Lorentz beams as a basis for a new class of rectangularly symmetric optical fields

Optics Communications 269 (2007) 274–284 www.elsevier.com/locate/optcom

Lorentz beams as a basis for a new class of rectangularly symmetric optical fields Omar El Gawhary b

a,*

, Sergio Severini

b

a Dipartimento di Fisica, Universita` ‘‘Roma Tre’’, Via della Vasca Navale 84, I-00146 Rome, Italy Centro Interforze Studi per le Applicazioni Militari, Via della bigattiera 10, 56010 San Piero a Grado (Pi), Italy

Received 8 May 2006; received in revised form 28 July 2006; accepted 8 August 2006

Abstract In this work we present a new and wide class of scalar, rectangular symmetrical optical fields, the free-space propagation of which can be given in a closed-form in the paraxial approximation. In particular it is shown how such fields can be expressed as a finite linear combination of the recently introduced Lorentz beams [O. El Gawhary, S. Severini, J. Opt. A: Pure Appl. Opt., 8 (2006) 409.] that, in this way, act as a basis for the newly introduced class. Because of their mathematical form, we call such fields super-Lorentzian beams. Some common features of the class are pointed out and the concept of order of the beam introduced. Moreover, by using these results, we demonstrate the existence of a new family of mutually orthogonal paraxial fields with a related new class of orthogonal polynomials.  2006 Elsevier B.V. All rights reserved. Keywords: Beam propagation; Orthogonal polynomials

1. Introduction The exact description for the propagation of any kind of optical field distribution, once it is known on a generical starting source plane (indicated, from now on, as z = 0 plane), still represents a very interesting challenge in optics. It is of great interest to do well to construct, at least in the paraxial limit approximation, the mathematical form assumed by the optical field on each plane, during free propagation. This fact can be useful for practical applications, since it was shown, for example, that there exist some sources (e.g., some double heterojunction lasers [1,2]) that produce fields with a special shape, that require different models than the ubiquitous TEM00 Gaussian beam. Probably it is not possible to obtain a mathematical closed form propagation of a field distribution for every kind of starting distribution, but the class of optical fields for which that can be done grows continually [4,3,5–10]. Actually it

*

Corresponding author. E-mail address: elgawhary@fis.uniroma3.it (O. El Gawhary).

0030-4018/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.08.007

is well known that such a problem can be solved, from a theoretical point of view, by using one of the orthogonal and complete family of fields that constitute a basis for any paraxial optical field (i.e., Hermite–Gauss, Laguerre– Gauss or Ince–Gauss beams [9,11,12]) to describe the special field distribution under analysis. For example, indicating as f(x, y, z) a generical paraxial field, one can put f ðx; y; 0Þ ¼

1 X 1 X

cmn HGmn ðx; y; 0Þ

ð1Þ

m¼0 n¼0

where cmn ¼

Z Z

f ðx; y; 0ÞHGmn ðx; y; 0Þ dx dy

ð2Þ

1

and HGmn is the Hermite–Gaussian beams of orders m, n. Obviously, analogous expansions are valid if Laguerre– Gauss or Ince–Gauss beams are utilized instead of Hermite–Gaussian ones. The expression for the field f(x, y, z) is valuable over each (x, y) plane, being well known in which fashion the Hermite–Gaussian beams propagate. Such an approach requires, to obtain an analytical

O. El Gawhary, S. Severini / Optics Communications 269 (2007) 274–284

expression for the field on each (x, y, z) point, that the coefficients (2) are calculable in closed-form. Moreover, (1) is a sum composed of an infinite number of terms, that, for every practical application, needs to be truncated. In the present work we introduce a wide class of realizable and scalar optical beams that, to the authors’ knowledge, have never been studied before today, for which the paraxial propagation can be given in closed-form. This can be done by utilizing a different kind of beam as basis of expansion, i.e. the recently introduced Lorentz beams [10]. Because of their shape, we call these fields as super-Lorentzian beams. We will show that an optical beam belonging to the class in subject can be represented by means of a linear combination of a finite number of elementary constituent fields of a Lorentz beam. Once the aforesaid expansion is pointed out, we will demonstrate the existence of a new family of orthogonal paraxial fields; in particular it deals with a set of mutually orthogonal fields containing a finite number of elements. The paper is organized as follows: in Section 2 we introduce the generical form for a super-Lorentzian beam; in Section 3 we recall the propagation features of a Lorentz beam; in Section 4 we discuss the paraxial propagation of super-Lorentzian beams; in Section 5 we introduce the aforesaid new family of orthogonal optical fields, defining a new class of orthogonal polynomials and discussing some applications of the theory. 2. Super-Lorentzian beams Let us suppose we have, on the (x, y, 0) plane, the following real field distribution: V ðx; y; 0Þ ¼ gðxÞhðyÞ

