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Exact and geometrical-optics energy trajectories in

M V Berry1

& K T McDonald2

1H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8

2Joseph Henry Laboratories, Princeton University, Princeton, N

Abstract

Energy trajectories, that is, integral curves of the Poynting (cu

calculated for scalar Bessel and Laguerre-Gauss beams carryin

momentum. The trajectories for the exact waves are helices, w

cylinders for Bessel beams and hyperboloidal surfaces for Lag

beams. In the geometrical-optics approximations, the trajectoriof beam are overlapping families of straight skew rays lying on

surfaces; the envelopes of the hyperboloids are the caustics: a c

Bessel beams and two hyperboloids for Laguerre-Gauss beams

PACS numbers: 42 15 Dp 42 25 Fx 42 25 Hz


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1. Introduction

One of several ways to depict optical fields is by the trajectorie

that is, the lines everywhere tangent to the Poynting vector. He

describe light by a scalar wave (r), constructed for example f

potential [1, 2], or representing a single field component, or sim

a physical model in which polarization effects are neglected. T

vector can be chosen parallel to the current, that is, the expecta

local momentum operator, namely

P r( ) = Im* r( ) r( ) = r( )2 arg r( ) .

The second equality expresses the fact that in vacuum P(r) is o

wavefronts (contour surfaces of phase arg). P(r) is an import

calculating the orbital angular momentum of the field [3], and

particles in the field. In quantum physics, the trajectories are th

the Madelung [4] hydrodynamic interpretation, later regarded

quantum particles in the Bohm-de Broglie interpretation [5].

For optical beams, where there is a well-defined propag

it is natural to represent the trajectories using z as a parameter,

using cylindrical polar coordinates {r,,z}. Then, writingP(r)


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dr z( )dz

r z( ) = vr r z( )( ), dz( )dz

z( ) = v r z( )( )r z( )

.

Our aim here is to understand the energy trajectories for

Laguerre-Gauss beams carrying orbital angular momentum (t

which are of current interest theoretically and experimentally,

extending previous studies [6, 7]. We emphasize a fundamenta

the understanding of energy flow: the equation (1) can be inter

quite different ways, both of which we will employ in the follo

In the first (sections 2 and 3), (r) is the exact solution o

wave equation; then P(r) has the advantage that it represents w

approximation the flow described by the wave equation.

In the second (sections 4 and 5), the lines ofP(r) represe

geometrical optics, which although approximate carry the intu

their envelopes are the caustic surfaces on which the field is m

this case, (r) represents one of possibly several locally plane

superposed to create the total field. When points in the field ar

several geometrical rays, the corresponding pattern of trajector

unlike the exact Poynting trajectories of the total field, which a

defined at each point. P(r) depends nonlinearly on (r), so the

j i i diff f h j i f h i i


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showing that they satisfy the following equations, expressing tthe curvature

r r / r :r + 2 r = 0, r r 2 = 0.

By contrast, the trajectories of the exact Poynting vector are us

For the beams we study here, it turns out, unexpectedly,

calculate the exact Poynting trajectories than the geometrical r

require knowledge of the asymptotics of Bessel and Laguerre f

2. Bessel beams: exact Poynting flow lines

These beams are exact solutions of the Helmholtz equation, de

l r( ) = exp i k2 q2z + l

Jl qr( ),

in which l is the angular-momentum quantum number, kthe fr

wavenumber and q the magnitude of the transverse component

wavevector of the plane waves comprising the beam.


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The trajectories are determined by the differential equations (1trivially solved to give

r= constant, = 0 +l

r2 k2 q2z .

This describes a two-parameter family of helices (figure 1) fill

helix can be defined by the point {r,0} where it pierces the pl

of the helices with radius r is

z =2r2 k2 q2

l,

so the wider helices are more longitudinally stretched.

3. Laguerre-Gauss beams: exact Poynting flow lines

These beams are exact solutions of the paraxial wave equation

l,p r( ) = exp i kz + l( ){ }exp 2

2w ( )

l

w ( )l+1

l w ( )

w ( )

p

L

Here Ll

denotes the Laguerre polynomial [10] with indices re


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and

w ( ) =1+ i.

