Production of optical phase space vortices with non-locally distributed mode converters

0.19 MB·English

by  R. M. Gomes

Production of optical phase space vortices with non-locally distributed mode
converters
R. M. Gomes,1 A. Salles,2 F. Toscano,1 P. H. Souto Ribeiro,1 and S. P. Walborn1
1Instituto de F´ısica, Universidade Federal do Rio de Janeiro,
Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil
2Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
Optical vortices have been observed in a wide variety of optical systems. They can be observed
directlyinthewavefrontofopticalbeams,orinthecorrelationsbetweenpairsofentangledphotons.
Wepresentanovelopticalvortexwhichappearsinanon-localplaneofthetwo-photonphasespace,
composed of a single degree of freedom of each photon of an entangled pair. The preparation of
thisvortexcanbeviewedasa“non-local”ordistributedmodeconverter. Weshowhowthesenovel
1
optical vortices of arbitrary order can be prepared in the spatial degrees of freedom of entangled
1
photons.
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2

n
I. INTRODUCTION
a
J
4 Optical vortices arise from phase singularities, typically associated to the orbital angular momentum (OAM) of
1 light. The OAM of light beams is a classical feature that has only been studied in detail recently [1]. Among
other applications [2], a key motivation for this study is related to applications in quantum information[3]. These

] potentialapplicationshavemotivatedthe studyoftheOAMinthecontextofthespatialcorrelationsbetweenphoton
h
pairs produced in spontaneousparametric down conversion(SPDC)[4–6]. Experimentally, entanglement in the OAM
p
- of photon pairs has been observed [7–13] and quantified in recent experiments [14, 15]. These correlations are a
nt consequence of the conservation of the OAM in the SPDC process, which arises from the transfer of the angular
a spectrum of the pump beam to the wave function of the down-converted photons [16, 17].
u InvestigationsofOAMcorrelationsbetweendown-convertedphotonsgenerallyrelyononeoftwotypesofdetection
q systems. In the first type, holographic masks and single mode optical fibers are used to project the down-converted
[
fieldsontomodeswithwell-definedOAM[7],suchastheLaguerre-Gaussian(LG)modes. Inotherexperiments,point-