M Y m¼1

N Y cm dn 2 2 ðx þ am Þ n¼1 ðy 2 þ b2n Þ

ð3Þ

where cm and dn are constant coefficients, g(x) and h(y) are real and well-behaved functions over the x and y axes, respectively, that apart from some fundamental hypotheses, can be chosen in a completely arbitrary way. The coefficients am, bn can even be complex but in this case they have to appear in complex conjugated couples, because of the reality of (3). Without losing generality, dealing with a rectangular symmetry, we can limit ourselves to study two-dimensional fields, by dropping out the y dependence, so that we concentrate us on the following scalar distribution on the source (z = 0) line: M Y cm V ðx; 0Þ ¼ gðxÞ ð4Þ 2 þ a2 Þ ðx m m¼1 A field like (4) is composed of a number of complex conjugate poles, that we suppose to be different one to another, multiplied by the function g(x) not singular by definition. Due to the form of each denominator appearing in the product, we can give the name of super-Lorentzian beam (SLB for short) to this kind of field. We can separate the odd and even terms in g(x) gðxÞ ¼ go ðxÞ þ ge ðxÞ

ð5Þ

275

where gðxÞ  gðxÞ 2 gðxÞ þ gðxÞ ge ðxÞ ¼ 2 so that (4) becomes

ð6Þ

go ðxÞ ¼

ð7Þ

V ðx; 0Þ ¼ V o ðx; 0Þ þ V e ðx; 0Þ

ð8Þ

with V o ðx; 0Þ ¼ go ðxÞ

N Y m¼1

V e ðx; 0Þ ¼ ge ðxÞ

cm ðx2 þ a2m Þ

ð9Þ

cm þ a2m Þ

ð10Þ

N Y m¼1

ðx2

We succeed to give the closed-form expression for the propagated field (8) in free space, by using the results obtained in a recent work about Lorentzian beams [10]. For this reason, in the next section we first recall briefly the main features of a Lorentz beam and then we will move on to the study of a SLB. 3. Lorentz beams A two-dimensional scalar Lorentz beam is an optical paraxial field which is a realizable field characterized by the following structure: expðikzÞ LBðx; zÞ ¼ C pffiffiffiffiffiffi ½V þ ðx; zÞ þ V  ðx; zÞ ikz

ð11Þ

where pffiffiffiffiffiffiffiffiffi V  ðx; zÞ ¼ exp½pðwx  ixÞ2 =ikzf1  erf½pðwx  ixÞ= ipkzg ð12Þ pffiffiffi with erf(x) the usual error function erfðxÞ ¼ ð2= pÞ Rx expðs2 Þds and C is a constant value. Its name origi0 nates from the fact that on the line z = 0 it reduces to a Lorentzian field distribution. This fact can be shown by expanding the error function in (11) for high values of its argument as we extensively showed in [10]. In the limit z ! 0 the field in Eq. (12) assumes the form V ðx; 0Þ ¼

C 1 wx ½1 þ ðx=wx Þ2 

ð13Þ

where we recognize a Lorentzian curve with the parameter that affects the curve width as wx. We note that the functions V+(x, z),V(x, z) play the role of constituent functions for a Lorentz beam. It is relevant to point out that expression (12) for complex values of wx is still valid. Before closing this brief reminder, as it will be important for the continuation of the present work, we also remember here the explicit form of the plane wave spectrum for such a field Z parax LBðx; zÞ expð2ippxÞdx Az ðpÞ ¼ 1

¼ Cp expð2pjpjwx  ipkzp2 Þ expðikzÞ

ð14Þ

276

O. El Gawhary, S. Severini / Optics Communications 269 (2007) 274–284

with real or complex wx. In the next section we will express a beam like in (4) as a combination of fundamental fields V+(x, z) and V(x, z) in the hypothesis of considering only real values for the coefficients ai and bi in (3) and (4). A discussion on complex values of these coefficients will be presented in Section 5.

Here we do not provide proof for this lemma because it can be found in any book on complex calculus (see, for example, [16] for details). The importance of the theorem resides in the fact that, if the function f(s) is analytic (or holomorphic) in the half plane Re(s) 6 0 except than in a finite number of poles (simple or multiple) all lying on the half plane Re(s) 6 0, one obtains

4. Propagation of super-Lorentzian beams We wish to show that, in free space, it is possible to give an analytical and closed form expression for the field in (4) in the paraxial propagation. To do this we find the plane wave spectrum for the field on z = 0, say A0(kx), and then we calculate it on z 5 0 by adding the Fresnel quadratic phase shift term. Once we know the spectrum on z, by means of an inverse Fourier transformation we can evaluate the propagated field V(x, z). For the spectrum A0(kx) we have Z 1 A0 ðk x Þ ¼ V ðx; 0Þ expðik x xÞdx ð15Þ 1

