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Optimal coupling of entangled photons into

single-mode optical fibers

R. Andrews

Department of Physics, Faculty of Agriculture and Natural Sciences, The University of the West Indies, St.

Augustine, Republic of Trinidad and Tobago, W.I..

randrews@fans.uwi.tt

E. R. Pike and Sarben Sarkar

Department of Physics, King’s College London, Strand, London WC2R 2LS, UK.

roy.pike@kcl.ac.uk, sarben.sarkar@kcl.ac.uk

Abstract:

We present a consistent multimode theory that describes the coupling of

single photons generated by collinear Type-I parametric down-conversion

into single-mode optical fibers. We have calculated an analytic expression

for the fiber diameter which maximizes the pair photon count rate. For a

given focal length and wavelength, a lower limit of the fiber diameter for

satisfactory coupling is obtained.

2004 Optical Society of America

OCIS codes: (190.0190) Nonlinear optics; (190.4410) Parametric processes

References and links

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18.

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23.

1. Introduction

The optical process of spontaneous parametric down-conversion (SPDC) involves the virtual

absorption and spontaneous splitting of an incident (pump) photon in a transparent nonlinear

crystal producing two lower-frequency (signal and idler) photons [1-3]. The pairs of photons

can be entangled in a multi-parameter space of frequency, momentum and polarization. In

type-I SPDC the photons are frequency-entangled and the signal and idler photons have

parallel polarizations orthogonal to the pump polarization. Entangled photons have been used

to demonstrate quantum nonlocality [4,5], quantum teleportation [6-8] and, more recently,

quantum information processing [9-11] and quantum cryptography [12,13]. One of the

challenges in recent practical schemes such as quantum cryptography and quantum

communication is to maximize the efficiency of coupling of single photons into single-mode

optical fibers. Several models and experiments have been developed to predict and measure

the coupling efficiencies of down-converted light into single-mode fibers [14,15,16]. In the

model and experiment of Kurtsiefer et. al. [14], the angular distribution of non-collinear type-

II down-converted light for a given spectral bandwidth is calculated. Their idea is to match the

angular distribution of the photon pairs to the angular width of the fiber mode. Bovino et. al.

[16] produced a more rigorous model in which the dependence of the coupling efficiency on

crystal length and walk-off were investigated. The importance of hyper-entanglement in type-I

down-conversion and its possible use as a resource in the field quantum information has been

recently discussed [22]. Single photon on-demand sources have also been designed using an

array of type-I down-converters [23]. In contrast to previous work, we present a detailed

theoretical model describing single-photon mode coupling in the simple situation of collinear

type-I down-conversion. The pair photon count rate is calculated and an analytic expression

which determines the condition for optimum single-photon coupling in single-mode optical

fibers is obtained in terms of experimental parameters.

2. Amplitude for pair detection in single-mode fibers

The amplitude for detecting photon pairs at conjugate space-time points

is defined by

A

11,tr

and

22,tr

2211

)2(

,,trEtrE

HH

(1)

where t1 and t2 are detection times of signal and idler photons and

Heisenberg electric field operators [18]. In the steady state the right-hand-side of Eq. (1) can

be expressed as

),(trE

iH

are the

Page 3

3

r

3

r

3

fr

)(,,,, 0 ,

0

k

222

0210

210

002211

2111

)(

I22

)(

I

2

1

0

2

1

0

2

1

0

3

*

k

*

k

2

3

1

3

3

3

kkkk

kkk

p

k

kktrEtrE

rUUUkdkdrd

(2)

),(trE

detection points 1r and

iI

are the interaction-picture electric field operators at some arbitrary but specified

2r and

fp

)(3rU

i

are plane-wave modes describing the electromagnetic

k

as the wave vectors of the signal and idler photons,

i are polarization indices. (As in our previous

studies we have not introduced the effects of a change in the linear refractive index between

the crystal and its surroundings. The incorporation of this difference does not qualitatively

change our conclusions). We have therefore assumed that the nonlinear crystal is embedded in

a medium whose linear refractive is the same as that of the crystal. Optical fibers used in the

analysis are assumed to have the same refractive index as the crystal.

, 0

3r

is a function which describes the shape of the pump

in the transverse direction;

ik

field in the crystal with

21,k

0k

is the

wave vector of the incident pump photon and

The initial state of the electromagnetic field

0

k

, consists of a coherent state with wave-

vector

take the quantized electric field in the fiber as

,

0 k (the monochromatic pump beam) with other modes in the vacuum state 0 . We

i

'

k

tf

kkkk

k

aerUkdkditrE

2

)('

',

2 / 1

0

33

'

(3)

where the coupling coefficient

kk

, ''

is a projection onto the appropriate fiber mode and is

defined as [19]

),(),(

*

'' , ''

yxUyxUdxdy

f

k

in

kkk

(4)