For convenience we will henceforth consider only positive l, a

the modulus signs. (The paraxial approximation to Bessel beam

k2 q

2in (2.1) is approximated by k-q

2/2kcan be obtained

limit p, using the identity (22.15.2) in [10].)

Since Lpl

is real, it does not contribute to the phase, whi

given by

argl,p = kz + l+2

2 1+ 2( )+G ( ) .

G() denotes the Gouy phase, which enters through the factors

powers ofw() and does not affect the shape of the trajectories

slight longitudinal stretching; in what follows, we will neglect

Identification of the Poynting components via (1.2) gives

v =

1+ 2, v =

l

,


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The solutions are easily found, and give the trajectory pa

point 0, 0 in the waist plane =0 as

( ) = 0 1+ 2, ( ) = 0 +

l

02tan

1.

These are curves wound on hyperboloids (figure 2); the windin

|| increases, and the total angle turned through between =-

=l

02

.

Note that this rotation does not involve the Gouy phase.

The trajectories are generally not straight, because one o

components in (1.4) is non-zero:

r + 2 r = 0, r r 2 = 1 l/02( )

2

1+ ( )3/2

.

However, the hyperboloid with 0l is exceptional: its trajecto

lines. For the special beam with p=0, this was noted in [6] and

terms of skew rays (see also figure 6 of [12]), which as we wil


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magnitude ||, which are not energy trajectories in the usual se

integral curves of the Poynting field.

4. Bessel beams: geometrical-optics rays

In the geometrical optics regime of large l, the Bessel function

exponentially small ifqrl, there are radial oscillatio

the Debye approximations [10]

Jl qr( ) 2

cos q2r2 l2 l tan1 q2r2 l2 /l

q2r2 l2( )l >>1,qr> l( ).

The oscillations correspond to the interference of geometrical

treat separately by regarding the cosine as the sum of two com

(this is equivalent to the decomposition into outgoing and ingo

functions: Jl = Hl1( )

+ Hl

2( )

/2).

From (2.1), the total phase of each contribution is

2 2 2 2 2 1 2 2


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and the trajectories are the solutions of

r z( ) = q2r z( )

2 l

2

r z( ) k2 q2, z( ) = l

r z( ) k2 q2.

It is convenient to specify the trajectories by the height z

radius, where r z0( ) = 0 , that is r=l/q. Thus

drr

q2r2 l2q/l

r z( )

=

z z0( )

k2 q2,

leading to the explicit solutions

r z( ) = q

k2 q2

z z0( )2 +l2 k2

q2

( )q4

z( ) = 0 + tan1 z z0( )q

2

l k2 q2

.

With the dimensionless variables

r l z

l k2q

2


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These expressions satisfy (1.4), confirming that the geometrica

straight lines. For fixed 0, the trajectories for different 0 agai

skew rays lying on hyperboloidal surfaces (figure 3). The total

hyperboloids fills the space outside the cylinder =1, that is r=

caustic (figure 4) where Jl(qr) is large for large l.

5. Laguerre-Gauss beams: geometrical-optics rays

In geometrical optics, interpreted as the regime where waves c

as the superposition of locally plane waves, l and p are simulta

study this,, we need some facts about the function

gl,p x( ) exp 12x( )x l/2Lpl x( ),

involving the Laguerre polynomialLpl

x( ); derivations are outli

Appendix. g(x) oscillates in a certain range x< x< x

+, and d

exponentially outside this range. The limits are given by

x = l + 2p +1 l + 2p +1( )2 l 1( )

2 lsp

l

,

in which the functions s(u) are


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Lpl

x( )

Xl,p x( )+

Xl,p x( )( )*

,

which are locally exponential in form. The rate of oscillation (

wavenumber) is

ddx

argXl,p x( ) = x+

x( ) x

x( )

2x.

Thus, introducing the physical dimensionless variables and Poynting vector components are

v =1

1+ 2

+2 2( ) 2 2( )

, v =

l

,

in which

2= ls 1+

2( ).