At this point we assume all the above mentioned functions extendable in the complex domain so that, if we let x = is, with s = sr + isi complex variable with real part sr and imaginary part si, we obtain Z i1 V ðis; 0Þ expðk x sÞi ds ð16Þ A0 ðk x Þ ¼ þi1

that is 1 A0 ðk x Þ ¼ 2pi

Z

i1

2pV ðis; 0Þ expðk x sÞds

ð17Þ

i1

It is easy to recognize in (17) the inverse Laplace transformation of the function 2pV(is, 0). This kind of integral can be evaluated, under certain hypotheses, by means of the residua theorem. In fact, let us refer to a generical complex integral Z cþi1 1 I¼ f ðsÞ expðtsÞds ð18Þ 2pi ci1 where we suppose t > 0 and c 2 R. If we choose the parameter c = 0 (by guiding us, in this way, just to a form like in (17)) and consider a semicircle c(R) of radius R centered in the origin of the complex plane and lying in the half plane Re(s) 6 0, we suppose the following condition fulfilled Lemma (Modified uniformly

Jordan’s

lim f ðsÞ ¼ 0

jsj!þ1

then it holds Z expðtsÞf ðsÞds ¼ 0 lim

R!þ1

lemma). If

tP0

and ð19Þ

ð20Þ

c

(If the parameter t is negative the curve c must lie in the half plane Re(s) P 0).

I¼

1 2pi

Z

þi1

f ðsÞ expðtsÞds ¼

i1

M X

Resm

ð21Þ

m¼1

where M is the number of poles and Resm is the residuum of the function exp(ts)f(s) in the mth pole. The condition just introduced limits the class of the functions g(x) (otherwise totally arbitrary) to those ones for which uniformly it holds lim gðisÞ

jsj!þ1

M Y

cm

m¼1

½ðisÞ þ a2m 

2

¼0

ð22Þ

On coming back to the integral (17), we have 1 2pi

Z

i1

M Y

cm expðk x sÞds 2 ½ðisÞ þ a2m  m¼1 Z i1 M Y 1 cm expðk x sÞds 2p½go ðisÞ þ ge ðisÞ ¼ 2 2 2pi i1 m¼1 ½ðisÞ þ am  Z i1 M Y 1 cm expðk x sÞds 2pgo ðisÞ ¼ 2 2 2pi i1 m¼1 ½ðisÞ þ am  Z i1 M Y 1 cm 2pge ðisÞ expðk x sÞds þ 2 2pi i1 ½ðisÞ þ a2m  m¼1 Z i1 M Y 1 cm 2pgo ðisÞ expðk x sÞds ¼ 2 2pi i1 a  s2 m¼1 m Z i1 M Y 1 cm 2pge ðisÞ expðk x sÞds ð23Þ þ 2  s2 2pi i1 a m¼1 m

A0 ðk x Þ ¼

2pgðisÞ

i1

To take into account the contribution, to the integral (23), of the generical residuum in the mth pole it is necessary to distinguish the two cases kx > 0 and kx < 0. First of all, we note that the field in (4) possesses only couples of complex conjugate poles (an obliged choice, to exclude the presence of real poles that would give rise to singularities in the field pattern which would become, in this way, physically unrealizable) so that the analytically prolonged optical beam will still have only couples of real simple poles, symmetric with respect to the imaginary axis. So, to assure the convergence of the relative integral, one has to consider only the contribution of poles with sr < 0, in the case kx > 0, while the contribution of poles with sr > 0 when kx < 0. Separating the even part of the spectrum from the odd one we can write Z i1 M Y 1 cm ðoÞ A0 ðk x Þ ¼ 2pgo ðisÞ expðk x sÞds ð24Þ 2  s2 2pi i1 a m¼1 m

O. El Gawhary, S. Severini / Optics Communications 269 (2007) 274–284

ðeÞ A0 ðk x Þ

1 ¼ 2pi

Z

M Y

i1

2pge ðisÞ

i1

a2 m¼1 m

cm expðk x sÞds  s2

ð25Þ

where we have let A0 ðk x Þ ¼

ðoÞ A0 ðk x Þ

277

To calculate the plane wave spectrum on z, within the paraxial approximation, one needs to introduce the quadratic Fresnel phase shift Az ðpÞ ¼ A0 ðpÞ expðipkzp2 Þ

ðeÞ A0 ðk x Þ

þ

ð26Þ

Without loosing in generality, we consider positive all the values of the coefficients ai i.e. ai P 0. For the calculation of the residua of functions in (24) we have to consider of the n-th couple of poles (an; an), the pole an when kx > 0, and the pole an when kx < 0. For this reason, the residuum of the odd and even part of the spectrum (26) for kx > 0 can be evaluated as M Y ðs þ an Þ cm Resn ¼ lim 2pgg ðisÞ expðk x sÞ s!an ðan  sÞðan þ sÞ m6¼n a2m  s2