)(

''

rUin

k

are spatial modes incident on the fiber and

)(rUf

k

are the fiber modes. For single-

and . We may mode fibers we assume a spatial mode profile that is independent of k

therefore suppress the fiber mode indices in Eq. (4) to simplify the notation. After performing

the integration over the volume of the crystal and evaluating the expectation value, Eq. (2)

simplifies to

2121

sinc

ttp

kkfkdkd

1

r

2

r

210

2

2

1

1

21

22112z1z

2/ 1

0

2/ 1

0

kk

33

22

2

~

kkk

titikk

ff

kk

kkee

UU

d

(5)

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where d is the length of the crystal,

zzkk

k z

ˆ

kkk

zz

210

21

and the signal/idler wave

vectors are defined as

iziy ixi

k y

ˆ

k x

ˆ

k

where i = 1,2 denote signal, idler modes and

iz iy ix

kkk,,

are components of the wave vector along the x,y,z directions respectively.

rfp

emitted at a cone angle

to the pump and for frequencies close to the degenerate frequency,

the sinc function in Eq. (5) can be approximated to unity in a first-order analysis. If we assume

that the divergence of the pump is negligible over the length of the crystal, i.e., the crystal is

sufficiently thin, and that the transverse shape of the pump is gaussian, we can take the shape

function as

kfp

~

is the two-dimensional Fourier transform of

. For degenerate signal and idler photons

*

p

22

w

exp)(

yx

rf

t

(6)

Thus,

)w

4

1

exp()(

~

fp

22

pkk

, where

p

w is the effective width of the pump beam. In

terms of photon frequencies, a first-order analysis gives

~

1tp

kf

) sin)(w

4

1

exp()(

*2222

p2

21

kkt

vk

(7)

where v is the first-order dispersion coefficient of the nonlinear crystal. The collinear situation

is obtained for

0

.

*

3. Calculation of the coupling coefficient

The coupling coefficient is defined in Eq.. (4) as the overlap integral of the input modes and

the fiber mode. For our calculations in the collinear geometry the fibers are positioned along

the z-axis. In experiments one needs to use a beam splitter to separate the beams as is shown

in Fig. 1. A lens of focal length f is used to focus collinear signal and idler beams onto

identical single-mode fibers through a 50/50 beam splitter. The pair count rate is then

measured by single-photon detectors D1 and D2 connected to a correlator.

Since the beam splitter introduces constant phase factors into the pair detection amplitude, our

calculations will also be valid for this experiment.

Fig. 1: Schematic of the experimental setup to determine pair photon coupling in fibers

BS

D1

D2

lens

Single mode

fiber

Single mode

fiber

Nonlinear

Crystal

Page 5

Assuming that the input face of the fibers is at the image plane of the lens and that plane-wave

modes, described by the free-field interaction-picture electric field operators in Eq. (2), are

incident on the lens, we can employ diffraction theory to obtain the electric field modes at the

image plane of the lens [20]. We obtain

0

' ikd

2

0

'

sin

d'

cos

d

2'2

'2

)'(

2

22

1

,

R

r

yx

ik

r

f

k

d

k

i

yx

d

k

i

dd ikin

k

ee drrdeeyxU

(8)

where d is the distance from the center of the crystal to the lens and ' d is the distance from

the lens to the image plane. We are also assuming that a circular region of the lens of radius R

is being irradiated by the down-converted light. We take the fiber modes,

Gaussian modes [21] defined by

1

),(yxUf

w is the radius of the fiber mode. Substituting Eq. (8) and Eq. (9) in Eq. (4) we

kas

ikd

'w'

0

where

yxUf

,

, as

w

2

0

22

0

2

exp

w

yx

(9)

where

0

obtain the coupling coefficient

1

kk

R iBA

k

iBA

e

kd kd

d

w

kk

2

'w'w

4

0

22

4

0

22

0

(10)

k A and

k B are defined as

4

0

22

4

0

k

3

4

0

22

2

0

2

w''2

w

2

1

'2

1

d

w'2

w

k

dd

k

k

f

B

d

k

A

k

k

(11)

4. Two-photon count rate in single-mode fibers

Consider the situation in which the fiber is in the focal plane of the lens, i.e.,

assume that Gaussian filters are used to select frequencies close to the degenerate frequency.

Then for

0

fk

and to first-order in the dispersion of the crystal, the two-photon

amplitude in Eq. (5) approximates to

x

xi

)2(

fd '

. We

242w

1

dxeeA

bxa

2

)(

2

2

(12)

where

2

3

4

0

2*

2

2

0

*

2

3

4

0

3*

2

2

0

2*

2

w

2

3

w

cf

2

w

f2

w

f

R

cf

kn

i

nk

b

R

k

i

k

a

gg

(13)

is the time difference between the detection of signal and idler photons,

*

k is the

degenerate phase-matched wave number,

g n is the group refractive index and is the