1 likedetectorsarescannedinthetransversedetectionplane,andthecharacteristicdoughnutmodesareobservedinthe
v coincidencecountdistributions[9,11]. InRefs. [9,11],onedetectorwaskeptfixedwhiletheotherwasscannedinthe
5 two-dimensionaldetectionplane. The intensitydistributionofadoughnutmodedoesnotappearinthesingle-photon
2
counts but ratheronly inthe joint distributionof coincidence counts. The correlationsobservedwith these twotypes
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of detection systems have been shown to be characteristic of entanglement.
2
A different kind of non-local optical vortex has been recently observed [18]. Here the term “non-local” is related
.
1 to the two-dimensional space in which the optical vortex appears. A doughnut-shaped coincidence distribution and
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phase dislocation was observed, but the pump beam is not prepared in an LG mode and neither of the twin photons
1
are projected onto states with well-defined OAM. In all previous experiments, OAM correlations were observed only
1
if either the pump, or at least one of the down-converted photons is prepared or projected onto a state with OAM
:
v different from zero. This is not the case in Ref. [18], where the optical vortex appears in the coordinate plane
i
X composed by the position variable of photon 1 and the wave vector variable of photon 2.
In this paper, we generalize the scheme employed in Ref. [18] and show that the presence of the optical vortex
r
a is due to the implementation of a non local mode converter. We demonstrate that it is possible to create non-local
optical vortices of arbitrary order through manipulation of the pump laser beam. In addition, we show that this can
be done separately and simultaneously in two spatial dimensions.
II. NON-LOCAL OPTICAL VORTEX
Fig. 1 shows a sketch of the experimental set-up used in Ref.[18] to produce a nonlocal optical vortex. The
pump laser is prepared in a first order Hermite-Gaussian mode (HG) HG , which is achieved by passing half of the
10
Gaussian laser beam through a thin glass slide so that a phase difference of π is produced between the two halves
of the pump beam. Further propagation in free space provides spatial filtering necessary to clean up the beam,
producing a goodquality HG mode. This beam is then focused onto a nonlinearcrystalwith a cylindricallens and
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produces entangled photons through spontaneous parametricdown-conversion(SPDC). Twin photons with the same
wavelength are collected and detected through narrow band interference filters. For photon 1, an imaging system
is used to map the source distribution (x,y) onto the detection plane. For photon 2, a Fourier lens maps the wave
vector (q ,q ) distribution of the source onto the detection plane. Both detectors are scanned in a line along the
x y
2
x x
2 2
x q
1 1
Coincidences Coincidences
D1 position D1 position
gaussian pump beam
classical beam
gaussian b)
double
slit
fringes shifted fringes
FIG. 1: Sketch of the experimental set-up (left) and results (right) for the observation of a non-local optical vortex [18]. The
toprightillustratesthecoincidencedistributionsobtainedinthex1,x2 andq1,x2 planes,aswellasthedoubleslitinterference
obtained in coincidence counts. The blue and red curves correspond to the interference patterns obtained with different
positions of detector 2. In the case of a HG pump beam, a shift is observed, while in the case of a Gaussian pump beam,
no shift is observed (blue box). For comparison, the grey box illustrates double slit interference with a a) Gaussian and b)
Laguerre-Gaussian beam LG10.
vertical(eitherxorq )detectionaxis,givingatwo-dimensionalnon-localdistributionofcoincidencecounts. Atypical
x
distribution is sketched in Fig. 1, showing the doughnut shape that is an indication of the presence of an optical
vortex. For comparison, Fig. 1 also sketches the coincidence distribution when both photons are measured in the
position basis x. We see that in this case the coincidence distribution mimics the pump beam inside the crystal [16].
It is important to notice that the pump beam is not prepared in an LG mode and the down-converted photons are
not projected onto LG modes.
The simple observation of a doughnut shape is not enough to prove the existence of the optical vortex. It is also
necessary to obtain information about the azimuthal phase distribution. In Ref.[18], this was done through a double
slit interference experiment. For a classical light beam, the phase singularity associated to an LG beam can be
evidenced by observing a shift in the interference fringes [19], as illustrated in the inset of Fig. 1. This method was
used to evidence the phase dependence of the nonlocalvortex in Ref. [18]. As shownin Fig. 1, a double slit aperture
is insertedinto the path ofphoton 1 andthe detectoris scannedto registerthe interferencefringes in the coincidence
distribution for two different positions of detector 2. The measurements are performed for two configurations of the
pump beam. First pumping with a zero orderGaussian beam and second with the HG mode. In the first case, the
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sets of observedfringes are in phase,while in the secondthe observedfringes are out ofphase (see figure 1), as in the
case of a classical LG1 beam.
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We will show in section IV that this experiment can be viewed as the implementation of mode conversionbetween
a nonlocal HG and LG mode. To do so, let us briefly review mode conversionof HG and LG beams.
III. MODE-CONVERSION BETWEEN HERMITE-GAUSSIAN AND LAGUERRE-GAUSSIAN MODES
In classical optics, the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams are solutions to the paraxial
Helmholtzequation[20]incartesianandcylindricalcoordinates,respectively. WewillwritetheHGmodesasHG =
nm
HG (x)HG (y), where n (m) is the number of zeroes of the Hermite polynomials H (x) (H (y)). The order of the
n m n m
HG modes is n+m. The LG modes can be written as LGℓ(x,y), where the order is ℓ +2p. It is well known that
the Laguerre-Gaussianbeams carry orbital angular momentpum of ℓ~ per photon [1]. | |
Beijersbergenet al. showed how one can exploit the Guoy phase of a paraxialbeam to convertHG modes into LG