ð35Þ

where p = kx(2p)1 and k is, as usual, the wavelength. At this point we can give the field expression V(x, z) by means of an inverse Fourier transformation V ðx; zÞ ¼ I 1 þ I 2 ¼ expðikzÞ

M Z X

1

h

ðo;mÞ

A0

1

m¼1

ðe;mÞ

ðpÞ þ A0

   exp ipkzp2 expð2ippxÞdp

ðpÞ

i

 ð36Þ

On performing the calculation, we obtain

m¼1

¼

An p g ðian Þ expðk x an Þ an g

ð27Þ

I 1 ¼ expðikzÞ

M Z X m¼1

and for kx < 0 we have Resn ¼  lim 2pgg ðisÞ s!þan

M Y ðs  an Þ cm expðk x sÞ 2 ðan  sÞðan þ sÞ m6¼n am  s2 m¼1

ð28Þ

where g = (o, e) it is referred to odd or even. In (27) and (28) we have defined the quantities An as follows: An ¼

M Y m6¼n m¼1

ð29Þ

ðe;nÞ A0 ðk x Þ

On introducing the quantities and fined as  An pgo ðian Þ expðk x an Þ=an ; k x > 0 ðo;nÞ A0 ðk x Þ ¼ An pgo ðian Þ expðk x an Þ=an ; kx < 0  A pg ðia Þ expðk a Þ=a ; k n n x n n x > 0 e ðe;nÞ A0 ðk x Þ ¼ An pge ðian Þ expðk x an Þ=an ; kx < 0

de-

ðo;mÞ

½A0

ð30Þ ð31Þ

ðe;mÞ

ðk x Þ þ A0

ðk x Þ

ð32Þ

m¼1

with ðoÞ

M X

ðeÞ

m¼1 M X

A0 ðk x Þ ¼ A0 ðk x Þ ¼

m¼1

ðo;mÞ

A0

ðe;mÞ

A0

0

M

expðikzÞ X Am p ¼ pffiffiffiffiffiffi go ðiam Þ V ðþ;mÞ ðx; zÞ  V ð;mÞ ðx; zÞ x x ikz m¼1 am

and I 2 ¼ expðikzÞ

M Z X m¼1

where we used parity properties of go(x) and ge(x) i.e., go(x) = go(x) and ge(x) = ge(x), the spectrum (26) becomes M X

M Z X

ðpÞ expðipkzp2 Þ expð2ippxÞdp

ð37Þ

cm a2m  a2n ðo;nÞ A0 ðk x Þ

A0 ðk x Þ ¼

1

ðo;mÞ

A0

Am pgo ðiam Þ expð2ppam Þ 1 am m¼1 Z 1 Am  expðipkzp2 Þ expð2ippxÞdp  pg ðiam Þ am o 0  2  expð2ppam Þ expðipkzp Þ expð2ippxÞdp

¼ expðikzÞ

An p g ðian Þ expðk x an Þ ¼ an g

1

ðk x Þ

ð33Þ

ðk x Þ

ð34Þ

1

1

M Z X

ðe;mÞ

A0

ðpÞ expðipkzp2 Þ expð2ippxÞdp

0

Am pge ðiam Þ expð2ppam Þ expðipkzp2 Þ 1 am m¼1 Z 1 Am  expð2ippxÞdp þ pg ðiam Þ expð2ppam Þ am e 0   expðipkzp2 Þ expð2ippxÞdp

¼ expðikzÞ

M

expðikzÞ X Am p ¼ pffiffiffiffiffiffi go ðiam Þ V ðþ;mÞ ðx; zÞ þ V ð;mÞ ðx; zÞ x x a ikz m¼1 m

ð38Þ  where the expressions of V þ x and V x are given in Eq. (12). Eqs. (37) and (38) represent the result we were looking for. In particular we were able to express a very generical field distribution, as that in (4), as a combination of a finite number of the elementary constituents of a Lorentz beams,  namely the function V þ x ðx; zÞ and V x ðx; zÞ. On considering the (37), (38) we can rearrange (36) as follows:

278

O. El Gawhary, S. Severini / Optics Communications 269 (2007) 274–284

( M expðikzÞ X Am p ðþ;mÞ V ðx; zÞ ¼ pffiffiffiffiffiffi V ðx; zÞ½ge ðiam Þ þ go ðiam Þ am x ikz m¼1 ) Am p ð;mÞ þ V ðx; zÞ½ge ðiam Þ  go ðiam Þ am x

main. If we indicate with coefficients D0 and D1 the norms of the two beams SLB0 and SLB1 respectively, we have 1=2