Thus, the equations determining the trajectories are

( ) = 11+ 2

( )+2 ( )

2( ) ( )2 2( ) ( )

,


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( ) = l

Ap

l,0

Apl,0

2

0( )2 +1

( ) = 0 + tan1

Ap

l,0

0( )

,

in which

A u,0( ) =s+ + s s+ s( )

2 40

2

2 1+ 02( )

=1+ 2u 4u 1+ u( )0

2

1+ 02( )

.

It is clear that all trajectories have closest approaches to the ax

heights 0 < 2p

l1+

p

l

. Again, use of (1.4) confirms that th

straight.

Figure 6 shows the + and - ray families for a typical 0.

hyperboloids, with the difference from the Bessel beam that no

hyperboloids for each 0, and the dimensions of the hyperbolo

The envelope of all the hyperboloids is a caustic surface with t


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corresponding to large values of the Laguerre polynomials at t

of the oscillatory range. The totality of all the skew rays on all

fills the space between the two caustic surfaces.

6. Concluding remarks

The results reported here show that the energy trajectories of B

Laguerre-Gauss beams can be calculated exactly, and show so

behaviour than might have been anticipated. Especially signifi

differences between the Poynting vector lines of the exact beam

curved) and those of the component ray families in the geomet

approximation (overlapping families of straight skew rays).

Nevertheless, the behaviour is not typical. The Bessel fu

Laguerre polynomials are real and therefore have zeros on nod

(hyperboloidal in form), in contrast to typical complex wavefu

zeros are lines (of phase singularity, also called wavefront dis

optical vortices). Associated with this is the fact that the locall

that are superposed in the oscillatory regions of the beams are

amplitude. Typically, the waves are of unequal amplitude, and

Poynting trajectories are wavy. A simple - almost trivial - exam


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possesses maxima (figure 8a) along lines parallel to the comm

direction y. This contrasts with the Poynting trajectories, which

parametrised by the height y0 where they intersect the y axis ar

y y0 =1

a 1 b2

( )1+ b

2( )x +b

a

sin2ax

.

These are wavy lines (figure 8b) slanted towards the direction

intense plane wave component in (6.1). And of course these in

different from the superposition of the two families of geometr

(families of parallel straight lines, figure 8c) representing the t

waves. The particular wave (6.1) has no vortices. When vortic

trajectories generally spiral slowly into and out of them [14] -

that does not occur for the axial vortices in the beams consider

We expect other families of beams (for example, Hermi

Mathieu) to exhibit similar behaviour, that is, curved Poynting

the exact waves and, in the geometrical-optics approximation,

straight rays enveloping caustic surfaces.

Acknowledgments. MVBs research is supported by the Roya


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We include this section to present the most compact derivation

need; for a rigorous treatment, and references to the extensive

literature on Laguerre polynomials, see [15]

From the Rodrigues formula [10] for the Laguerre polyn

function gl,p(x) in (5.1) can be written as a pth derivative, whic

expressed as a contour integral:

gl,p x( ) exp 12

x( )x l/2Lpl x( ) =exp 1

2x( )xl/2

p!

dp

dxp

exp(

=

exp 12

x( )xl/2

2i

dz

z x( )p+1

exp z( )z

p+l

=

exp 12

x( )xl/2

2idz exp F z,x( ){ },

where the contour is a loop surrounding z=x, and

F z,x( ) z + p + l( ) logz p +1( ) log z x( ).

In the geometrical-optics regime (large l and p), Fis largintegrand is fast-varying, the contour can be deformed so as to

saddle points, where the exponential is locally stationary; the i

dominated by contributions from these points There are two s


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The function gl,p(x) is now given by the standard saddle

[16, 17] (local gaussian expansion of the integrand) as

gl,p x( ) exp12

x( )xl/22

Imexp F z

+x( ),x( ){ }

2F z = z+

x( ),x( )/

x < x < x+( )

or, more explicitly (after some algebra),

gl,p x( ) 2

p+

l( )

12

p+l( )+ 14

exp1

2 l{ }p

12p+ 1

4 x+

x( ) x x

( )[ ]1/4

sin x


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the radial oscillations of the Bessel beams. This is an example

phenomenon that high derivatives of almost every smooth func

sinusoidal; for a general theory, see [18].

References

1. Green, H. S. & Wolf, E.,1953, A scalar representation o

fields Proc. Phys. Soc. A66, 1129-1137.

