HypergeometricGaussian Modes
Abstract
We studied a novel family of paraxial laser beams forming an
overcomplete yet nonorthogonal set of modes. These modes have a
singular phase profile and are eigenfunctions of the photon orbital
angular momentum. The intensity profile is characterized by a single
brilliant ring with the singularity at its center, where the field
amplitude vanishes. The complex amplitude is proportional to the
degenerate (confluent) hypergeometric function, and therefore we
term such beams hypergeometric gaussian (HyGG) modes. Unlike the
recently introduced hypergeometric modes (Opt. Lett. 32,
742 (2007)), the HyGG modes carry a finite power and have been
generated in this work with a liquidcrystal spatial light
modulator. We briefly consider some subfamilies of the HyGG modes
as the modified Bessel Gaussian modes, the modified exponential
Gaussian modes and the modified LaguerreGaussian modes.
ocis: 050.1960, 230.6120
In the recent years there has been an increasing interest in laser beams especially tailored to experiments. In particular, experimentalists are looking for laser beams which are either nondiffracting or have a definite value of the photon orbital angular momentum (OAM) along the propagation direction. Such special laser beams have found useful applications in optical trapping, image processing, optical tweezers, metrology, microlithography, medical imaging and surgery, wireless and optical communications he95 (); durnin87 (); salo99 (); durnin87a (); davis00 (). Moreover, beams carrying definite photon OAM present novel internal degrees of freedom that are potentially useful for quantum information applications molinaterriza02 (); molinaterriza07 (). Motivated by these issues, there has been an increasing interest in generating and studying laser beams corresponding to particular solutions of the scalar Helmholtz paraxial wave equation other than the well known HermiteGaussian (HG) and the LaguerreGaussian (LG) modes. Miller has solved the 3D Helmholtz paraxial wave equation in 17 coordinate systems, 11 of which were based on orthogonal coordinates miller (). Other examples of recently investigated paraxial beams are Parabolic, Mathieu, InceGaussian, HelmholtzGaussian, LaplaceGaussian, and pure light vortices (see kotlyar07 () and references therein).
In this Letter we introduce a novel family of paraxial beams which are solutions of
the scalar Helmoltz paraxial wave equation and are also eigenstates
of the photon OAM. The field profile of these beams is proportional
to the confluent hypergeometric function so we call them
HypergeometricGaussian modes (HyGG). Unlike the hypergeometric
modes studied in kotlyar07 (), our HyGG modes carry a finite
power so that they can be realized in practical experiments.
The
field of the HyGG modes is given by
(1)  
where is integer, is real valued, is the
gamma function and is a confluent
hypergeometric function abramowitz (). In Eq. (1) we
used dimensionless cylindrical coordinates , ,
, where is the beam waist and is the beam Rayleigh range. We notice the
characteristic factor in the field, so that the integer
values of are identified with the eigenvalues of the photon OAM
in units of . The normalization
condition inner_product () fixes the constant to
up to a constant
phase factor. The HyGG modes are an overcomplete not orthogonal set
of modes. The inner product of two normalized HyGG modes is given by
.
The asymptotic behavior of the intensity
of the HyGG modes as
at fixed is
in general, and changes into
the Gaussian behavior
when
is a non negative even integer (see below). At the beam center
() the field of the HyGG modes vanishes as
as expected for the eigenmodes of the photon OAM (for
). Because all zeroes of the hypergeometric function
occur for real values of , a characteristic
feature of the HyGG modes given by Eq. (1) is that their
intensity never vanishes in the transverse plane,
except at the beam axis and at infinite. This confers to
the intensity of the HyGG modes the typical doughnut shape (a single
brilliant ring) for any values of and , except when
the HyGG mode reduces to the TEM Gaussian mode.
The limit
of the field at the pupil plane
is given by
(2) 
This feature is very useful because this pupil function can be
generated by applying the singular phase factor to a
Gaussianparabolic transmittance profile of the order . In
particular, the HyGG modes with are simply generated by
applying the phase factor to a TEM laser field at
its waist plane.