D0 ¼ hSLB0 ðx; 0Þ; SLB0 ðx; 0Þi Z 1 1=2 ¼ SLB0 ðx; 0ÞSLB0 ðx; 0Þdx 1

ð39Þ In the next section we apply this result to obtain the closedform expression for the propagation of some special beams obtained by fixing the form of the function g(x). 5. A set of mutually orthogonal SLBs We have shown, in the previous sections, that the propagation of the generic field (4), we named SLB, can be divided into even and odd contributions M expðikzÞ X Am p V e ðx; zÞ ¼ pffiffiffiffiffiffi ge ðiam Þ½V ðþ;mÞ ðx; zÞ þ V ð;mÞ ðx; zÞ x x ikz m¼1 am

ð40Þ M X expðikzÞ Am p go ðiam Þ½V ðþ;mÞ ðx; zÞ  V ð;mÞ ðx; zÞ V o ðx; zÞ ¼ pffiffiffiffiffiffi x x ikz m¼1 am ð41Þ From now on, we define the quantity M as order of a SLB. Once the order M is fixed and consequently the coefficients ai, i = 1, . . . , M are fixed, it remains associated, to each SLB, a family of orthogonal polynomials of orders {0,1, . . . ,2M  1}. The maximum order for the polynomials is 2M  1 because of the condition imposed from Jordan’s Lemma in Section 4. With the space of polynomials {1,x, x2, . . . , x2M1} we can construct a family of mutually orthogonal SLB, by making use of the Gram–Schmidt method [17]. In order to do that, let us begin with taking in consideration the following couple of functions g0 ðxÞ ¼ 1 g1 ðxÞ ¼ x

ð42Þ ð43Þ

to which correspond the SLBs SLB0 ðx; 0Þ ¼ g0 ðxÞ

M Y i¼1

SLB1 ðx; 0Þ ¼ g1 ðxÞ

M Y i¼1

1 ðx2 þ a2i Þ

ð44Þ

1 ðx2 þ a2i Þ

ð45Þ

We wish that the SLBs in (44) and (45) be orthogonal according to the usual definition of scalar product, i.e., Z 1 hSLB1 ðx; 0Þ; SLB0 ðx; 0Þi ¼ SLB0 ðx; 0ÞSLB1 ðx; 0Þdx 1

¼

"Z

1

M Y

1 m¼1

#1=2

1 ðx2 þ a2m Þ

2

ð47Þ

dx

and the expression of the normalized SLB0 is M Y 1 g 0 ðx; 0Þ ¼ g0 ðxÞ SLB D0 i¼1 ðx2 þ a2i Þ "Z #1=2 M M 1 Y Y 1 1 ¼ dx  2 þ a2 Þ 2 2 2 ðx 1 m¼1 ðx þ am Þ i i¼1 ð48Þ where the tilde stands for Normalized. Likewise for SLB1(x) and D1 1=2

D1 ¼ hSLB1 ðx; 0Þ; SLB1 ðx; 0Þi Z 1 1=2 ¼ SLB1 ðx; 0ÞSLB1 ðx; 0Þdx 1

¼

"Z

1

x 1

2

#1=2

M Y

1

m¼1

ðx2 þ a2 Þ

2

ð49Þ

dx

and the normalized SLB1 is g 1 ðx; 0Þ ¼ SLB1 ðx; 0Þ SLB D1 "Z #1=2 M M 1 Y Y x2 x2 ¼ dx  2 2 2 ðx2 þ a2i Þ 1 m¼1 ðx þ am Þ i¼1 ð50Þ Values of coefficients Di, representing the norms of the super Lorentz Beams SLBi, depend on the order M and the associated poles ai, i = 1, 2, . . . , M. At this point one goes on with the higher order beams. For the field SLB2(x,0) the g(x) will be g2 ðxÞ ¼ ðC 20 þ x2 Þ

ð51Þ

and it must hold hSLB2(x,0),SLB0(x,0)i = 0 and hSLB2(x, 0), SLB1(x,0)i = 0. For this beam with D2 we indicate the normalization coefficient hSLB2jSLB2i1/2. Still for parity matters it results automatically that hSLB2(x, 0), SLB1(x,0)i = 0. For the first condition we put Z 1 M Y 1 ðC 20 þ x2 Þ dx ¼ 0 hSLB2 ðx;0Þ; SLB0 ðx; 0Þi ¼ 2 2 2 1 m¼1 ðx þ am Þ ð52Þ