2. Wolf, E.,1959, A scalar representation of electrromagne

Phys. Soc. 74, 269-280.

3. Allen, L., Padgett, M. J. & Babiker, M.,1999, The orbita

momentum of light Progress in Optics39, 291-372.

4. Madelung, E.,1926, Quantentheorie in hydrodynamische

Phys.40, 322-326.

5. Holland, P.,1993, The Quantum Theory of Motion. An A

Broglie Bohm Causal Interpretation of Quantum Mecha

Press, Cambridge).


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8. Durnin, J.,1987, Exact solutions for nondiffracting beam

theoryJ. Opt. Soc. Amer.4, 651-654.

9. Durnin, J., Miceli Jr, J. J. & Eberly, J. H.,1987, Diffract

Phys. Rev. Lett.58, 1499-1501.

10. Abramowitz, M. & Stegun, I. A.,1972,Handbook of ma

functions (National Bureau of Standards, Washington).

11. Leach, J., Keen, S., Padgett, M. J., Saunter, C. & Love,

Direct measurement of the skew angle of the Poynting v

helically phased beam Opt. Express14, 11919.

12. Courtial, J. & Padgett, M. J.,2000, Limit to the orbital an

mnomentum per unit energy in a light beam that can be

small particle Opt. Commun.173, 269-274.

13. Allen, L., Beijersbergen, M. W., Spreew, R. J. C. & Wo

P.,1992, Orbital angular momentum and the transformat

Laguerre modes Phys. Rev. Lett. A45, 8185-8189.

14. Berry, M. V.,2005, Phase vortex spiralsJ. Phys. A, L745

15. Bosbach, C. & Gawronski, W.,1998, Strong asymptotic

polynomials with varying weights J Comp and Appl M


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18. Berry, M. V.,2005, Universal oscillations of high deriva

Soc. A461, 1735-1751.

Figure captions

Figure 1. Exact Poynting trajectories for Bessel beams. The tr

fill space, are helices wound on cylinders corresponding to diff

the helices on different cylinders have different pitches z, acc

Figure 2. Exact Poynting trajectories for Laguerre-Gauss beam

trajectories, which fill space, wind on hyperboloidal surfaces c

different values of02/l . (a): 0

2/l=0.5; (b): 0

2/l=1;(c) 0

2/l=

rotation angle of the trajectory, (equation (3.8)) is unrelated

phase. The trajectories are all curved except when 02/l=1 (as

Figure 3. Geometrical rays for Bessel beams, calculated from

and 0=-3 for a range of starting angles 0. All rays are straigh

betweeen . The totality of rays for all 0, 0 fill space outs

=1 that is r=l/q


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approximation (A.4-A.6) in oscillatory region; dashed lines: th

limits x-=19.62 and x+=122.38.

Figure6. Geometrical rays {(),()} for Laguerre-Gausscalculated from (5.9) and (5.10) with 0=1.5 and a range of sta

and x+/l=4, x-/l=0.25. All rays are straight, and turn bybetwe

totality of rays for all 0, 0 fill space between the two caustic

in figure 7.

Figure 7. (a) Geometrical Laguerre-Gauss rays: radial coordin

different 0, showing the caustics at =x

l1+ 2( ) . (b) Ca

Figure 8. Representations of the unequal superposition (6.1) o

waves with a=b=1/2. (a) Intensity ||2. (b) Thick curves: Poyn

Im*; dashed curves: wavefronts arg=constant. (c) Geom

rays: thick lines represent the right-travelling (stronger) wave a

represent the left-travelling (weaker) wave.


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-1 0

1

-1

0

1

0

2

z

y x


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figure

2

-2

0

2

0

-2

-1

-2

-1

-2

-1

01

2

0

2

-2

-2

0

01

2

0

2

0

0

12

a

b

c

-2


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0

0

5

-5

-5

-5

0

5


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figure 4

-3 -2 -1 0 1 20

1

2

3

4


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figure 5

0 50 100

g50,10(x)


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-20

-2

0

2

-2

1

0

1

2


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fig

ure

8

10

5

0

5

10

10505

10

10

5

0

5

10

10505

10

10

5

0

5

10

10505

10

a

b

c

x

y

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