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Broadband nulling of a vortex phase mask

Grover A. Swartzlander, Jr.

College of Optical Sciences, University of Arizona, Tucson, Arizona 85721

Received May 24, 2005; revised manuscript received June 15, 2005; accepted July 18, 2005

A pulse transmitted through a helical vortex phase mask undergoes a temporal Hilbert transform. The flu-

ence transmitted into the unfavorable plane wave mode is found to increase as the square of the bandwidth

and, to first order, is independent of the topological charge. © 2005 Optical Society of America

OCIS codes: 100.5090, 070.6610, 260.2030, 350.1260.

Optical vortices enjoy a wide range of applications for

quasi-monochromatic light, including optical spatial

filtering,1optical tweezers,2quantum cryptography3

and communication,4high-resolution spectroscopy,5

and semiconductor patterning.6This broad range of

uses is afforded by robust attributes: vortices exist as

natural modes in many systems, they exhibit a dis-

tinct point of destructive interference in coherent

light, and they may be readily produced in the labo-

ratory by various techniques. Additional applications

may be possible once the broadband properties of vor-

tices are fully explored. Colored vortices and other

optical singularities have been investigated.7Spec-

tral anomalies have been reported,8–10and temporal

coherence properties under broadband illumination

have been measured.11Here we explore the broad-

band transmission properties of a vortex phase mask,

giving special attention to the amount of light trans-

mitted into the nonvortex mode. This latter mode

limits the use of a vortex phase mask for spatial fil-

tering applications such as the search for extrasolar

planets.12The topological dispersion11of the vortex

phase mask provides a means to both null the peak of

a pulse via a temporal Hilbert transform and signifi-

cantly attenuate the on-axis fluence.

Optical vortices are modes that include a broad

family of separable cylindrical wave functions.13The

complex amplitude of the scalar field of such modes

may be expressed as a separable function in cylindri-

cal coordinates ?r,?,z?:

Fm?r,?,z? = Am?r,z?exp?im??,

?1?

where m is a mode number called the topological

charge. A lossless phase mask is frequently used to

transform an initial distribution, Fm?, into Fm

=t???Fm?where the azimuthally dependent transmis-

sion function is given by t???=exp?i?m−m????. Typi-

cally the mask is used to create a vortex of charge m

on a beam having m?=0. For example, the initial dis-

tribution may represent a uniform plane wave or a

Gaussian beam. We thus assume m?=0 and write

t??? = exp?im??.

?2?

A single mask that satisfies Eq. (2) across a broad

spectrum does not yet exist. On the other hand, it is

simple to design such a mask for a single wavelength

or over a small band of wavelengths. For example, a

vortex mask may be made by azimuthally varying

the thickness of a substrate by an amount

d??? = dbase+ ?d?1 − ?/2??,

?3?

where ?d=m0?0/?ns??0?−n0??0??, dbaseis the mini-

mum thickness, m0is the topological charge pro-

duced at the design wavelength ?0, and the refractive

indices of the substrate and the surrounding media

at the design wavelength are ns??0? and n0??0?, re-

spectively. The effective topological charge generated

at any frequency ? is given by

?0?

m??? = m0

?

ns??? − n0???

ns??0? − n0??0??,

?4?

where ?0=2?c/?0is the design frequency and ? is

the angular frequency associated with the light

source. We note that m??? is not necessarily an inte-

ger. For most materials the factor in brackets in Eq.

(4) differs much less than the ratio ?/?0. For math-

ematical convenience we therefore assume ns???

−n0????ns??0?−n0??0?, and we write

m??? = m0?/?0.

?5?

If m0is an integer, then m=m0p is also an integer

whenever

?=p?0, where

1,2,3,... . The frequencies satisfying ?=p?0will be

called principal frequencies.

Although the topological charge varies continu-

ously acrossthespectrum,

advantageous11,14to rewrite the field as a Fourier se-

ries of modes having an integer topological charge.

For this purpose we write

p=1/m0,2/m0,...,

it is sometimes

t??,?? =?

l=−?

?

Cl???exp?il??,

?6a?

Cl??? = ?2??−1?

−?

?

t??,??exp?− il??d?,

?6b?

where Cl??? shall be called the lth order vortex spec-

trum. Inserting Eq. (5) into Eq. (2), one may readily

show that

Cl??? = sinc?m0??/?0− l??.

?7?

The vortex spectra, C0??? and C±1???, are plotted in

Fig. 1 for the case m0=1. As expected, at ?=?0only

C+1has a nonzero value. We also see that C0=0 at the

principal frequencies, varying linearly in the vicinity

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0146-9592/05/212876-3/$15.00© 2005 Optical Society of America

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of ?−l?0. The discussion below makes use of this lin-

ear variation to null the peak of the transmitted

pulse along the optical axis.