¼0 ð46Þ where the asterisk indicates a complex conjugation. The integral (46) is null since the product of SLB0(x)SLB1(x) is an odd function that is integrated on a symmetric do-

that yields Z 1Y M C 20 ¼  1 m¼1

x2 ðx2 þ a2m Þ2

"Z dx

1

M Y

1 m¼1

1 ðx2 þ a2m Þ2

#1 dx

ð53Þ

O. El Gawhary, S. Severini / Optics Communications 269 (2007) 274–284

The process can be extended up to the beam of maximum order. For p > 1 we will obtain SLB2p ðx; 0Þ ¼ g2p ðxÞ

M Y m¼1

1 2 ðx þ a2m Þ

ð54Þ

where g2p ðxÞ ¼ x2p þ

p1 X

C ð2pÞð2kÞ x2k

ð55Þ

k¼0

and M Y

1 2 ðx þ a2m Þ

ð56Þ

C ð2pþ1Þð2kþ1Þ x2kþ1

ð57Þ

SLB2pþ1 ðx; 0Þ ¼ g2pþ1 ðxÞ

m¼1

279

5.1. Case M = 1 In this case there are only two orthogonal SLBs. On performing the calculations as indicated in the previous section, i.e., in (48) and (49), we obtain rffiffiffiffiffiffiffi 2a3 1 g ð59Þ SLB0 ðx; 0Þ ¼ 2 p ðx þ a2 Þ rffiffiffiffiffi 2a x g ð60Þ SLB1 ðx; 0Þ ¼ p ðx2 þ a2 Þ In Fig. 1 we report the form for the fields and the intensities, on z = 0. 5.2. Case M = 2

where g2pþ1 ðxÞ ¼ x2pþ1 þ

p1 X k¼0

Obviously for all these fields the normalized version are g i ¼ SLBi =Di simply. In Appendix A we report some SLB useful analytical statement for the exact closed form calculations for the generic coefficients Di and coefficients Cn, m. One could now ask oneself if the family of orthogonal beams maintains the orthogonality property under free propagation. In other words, we have derived a class of functions that are mutually orthogonal on the line z = 0, but are we sure that this condition is maintained on every line z 5 0? It is easy to verify that the field parity is preserved in paraxial propagation so that the orthogonality is assured for fields with opposed parity. But for beams endowed with the same kind of parity, can we say something? The answer is positive. In fact, one can prove that the orthogonality is preserved on every line z 5 0, independently from parity matters. To prove that, it is sufficient to recall Parseval’s theorem. On indicating as V1(x, z) and V2(x, z) two paraxial optical fields it states Z 1 hV 2 ðx; zÞ; V 1 ðx; zÞi ¼ V 1 ðx; zÞV 2 ðx; zÞdx 1 Z 1 ðzÞ ðzÞ  A1 ðpÞðA2 ðpÞÞ dp ¼ 1 Z 1 ð0Þ ð0Þ  A1 ðpÞðA2 ðpÞÞ dp ¼ 1 Z 1 V 1 ðx; 0ÞV 2 ðx; 0Þdx ¼

For this value of M four orthogonal SLBs will exist. In expression (4) there are two parameters a1 and a2 that we consider both reals. On performing calculations similar to the ones indicated in the previous section, we obtain 1 D0 ðx2 þ a21 Þðx2 þ a22 Þ x g 1 ðx; 0Þ ¼ SLB D1 ðx2 þ a21 Þðx2 þ a22 Þ C 20 þ x2 g 2 ðx; 0Þ ¼ SLB D2 ðx2 þ a21 Þðx2 þ a22 Þ C 31 x þ x3 g 3 ðx; 0Þ ¼ SLB D3 ðx2 þ a21 Þðx2 þ a22 Þ g 0 ðx; 0Þ ¼ SLB

ð61Þ ð62Þ ð63Þ ð64Þ

g i j SLB g j i ¼ dij , with where using the conditions h SLB i, j = 0, 1, 2, 3 and dij the Kronecker symbol, one finds the correct values of coefficients C20, C31, D0, D1, D2 and D3 (see Appendix A for an analytical facilities) C 20 ¼ 

a21 a22 a21 þ 3a1 a2 þ a22

C 31 ¼ a1 a2 D20 ¼ D21 ¼ D22 ¼ D23 ¼

pða21 þ 3a1 a2 þ a22 Þ 2a31 a32 ða1 þ a2 Þ3 p

ð65Þ

3

2a1 a2 ða1 þ a2 Þ pða21 þ 6a1 a2 þ a22 Þ 3

2ða1 þ a2 Þ ða21 þ 3a1 a2 þ a22 Þ pða21 þ 6a1 a2 þ a22 Þ ða1 þ a2 Þ

3

1

¼ hV 2 ðx; 0Þ; V 1 ðx; 0Þi

ð58Þ

ð0Þ 2 where AðzÞ r ¼ Ar expðikpzp Þ is the plane wave spectrum ð0Þ on z and Ar is that on z = 0, with r = 1, 2. Obviously, Eq. (58), which states that scalar products are preserved in the evolution on z, is a direct consequence of the unitary character of such an evolution [13,14]. In the next two subsections we analyze in detail the cases M = 1 and M = 2.