modes or vice-versa [21]. The LG modes, for example, can be described as a superposition of HG modes of the same
3
order. A simple example is the mode LG1, which is given by the superposition of HG and HG with a relative
0 10 01
phase of π/2: LG1 = 1 (HG +iHG ). At the same time, the diagonal HG mode DHG is defined as
0 √2 10 01
x+y x y
DHG (x,y)=HG , − (1)
nm nm
(cid:18) √2 √2 (cid:19)
and DHG is given by DHG = 1 (HG +HG ). One can see that the DHG mode can be converted into
01 01 √2 10 01 01
the LG1 mode by introducing a relative phase between the HG components. This relative phase can be introduced
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using a mode converter, which is a set of cylindrical lenses which produces an astigmatic region [21], giving rise to
the relative phase shift.
In general, the DHG modes are given by [21]
N=n+m
DHG (x,y)= b(n,m,j)HG (x,y), (2)
nm N j,j
Xj=0 −
while the higher order modes LGℓ are given by
p
N=n+m
LGℓ(x,y)= ijb(n,m,j)HG (x,y), (3)
p N j,j
Xj=0 −
1/2
with ℓ=n−m, p=min(n,m) and b(n,m,j)=h(2NN−nj!m)!j!!i 1jddtjj[(1−t)n(1+t)m]|t=0. For the higher-orderbeams,
eachcomponentHG inexpansion(2)picksupaphaseij uponpropagationthroughthe astigmaticregionofthe
N j,j
mode converter [21]. I−n this way, one can transform a general DHG mode into an LGℓ mode of the same order.
nm p
IV. NON-LOCAL CONVERSION OF HIGHER-ORDER MODES IN TWO DIMENSIONS
Inthis section,wewillshowageneralmethodofmode conversionusingentangledphotons,suchasthoseproduced
in SPDC. This method is a “non-local”or distributed mode converter,since the LG mode is generatedin a non-local
phase space of the spatial degrees of freedom of the entangled photons. This generalizes our previous result reported
in Ref. [18] to the case of higher-order modes and two spatial dimensions.
A fundamental characteristicof the SPDC process is that the transverseprofile of the pump field is transmitted to
the two photons state [16]. In the monochromatic and paraxial approximations, the two photon state produced by
SPDC can be written as
ψ = dq dq Ψ(q ,q )q q , (4)
1 2 1 2 1 2
| i Z Z | i| i
whereq isthetransversewavevectorofphotonj,and q representsasingle-photonstatewithwell-definedfrequency
j j
| i
and polarization. The wave function in wave vector space is given by
Ψ(q ,q )=v(q +q )γ(q q ), (5)
1 2 1 2 1 2
−
where v is the angular spectrum of the pump beam and γ is the phase matching function [16]. Let us assume that
thedown-convertedphotonsaredegenerateandquasi-collinear,sothatthephasematchingfunctionsimplifiesto[17]:
γ(q) sinc[λL(q2+q2)/8π], where λ is the wavelengthof the pump beam andL is the length of the crystal. Using a
∝ x y
Gaussianspatialfilter onthe down-converterphotons describedin Ref.[18] wecantailorthe phasematching function
γ(q) so it can be well approximated by a Gaussian: γ(q) HG (q,λ,w), where the waist is given by w 8π/λL.
00
≈ ≈
LetusfurtherassumethatthepumpbeamispreparedinanHGmodev(q)=HG (q,λ,w ),whichischaracterized
nm 0
by the wavelength λ and waist w of the pump beam. In wave vector representation, the HG beams are
0
w q w q w02(qx2+qy2)
HGnm(qx,qy)=DnmHn 0 x Hm 0 y e(cid:18)− 4 (cid:19)e[−i(n+m+1)η(z)], (6)
(cid:18) √2 (cid:19) (cid:18) √2 (cid:19)
where w is the beam waist, H is a Hermite polynomial and D is a constant [22]. Mathematically, the HG beam
0 n nm
with wavelength λ and waist w can be rewritten as HG (q,λ,w ) = HG (q/√2,2λ,√2w ). In this case the
0 nm 0 nm 0
wave function is given by
q +q q +q q q q q
Ψ(q ,q )=HG x1 x2, y1 y2 HG x1 − x2, y1 − y2 . (7)
1 2 nm 00
(cid:18) √2 √2 (cid:19) (cid:18) √2 √2 (cid:19)
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From here on, all HG and LG functions are characterizedby the wavelength of the down-converted photons λ =2λ
c
and waist w =√2w . Using HG (x,y)=HG (x)HG (y) we can rewrite
c 0 nm n m
q +q q q q +q q q
Ψ(q ,q )=HG x1 x2, x1− x2 HG y1 y2, y1− y2 . (8)
1 2 n0 m0
(cid:18) √2 √2 (cid:19) (cid:18) √2 √2 (cid:19)
By the definition of the DHG modes (1), we recognize that wave function (8) is equivalent to
Ψ(q ,q )=DHG (q ,q )DHG (q ,q )
1 2 n0 x1 x2 m0 y1 y2
n m
= b(n,0,j)HG (q ,q ) b(m,0,t)HG (q ,q )
n j,j x1 x2 m t,t y1 y2
Xj=0 − Xt −
n m
= b(n,0,j)HG (q )HG (q ) b(m,0,t)HG (q )HG (q ). (9)
n j x1 j x2 m t y1 t y2
Xj=0 − Xt −
Wave function (9) is a product of DHG modes which appear in non-local coordinate planes composed of one spatial
dimension of each down-converted photon. Using expansions (2) and (3), we see that introducing relative phases of
in and im in the x and y spatial dimensions of photon 2, these DHG modes can be converted into LG modes. These
relative phases can be introduced by a Fourier transform operation on photon 2 as was done in Ref. [18]. Indeed, as
theHermite-GaussianfunctionsareeigenfunctionsoftheFouriertransform ,i.e. [HG ]=inHG ,thetwo-photon
n n
F F
wave function is transformed to
n m
Φ(q ,ρ )= ijb(n,0,j)HG (q ,x ) itb(m,0,t)HG (q ,y )
1 2 n j,j x1 2 m t,t y1 2
Xj=0 − Xt=0 −
=LGn(q ,x )LGm(q ,y ). (10)
0 x1 2 0 y1 2
Thus, performing a bidimensional Fourier transform on photon 1 implements the required relative phases. This
operation can be viewed as a nonlocal mode converter, where the astigmatism is introduced between the spatial
degrees of freedom of photons 1 and 2, rather than the x and y coordinates of a single classical beam.
V. CONCLUSION
We have presented a method to produce nonlocal optical vortices using entangled photon pairs produced in para-
metric down-conversion. The vortex appears in a coordinate plane composed of one spatial degree of freedom of one
photonand the correspondingtransversewave vectordegree offreedom ofthe other photon.We have shownthat this
kind of vortex is obtained from a nonlocal or “distributed” mode converter in which a single lens is placed in each
of the down-converted photons. Pumping a nonlinear crystal with an appropriate HG beam, produces a pair of
nm
nonlocal optical vortices of order n and m in different spatial directions.
Acknowledgments
Thisworkwasperformedwithsupportfromthe funding agenciesCNPq,CAPES,FAPERJ,the BrazilianNational
Institute for Science and Technology of Quantum Information, (INCT-IQ), and EU STREP COQUIT under FET-
Open grant number 233747.
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