To describe the spatiotemporal effects of the vortex

mask, we write the scalar input field

Ein?r,?,z;t? = G?t?F?r,z?exp?i?t?exp?− ikzz?,

?8?

where G?t? describes the temporal profile of the

beam, ? is the carrier frequency, and kzis the projec-

tion of the carrier wave vector along the optical axis.

Applying the paraxial approximation, we set kz

=n?/c, where n is the refractive index. The temporal

pulse and its spectrum are related by the Fourier

transform pair:

g??? =?

−?

?

G?t?exp?i?t?exp?− i?t?dt,

?9a?

G?t?exp?i?t? =

1

2??

−?

?

g???exp?i?t?d?.

?9b?

If the maximum thickness of the mask, dbase+?d, is

much less than the characteristic diffraction length,

the field at the output plane of the mask may be writ-

ten as

Eout?r,?,z = 0;t? =?

l=−?

?

El?r,?,z = 0;t?,

?10?

where we have assigned the plane z=0 to the output

plane of the mask. The transmitted vortex modes are

given by

El?r,?,z = 0;t? = Gl?t?F?r,z = 0?exp?il??exp?i?t?,

?11?

where

Gl?t? = ?2??−1exp?− i?t??

−?

?

g???Cl???exp?i?t?d?,

?12a?

gl??? = Cl???g???.

?12b?

Material dispersion in the thin layer has been ig-

nored in Eq. (12). When the source spectrum is band

limited and Cl??? is an odd function over the spectral

band, Eq. (12a) may be interpreted as a temporal Hil-

bert transform. In this case the peak of the pulse be-

comes zero valued.

The zeroth-order vortex spectrum, C0???, is of spe-

cial interest because it is the only mode whose inten-

sity does not vanish along the optical axis. From a

spatial filtering point of view, this limits the nulling

ability of a vortex mask. Thus we see that the Fourier

series decomposition in Eq. (6) provides a means of

quantifying the nulling efficiency of a vortex mask

without the need to determine the net field. The net

field is complicated and may contain vortices in dif-

ferent locations.14,15Below we determine the relative

fluence, ?, transmitted into this mode. For conve-

nience we assume that the input field is planar,

F?r,z=0?=1:

? =?

?

−?

=?

?

−?

−?

?

?El=0?r,?,z = 0;t??2dt

?

?Ein?r,?,z = 0;t??2dt

=?

−?

?

?

?Gl=0?t??2dt

−?

?

?G?t??2dt

−?

?

?gl=0????2d?

?

?g????2d?

.

?13?

Parseval’s theorem has been invoked to obtain the

right-hand side of Eq. (13).

The following analytical and numerical calcula-

tions establish important characteristics of Eq. (13).

Analytical results are made possible by assuming a

uniform band-limited spectrum. Let g???=?2???−1/2

for ?−??????+?? and g???=0 otherwise, such

that ?−?

ably simplified by assuming that ?=?0, integrating

over the variable ??=?/?−1, and expanding the in-

tegrand to O???3?, assuming that ???1 and m0is an

integer:

??g????2d?=1. An evaluation of ? is consider-

Fig. 2.

of the plane-wave mode through a vortex mask (design pa-

rameters, m0=1 and ?0), normalized to the net fluence of

the input pulse having a uniform band-limited frequency

distribution of half-width ?? and center ?.

Transmitted fluence (joules per square centimeter)

Fig. 1.

case m0=1.

Vortex spectra for the orders l=0,+1,−1 for the

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? =

1

2???

=?

?−??

?+??

sinc2?m0??/??d?

1

2?m0??2????

−??/?

??/?sin2?m0????

??? + 1?2

d?? ?

1

3?

??

??

2

.

?14?

Numerical integration of Eq. (13), whose values are

plotted in Fig. 2, reveals that the right-hand side of

relation (14) is an excellent approximation. Further-

more this result is seen in Fig. 2 to be valid in the

broad bandwidth regime when ???0. On the other

hand, when the bandwidth is small, direct integra-

tion shows that ???/?→0=C0

An ideal vortex-nulling filter requires that ??1

over a broad bandwidth. In the small-bandwidth re-

gime design parameter errors (?0or m0) significantly

affect the nulling efficiency. An extinction of more

than 8 orders of magnitude is shown in Fig. 2 when

m0=1, ??=0.1, and ?0is selected to coincide with the

center frequency of the source, ?. Errors in ?0/? as

small as 0.1% limit the extinction to, at best, 6 orders

of magnitude. The limited performance of the mask is

attributed to the linear frequency dependence of the

effective topological charge shown in Eq. (4). In prin-

ciple an ideal frequency-independent vortex mask

may be produced by using a high-dispersion material.

2??0?=sinc2?m0??.

The author is grateful to Greg Foo (University of

Arizona) and David Palacios (Jet Propulsion Labora-

tory) for their comments. This work was supported by

the U.S. Army Research Office and the State of Ari-

zona.G.A. Swartzlander’s

grovers@optics.arizona.edu.

e-mail addressis

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