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Propagation of Photons in a Waveguide

Array

Ayushman Shukla 2010PH10831

Kartikay Bansal 2010PH10847

Supervisor: Professor K. Thyagarajan

Abstract: We study the propagation of different quantum states of light through a linear array of optical waveguides. Propagation of a single photon and pairs of photons is studied. Waveguide arrays are very interesting for simulation of various condensed matter phenomena and in this context we simulate Bloch oscillations through the waveguide array by injecting two photons in a periodic array consisting of non-identical waveguides. Email id: ayushmanshukla11@gmail.com kartikaybansal@gmail.com

INTRODUCTION

In recent years, there has been a lot of interest in the study of propagation of

photons through periodic waveguide lattices. This is because the

mathematical structure of the problem is quite similar to structure of many of

the experiments in solid state physics which are described by the tight-binding

models. Studying this phenomenon will provide an easy simulation to such

problems which are otherwise quite hard to solve.

An important reason for the growing interest in this field is also that recently

experimentalists have been successful in fabricating such optical circuits with

varying geometries. Techniques of fabricating waveguides upto 1 m in width

have been developed[1].

In 1929, Bloch predicted that the motion of electrons in a periodic lattice under

a uniform electric field (and thus a uniform force) would be oscillatory instead

of uniform. Bloch oscillations are not observed in normal lattices because the

scattering due to lattice defects disturbs the quantum coherence of their

motion. These oscillations in superlattices, cold atoms in an optical potentials

and small Josephson junctions. These Bloch oscillations can also be

simulated by the motion of two photons in a periodic waveguide array.

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In our project so far we done a theoretical reading of the quantum mechanics

of a beam splitter, motion of photons in a directional coupler and a periodic

waveguide. We have studied the motion of a single photon through a periodic

waveguide, the motion of two photons through a waveguide array and plotted

the probability correlations and in the end we have simulated Bloch

oscillations in a waveguide and plotted the oscillatory motion of photons.

THEORY

Motion of photons in waveguides:

Fig 1: Directional coupler in which two waveguides are at close proximity over a length L [2]

For the case of two coupled waveguides, the coupled equations are given

by:[3]

where

here, is the propagation constant for the ith mode

ai is the amplitude of the ith mode

C12 and C21 are coupling constants, whose explicit form is given in the

next section.

When generalized to n-coupled waveguides, the equations are found to be

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where

Here, nj is the refractive index of the jth waveguide

uj is the transverse distribution in the jth waveguide

and

Motion of two photons in an array of identical waveguides:

We now move into the quantum picture. The amplitude of the ith mode is now

replaced by creation operator . We also simplify the case by assuming that

the coupling constants between any two waveguides is the same,

and since the waveguides are assumed to be identical, they have

the same propagation constant .

The coupled equations now become,

The solution then is written as,

where describes the amplitude for the transition of a photon from

waveguide l to waveguide k and is given by[4]

where is the Bessel function of the first kind of order k.

Motion of photons in non-identical waveguide array:

In this case we assume two types of waveguides (with different refractive

indices) arranged in a periodic manner ABABABAB… In such a case, the

parameter that assumes importance is the difference between the

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propagation constants of adjacent waveguides, . We still

assume that the coupling constants between adjacent waveguides is the

same,C.

The coupled equations now become,

The form of the solution remains the same,

However, the unitary transformation function changes to,

Calculation of multiple detection probability:

As we have defined, gives the transitional amplitude of the photon starting

from the lth waveguide and ending up at the kth waveguide. Hence, according

to quantum mechanics, the probability of a photon, which was injected in the

lth waveguide, to be found in the kth waveguide is given by | |2.

For the case of two photons, calculating this probability is more intricate. This

is because these two photons are indistinguishable bosons and follow Bose

Einstein statistics. The multiple detection probability is given by,

where

and

For our study, we have taken N=2 (two photon propagation).

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SIMULATIONS AND DISCUSSION

Value of parameters

For all the simulations given below the following parameter values were

chosen/calculated:

Width of a waveguide, a = 4

Distance between two waveguides, d=8

Refractive index of waveguide (in case of identical waveguides), n1= 1.45

Refractive index of medium between two waveguides, n2 =1.44

Refractive index of other type of waveguide (in case of non-identical

waveguides), n3= 1.455

Wavelength of light = 600 nm

Propagation constants,

Coupling constant, C= 38774 m-1

Motion of a single photon in an array of identical waveguides

Fig 2: Propagation of a single photon through a waveguide array

This graph shows the simple ballistic propagation of a single photon through

the waveguide array. The photon was launched in the center of the array and

by about 6 coupling lengths, it spreads to the entire array of 41 waveguides.

The energy propagation is mainly along two lobes.

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Motion of two photons in an array of identical waveguides

For i.e. both photons launched in the centre waveguide

Fig 3: Probability correlation at z=1 coupling length

Fig 4: Probability correlation at z=2 coupling lengths

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Fig 5: Probability correlation at z=3 coupling lengths

Fig 6: Probability correlation at z=4 coupling lengths

Fig 7: Probability correlation at z=5 coupling lengths

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As we can see from the surface plots (figure 3-7), when two photons are

injected in the same waveguide, there are no interference effects. The

probability just spreads out as we move along the z-axis, which indicates the

spreading out of the photons as they move along the array. The fact that we

have four ‘pillars’ of high probability at each corner shows that there is no

interference effect. It is just an addition of probabilities , two

photons conducting their ballistic propagation independently.