In Fig. 2 the behaviour for the four orthogonal SLBs on z = 0 is displayed. Figs. 3 and 4 show the propagation of the two fields SLB0 and SLB2 for a generic z value. For the two poles used in the simulation the two values of Rayleigh distances are zR1 ¼ 2pa21 =k  3:7 m and zR2 ¼ 2pa22 =k  15 m (the value of Rayleigh distance for a Lorentz beam is given in [10]). In viewing Figs. 1 and 2 one can note some similarities between these fields and those associated to a Hermite–Gaussian beam of the same order,

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a

b

Fig. 1. Field amplitude (dotted line) and intensity (continuous line) for the first (a) and second (b) super-Lorentzian orthogonal fields corresponding to M = 1 and a = 1.

the only fundamental difference residing in the number of orthogonal modes. As a matter of fact, the family of the newly introduced orthogonal SLBs is composed of a finite number of beams because of the finiteness of the order M, which is determined by the super-Lorentzian term M Y m¼1

1 ðx2 þ a2m Þ

ð66Þ

that acts as the weight term in all the scalar product for the fields. For a Hermite–Gaussian beam, instead, the order is due to the Gaussian term (that now represents the weight function), and, as well known, is infinite. Consequently the number of functions belonging to the class becomes too. Besides, one can imagine that, once M ! 1, the two class of modes tend to be ever closer in form. We can justify this consideration by observing that the Fourier transformation of (66) is a convolution of M positive terms, each of which of the form like in (14), that, for the Central Limit Theorem [15], tends to a Gaussian distribution when M ! 1. In the next subsection we show the possibility of using a SLB as a tool to describe other kind of optical fields, within an a priori determined accuracy. 5.3. An application for SLBs with complex Since we are interested in the representation and propagation of real field functions on z = 0, if we consider the possibility of having complex values for the coefficients ai and bi in (3) (and consequently in (4)), it is immediately

evident that the poles have to appear in a double symmetric form: with respect to the imaginary axis and to real axis. In particular, if expression (4) admits an an as a complex pole, then the same expression have to admit an ; an and an other poles. All the relations just introduced can be used even if we consider complex values for ai but the calculation of the residua needs particular attention. In the case kx > 0, at the expression (27) we have to add the residuum calculated in an ; in the other case kx < 0, similar considerations must be repeated for the relation (28), in which we have to add the residuum calculated in an . On keeping this in mind, we wish to show that an SLB can also represent a good (and simple) approximated model to estimate the propagation of other kinds of realizable fields, for which the form of the analytical propagation is still lacking. As a test, we write out the approximation of a Gaussian field by means of a finite superposition of SLBs. In particular let us suppose we want to approximate the propagation of the Gaussian field V ðx; 0Þ ¼ expðx2 =w20 Þ, with w0 = 100 lm, by using a field distribution like (4). Expanding V(x, 0), as in the following, and accepting a maximum error of about 0.5% on the representation, we obtain " #1 5 X 1 ðx=w0 Þ2n ’ V ðx; 0Þ ¼ expðx2 =w20 Þ n! n¼0 ¼

1 2

1 þ ðx=w0 Þ þ

ðx=w0 Þ4 2

6

8

0Þ 0Þ þ ðx=w6 0 Þ þ ðx=w þ ðx=w 24 120

10

ð67Þ

O. El Gawhary, S. Severini / Optics Communications 269 (2007) 274–284

a

b

c

d

281

Fig. 2. The family of four orthogonal SLBs, when M = 2 . We report the field amplitude (dotted line) as well as the intensity (continuous line) on z = 0 for a1 = 1 and a2 = 2.

Fig. 3. Modulus of SLB0(x, z) in arbitrary units, plotted for the following values k = 6 · 107 m, a1 = 103k, a2 = 2a1. Variables x and z are expressed in meters and the range of variability of z covers values from  zR1 ¼ 3:7 m to  zR2 ¼ 15 m.

that is V ðx; 0Þ ’

5 Y i¼1

c ðx2 þ a2i Þ

ð68Þ

where a21 ¼ w20 ð0:2398 þ 3:1283iÞ, a22 ¼ w20 ð0:2398  2 2 2 3:1283iÞ, a3 ¼ w0 ð1:6495 þ 1:6939iÞ, a4 ¼ w20 ð1:6495  1:6939iÞ, a25 ¼ 2:1806w20 and c ¼ 120w10 0 . Values of ai and c are numerical evaluations of analytically exact values. In this case g(x) = 1. In Fig. 5 we report the behaviours of the approximated Gaussian beam and the

Fig. 4. Modulus of SLB2(x, z) in arbitrary units, plotted for the same values of Fig. 3.

relative error with respect to the exact Gaussian. Fig. 6 shows the propagation of the approximated Gaussian field: it is evident that the model represents very well the features of a Gaussian beam. We conclude this section with a 3D example. We compare the propagation of the exact Hermite–Gauss beam of order (2,0) and the same function approximated by SLB of order M = 5. In Fig. 7 the absolute values of the moduli, for both fields and for different distances, are plotted. In conclusion we can say that, within the precision due to the accepted error in the expansion, one can evaluate the analytical paraxial propagation for the field by means of (39).