The HyGG modes can be expanded in the complete
basis of the LaguerreGauss (LG) modes. In general, the mode
HyGG is a superposition of the infinite LG modes with
same and any integer . In fact, when both the HyGG and
the LG modes are normalized, we have with coefficients
given by
(3) 
The HyGG modes can exhibit different features when the mode parameters and are changed. It is then convenient to separate the HyGG modes in a few subfamilies having similar properties.

This mode is the well known Gaussian mode TEM. 
, odd.
The pupil field of these modes at is . At planes the modes are linear combinations of the modified Bessel functions and , where . We may call this subfamily of modes the modified Bessel Gauss (MBG) modes. Unlike the well known Bessel modes, these modes carry a finite power and are not diffraction free. When at fixed , the intensity of these modes vanishes according to . 
, even.
The pupil field of these modes at is . At planes the modes are linear combinations of exponential dependent terms. We may call this subfamily of modes the Modified Exponential Gauss (MEG) modes. When at fixed , the intensity of these modes vanishes according to . 
, even.
The pupil field of these modes is given by Eq. (2). When is a non negative even integer, the confluent hypergeometric function reduces to a Laguerre polynomial. We will refer to these modes as to the modified LaguerreGauss modes (MLG). The asymptotic behaviour of the intensity of the MLG modes as at fixed is the same as for the usual LG modes (i. e. ). Unlike the LG modes, however, the MLG modes have a singlering intensity profile for any admitted value of . The MLG modes can be expressed as the linear superposition of a finite number of LG modes, namely, the LG modes having the same and integer . In fact, when is a non negative even integer, Eq. (3) reduces to(4) where . The quantities form the entries of a non singular matrix. It is then obvious that this subfamily of HyGG modes forms a complete, yet not orthogonal, set of functions in the transverse plane and that the full set of HyGG modes is therefore overcomplete.
In our experiment, a TEM linearly polarized laser beam from a frequency doubled Nd:YVO ( nm, Model Verdi V5, Coherent) was used to illuminate a grey scale computer generated hologram (CGH) sent onto the LCD microdisplay of a spatial light modulator (SLM) (HoloEye Photonics LCR 3000), with pixels in a rectangle 18.2411.40 mm wide. The SLM was located in the waist of the incident beam. We performed two series of measurements, according to the beam waist values at the SLM position. We measured by best fit between the error function and the integral curve of the intensity profile of incident beam as seen by CCD camera.The measured values were mm and mm, corresponding to Rayleigh ranges cm and cm, respectively. The theory was compared with the experimental data just using as parameters the measured values reported above, without further best fitting. We focused our attention on the HyGG modes with .
In accordance with the theoretical predictions, we observed an
intensity profile in the transverse plane essentially made of a
single bright annulus, whatever the value of we used or the
observation plane. Some instances of the observed intensity
profiles are shown in Fig. 1. The ring diameter of the
beam as a function of is reported in Fig 2. The
diameter of the ring was defined as the maximum distance between
any two opposite maxima of the intensity profile. The scaling law of
versus turned to be in good agreement with the theoretical
predictions. We measured also the ratio between the diameter
of the luminous ring and the gaussian beam size at different
planes. The measurements were made by switching on and off the
CGH to compare the intensity profile of the HyGG mode with the
gaussian profile TEM beam profile at the same plane.
Figure 3 shows that the ratio was the same
in all planes, as predicted by theory when . The constant
value of the ratio obtained from the experiment was
, which is close to the theoretical
prediction
.
In summary, we studied a novel family of paraxial beams having hypergeometric field profile. This set of modes is overcomplete and nonorthogonal and all modes carry a finite power. In spite of their complicated field profile, these mode have a simple profile at the pupil plane. Finally we have experimentally produced these modes for different values of indexes. The agreement between experimental results and theoretical predictions turned to be satisfactory.
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