For i.e. one photon in the centre waveguide and one in the

adjacent waveguide

Fig 8: Probability correlation at z=1 coupling length

Fig 9: Probability correlation at z=2 coupling lengths

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Fig 10: Probability correlation at z=3 coupling lengths

Fig 11: Probability correlation at z=4 coupling lengths

The first thing to notice in this case is the absence of two ‘pillars’ (see figure

11). This is a clear indication of interference. This can be looked as a

generalized Hong Ou Mendel interference[5].There is an absence of

coincidence, probability at (-x,y) and (x,-y) is zero whereas the peaks are at (-

x,-y) and (x,y). In this case the probability is the square of the sum of the

amplitudes. As expected, the probability density spreads across the array as

we move along the z-axis, as can be seen from figures 8-11.

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Observing Bloch Oscillations in Waveguide Array

In this case we have a periodic arrangement of two types of waveguides (with

different propagation constants), ABABABAB…. We launch two photons in

two different ways:

(1) Both the photons launched in the centre waveguide

(2) One photon launched in the centre and one in the adjacent waveguide

We plot the photon density as a function of the propagation distance, z.

Fig 12: Bloch oscillation in non identical waveguide for

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Fig 13: Bloch oscillation in non identical waveguide for

As can be seen from figures 12 and 13, we have oscillations for both the

cases. These oscillations stem from the fact that and thus we have a

sinusoidal term in the transition amplitude. The non-identical waveguide

structure of the array provides for a ‘localization’ of the photons and they do

not move ballistically to the ends of the array. The periodicity of these

oscillations is equal to one coupling length m.

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FUTURE SCOPE OF THE PROJECT

In the next semester we would like to continue our work as our Major Project

Part II. We would like to extend our study to the propagation of entangled

photons. Work on simulation of propagation of entangled photons through a

waveguide array is relatively new and many research groups around the world

are doing these simulations to understand a number of condensed matter

problems.

One of the condensed matter problems that we would like to delve into in

detail is Anderson localization. It is a phenomenon in which an electron is

localized in a semiconductor due to the high degree of randomness due to

high levels of impurity and defects. It was proposed by American physicist

P.W. Anderson, for which he was awarded the 1977 Nobel Prize in Physics.

When the density of impurities and defects is not very high, electrons

experience weak scattering by them. This leads to the concept of resistivity. It

is this weak localization that makes the electron move at a uniform velocity

under the influence of an electric field (otherwise, under the influence of an

electric fields, an electron should have an accelerated motion). But when the

density of these impurities and defects becomes very large, strong localization

(also called Anderson localization) takes place, which results in the

correlations of electron motion decaying exponentially and the electron being

localized.

In a waveguide array, this phenomenon can be studied by looking at the

propagation of entangled photons and studying their correlations and then

arranging the parameters so as to obtain a decay in the correlation. Work on

observing this Anderson localization is being done by many research groups

like Armando Perez-Leija et all [8]. We would also like to study this effect in

detail and find some variation if possible.

Furthermore, if time permits, we would also like to simulate other problems of

condensed matter physics that have nearest neighbor interactions.

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ACKNOWLEDGEMENTS

We would like to pay our sincerest gratitude to our supervisor Professor

Thyagrajan who guided us in our project work with extreme humility,

helpfulness and patience. In our interaction with him, he motivated us to work

in a professional manner with all sincerity and also inspired us to look for

research avenues ourselves.

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REFERENCES

[1] Sergiusz Patela, “Fabrication methods of Planar Waveguides and other related structures”, Wroclaw

University of Technology. www w pwr wroc pl spatela pdfy pdf

[2] Ajoy Ghatak and K Thyagarajan, Optical Electronics( Cambridge University Press)

[3]B.Saleh and M. Teich, Fundamentals of Photonics,2nd edition (Wiley,2007)

[4] Yaron Bromberg, Yoav Lahini and Yaron Silberberg, “Bloch Oscillations of Path Entangled Photons”,

Physical Review Letters, 22 Decemeber 2010, 105, 263604.

[5] C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of Subpicosecond Time Intervals between Two

Photons by Interference”, Physical Review Letters, 2 November 1987, 59, 18.

[5] Yaron Bromberg, Yoav Lahini, Roberto Morandotti and Yaron Silberberg, “Quantum And Classical

Correlations in Waveguides”, Physical Review Letters, 6 June 9, 102, 253904.

[6] Hagai B. Perets, Yoav Lahini, Francesca Pozzi, Marc Sorel, Roberto Morandotti and Yaron

Silberberg, “Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices”,

Physical Review Letters, 2 May 2008,100, 170506.

[7] Alberto Peruzzo, et all, “Quantum Walks of Correlated Photons”, Report, Science Vol 329, 17

Septmeber 2010, 329, pp 1500-1503

[8] Armando Perez-Leija, Giovanni Di Giuseppe, Lane Martin, Robert Keil, Alexander Szameit, Ayman

F. Abouraddy, Demetrios N. Christodoulides, and Bahaa E.A. Saleh, “Observation of Anderson co-

localization of spatially entangled photon pairs” Quantum Electronics and Laser Science Conference, 6-

11 May, 2012.