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a

b

Fig. 5. The fundamental Gaussian beam expðx2 =w20 Þ, represented by means of a SLB, as indicated in (68) (a) and error behaviour (b). The figures are plotted on z = 0. w0 = 100 lm.

this new class of realizable fields as Super-Lorentz Beams (SLB) because of their definition. Closed-form expression for the paraxial propagation of this kind of fields has been given. Within this new class of realizable fields, we have also derived a new family of mutually orthogonal fields, introducing a new class of orthogonal polynomials. As application of the theory we have moreover demonstrated the possibility of approximating Cartesian symmetric optical beams in terms of SLBs; as a test function we discussed the SLB approximation of the well known Gaussian optical beam as well as an Hermite–Gauss beam of orders (2, 0). Fig. 6. Propagation, in arbitrary units, of the super-Lorentian beam of order M = 5, representing a Gaussian envelope of the kind expðx2 =w20 Þ, with w0 = 100 lm. Simulation was performed for k = 0.6328 lm. ‘‘x’’ and ‘‘z’’ values are expressed in meters.

6. Conclusion We have introduced a new class of scalar optical beams, for which the propagation under paraxial limit can be analytically given. These beams have a Cartesian symmetry and can be expressed by means of a finite summation of elementary constituents fields of the kind of the ones recently introduced with Lorentz Beams [10]. We named

Appendix A. Calculation of orthogonal Lorentz polynomial coefficients Starting from the observations of integral relation that defines Lorentz Polynomials coefficients, as in example (53), or viewing the relations that define the normalization coefficients, see in example (47) and (49), it is evident the importance of the resolution of the following integral: Z 1 Y M xk dx ðA:1Þ 2 2 2 1 m¼1 ðx þ am Þ The only condition we assume in the previous relation is that poles ai are different each other, but they can be

O. El Gawhary, S. Severini / Optics Communications 269 (2007) 274–284

283

Fig. 7. Propagation modulus, in arbitrary units, of the exact Hermite–Gauss beam (2,0) on the left side, and its approximated version done by means of super-Lorentian beams, on the right. Plot were made for z1 = 0, z2 = 0.5zR, z3 = zR, where zR is the Rayleigh distance. Simulation was performed for k = 0.6328 lm, w0 = 100 lm and M = 5. For all the graphs ‘‘x’’ and ‘‘z’’ are normalized on w0 value and they cover the range [5, 5] · [5, 5].

complex. Since the expression inside the integral in (A.1) must be real by definition, poles appearing in (A.1) must have a double axis symmetry, as discussed in Section 5.3. Since the value of integral (A.1) depends on the value of k, we can indicate the result of (A.1) as R(k). The first

observation we can do, by using parity considerations of function and integration range in (A.1), is that for odd values of k function R(k) = 0. For even values the integral can be easily solved by using the residua method. Rewriting integral (A.1) with 2k instead of k we obtain

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kþ1

Rð2kÞ ¼ ð1Þ

2p

X i

! M Y o x2k 1 lim x!d i ai os ðx þ d a Þ2 ða2j þ x2 Þ i i j¼1;j6¼i ðA:2Þ

where coefficient di is defined as follows:  i; if Re½ai  > 0 di ¼ i; if Re½ai  < 0

ðA:3Þ

For the definition of the coefficient R(k) the calculation of Lorentz Polynomial coefficients are extremely simple: in example coefficient C20 given by (53) is C20 =  R(2)/ R(0). Same consideration can be done for the normalization coefficients Di: as an example, relations pffiffiffiffiffiffiffiffiffiffi starting from pffiffiffiffiffiffiffiffiffi ffi (47) and (49) we have D0 ¼ Rð0Þ and D1 ¼ Rð2Þ. With similarity with other classical orthogonal polynomials [18] it is relevant to underline that even (odd) order Lorentz Polynomials contain only even (odd) powers of x, i.e., they can be of the kinds S 2n ¼

n X

C 2i x2i

ðA:4Þ

i¼0

S 2n1 ¼

n X i¼1

C 2i1 x2i1

ðA:5Þ

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