybe because at z > f , for the near-axis points (r ≪ z − f ), wecan write

E0�r ≪ z − f ;φ; z� ≈ iπA0

2H�1�

0

�kr2

4�z − f �

�

× J0

�k2

� ��r2 � �z − f �2

q− f � z

��

∼ −A0 ln�

kr2

4�z − f �

�

× J0

�k2

� ��r2 � �z − f �2

q− f � z

��; (37)

where “∼” sign means proportionality.

When r � 0, the logarithm takes infinite values for any z;however, for any other near-zero r, the logarithm decreaseswith increasing z. Thus, the simulated intensity wasshown to decrease along the optical axis behind the focalpoint z � f .

In a similar way, we simulated the propagation of a fieldwith the vortex n � 3. The modulus of the amplitude Ex inthe Oxz plane is shown in Fig. 7. The simulation parametersare the same. In this case, the axial focus shift also does notoccur, with the beam waist located in the plane z � f . Theintensity distribution in the transverse plane and the corre-sponding cross sections are shown in Fig. 8.

From Fig. 2 it might appear that the central intensity mini-mum can be made infinitesimally small, however this wouldhave required the use of the infinitely wide initial field inEq. (36). With this being actually unfeasible, the FDTDmethodresults in a pattern shown in Fig. 8. The diffraction ringFWHM is approximately equal to the wavelength of incidentlight.

7. CONCLUSIONAn exact analytical solution of the scalar Helmholtz equationto describe the propagation of the light beam in the positivedirection of the optical axis has been derived. The complexamplitude of such a beam is proportional to the product oftwo linearly independent solutions of the Kummer equation.Relationships for a particular case of such beams in the formof HB beams have been derived. The focusing properties ofthe HB beams have been studied.

ACKNOWLEDGMENTSThe work was financially supported by the federal program“Research and Educational Staff of Innovation Russia” (statecontract 14.740.11.0016), by the Russian Federation Presiden-tial grants for Support of Leading Scientific Schools (NSh-4128.2012.9), by a Young Researcher’s grant (MK-3912.2012.2), and by a Russian Foundation for Basic Researchgrant (12-07-00269).

REFERENCES1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon,

1986).2. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,”

Phys. Rev. Lett. 58, 1499–1501 (1987).3. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda,

“Alternative formulation for invariant optical fields: Mathieubeams,” Opt. Lett. 25, 1493–1495 (2000).

x/

y/

-8 -4 0(a) (b) (c)

4 8-8

-4

0

4

8

x/

I

-8 -4 0 4 80.00

0.25

0.50

-8 -4 0 4 80.00

0.25

0.50

x/

I

Fig. 8. Intensity distribution in the plane z � f : (a) two-dimensional pattern and its (b) vertical and (c) horizontal profiles.


4. M. A. Bandres, J. C. Gutierrez-Vega, and S. Chavez-Cedra, “Para-bolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46(2004).

5. V. Kotlyar, R. Skidanov, S. Khonina, and V. Soifer, “Hypergeo-metric modes,” Opt. Lett. 32, 742–744 (2007).

6. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato,“Hypergeometric-Gaussianmodes,”Opt.Lett.32, 3053–3055(2007).

7. V. Kotlyar and A. Kovalev, “Family of hypergeometric laserbeams,” J. Opt. Soc. Am. A 25, 262–270 (2008).

8. M. Bandres and J. Gutiérrez-Vega, “Circular beams,” Opt. Lett.33, 177–179 (2008).

9. V. V. Kotlyar and A. A. Kovalev, “Nonparaxial hypergeometricbeams,” J. Opt. A 11, 045711 (2009).

10. M. Abramovitz and I. A. Stegun, Handbook of MathematicalFunctions, Applied Mathematics Series (National Bureau ofStandards, 1965).

11. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, SpecialFunctions,Vol. 2 of Integrals and Series (Gordon & BreachScience, 1990).

12. W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).


hankel–bessel laser beams

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