The quantum description of light propagation inside a planar waveguide is given. In particular, the description describes the behavior of the field inside a directions coupler. Nonclassical effects are presented and discussed

Bertolotti, M.

Jansky, J.

Perina, J.

Pernova, V.

Sibilia, C.

English

NASA Technical Reports Server

N94-10574
NON CLASSICAL EFFECTS IN PLANAR WAVEGUIDES
M.Bertolotti* ,J.Jansky** ,J.Pedna# ,V.Pednova# and C.Sibilia*
* Dip.Energetica, Univ.Roma I,Vla Scarpa 16,00161 Roma, ltaly
# Joint Laboratory of Optics,Palacky Univ.,Olomouc,Czechoslovakia
• * Research Lab. for Crystal Phys.,Budapest,Hungary
Abstract
The quantum description of ligth pmpagmJon inside a planar waveguide is
given , looking in particular at the behaviour of the field inside a directional
coupler. Nonclassical effects are presented and discussed.
Introduction
Electromagnetic fields in optical guided wave systems are usually
described simply by using classical Maxwell's equations, but there are cases in
which a quantum treatment is necessary.Three purely quantum phenomena are
known having no classical analogous; namely photon antibunching,
sub-poissonian photon statistics, and squeezing of optical fields. If problems
89
https://ntrs.nasa.gov/search.jsp?R=19940006119 2020-06-16T21:09:45+00:00Z
connected with these phenomena or the evolution of photon statistics are to be
dealt with,a quantum mechanical treatment must be used.
Of course a in(mr system is not able to produce or change these
properties but a nonlinear one is. For this reason,in the following the
Harniltonian for a nonlinear optical wavaguide will be derived,and its application
to some propagation problems will be considered.Although the Hamiitonian is
quite ganerai,emphasis is given to planar structures only and a more suitable
approach to descdbe propagation fenomene is discussed.
One of the results of having the propagation problem treated in quantum
mechanical form is to allow for the possibility of studying how purely quantum
effects propagate in linear systems.We will show ,for example,that a quantum
effect as squeezing is affected by the operation of switching in a linear structure
because of the phase changes involved in the operation.
2. Ouantization of the radiation field
The recent expadments on nonclassical states of light have called for a full
quantum analysis of the electromagnetic field [1] especleJly in the cases of
propagation of the fields inside dipersive media.
We remember that the standard quantization method consists of writing the
Hamiltonlan in a given volume V , demanding periodicity in space. For
propagating fields, the space evolution is then replaced by a time evolution, by
linking the space and time variables by the equation z,, ct. The length of the
nonlinear medium is then replaced by an effective interaction time. Of course this
method has two main imitations. The flrt one is that, by identifying the space
evolution with time evolution we lose one vadable and this formalism can
descdbe only c.w. operation : the second problem is that this procedure cannot
9O
be applied rigorouslyto a dispersive medium, where each frequency propagates
with different velocity.
However we remember that by using the Hamiltonlan formaJism and working in
the Heisenberg picture, the time evolution of _z,t) operator is given by
at =
(1)
SO that the aeneratar for time evolution is tha Harniltopiarl _-/_
_aenerator for soace omoa_aation is tha momentum oDemtor
(2)
The _; operator is related to the wave flux of the PoyntJng vector [2].
Therefore a suitable way to quantize the radiation field to describe the
propagation fenomena is the one starting from the flux of the Poynting vector.
This leads us to the rearBation that the important quantity is the flux and not ,
as usuaJly is assumed with the Hamiltonlan formalism, the energy density.
In this way Instead of quantizing the field in a large volume and demanding for
spatial periodicity, it is necessary to assume a time periodicity T of the field,
with the requirement that T must be large with respect to any relevant time.
Then instead of writing the field in term of spatial modes ( thus performing a
Fouder analysis of the space variable z into the wave vector K m ) it is possible
to write it in term of temporal modes ( thus performing a Fourier analysis of the
time variable t into discrete frequencies (o m ) and space dependent
operators. The advantage is that the temporal modes remain the same inside
and outside the dielectric medium [3] and the space evolution of the mode
operator can now be obtained by means of the momentum operator; moreover
dispersion of the material can be included.
9]
By the help of this formulation the expression of the ek)ctrk::field is ( Inside a
dielectric )
E (z,t) - _" (z,t) + E (z,t),
where the cross stands for c.c. and
_'+ (z,t) = m_[ 2_c'_(o)hO m ]1_ [_(z,(_n)eXp(-/O_m_] (3)
being _ (z,(om) and their conjugates form a set of localized creation and the
annihilation operators ,eOm the field fmquancy, n(o)) the refractive index at the (o
frequency, e 0 the dielectric constant and c the iigth velocity.
The number operator for the field becomes
A A
N(z0, o)m)- a+ (z0,o)m) a(z0,mm) (4)
which represents the number operator of the photons of frequency o) m
passing through the plane z,, z0 during a period T , and the commutation
rules now become commutation at "equal space':
+ . si d :) (5)
and the _; operator is detined as
A A
/11
(6)
(Dm
where kin,= n (O_m)_ is the wavevector of the field.
92
3. Quantum mechanical deecripUon of propagation In a planar
wavegulde
A planar dielectric waveguide is a medium whose dielectric permittivity
depends on one diraction,paraliel to which we shall assume the x-axis (s.F,g.1).
If this medium does not contain absorbing centers,if there is no
amplification of radiation,and if the permittivity is weakly dependent on the field
frequency (o i ,the electromagnetic field inside the guide is expressed in terms
of normal modes in the following form [4,5]
A (r) = _ 0 f (x) exp( I _ .r), (s)
where _ I is the wave-vector with components y and z (_ j.r= ky y + kz z) of the
j-th mode propagating inside the waveguide and f (x) is a function dependent
only on x, defined over all space, and determined by the waveguide structure.
Therefore each guided mode is defined by a _ vector at each col frequency.
From the quantum then/ point of view ( as pointed out in the previous
paragraph ,if the operator describing the field mode in a free space is given by
( note the operator is the one thai obeys at the equal space commutation rules)
(7)
in a guided strucure it can be described as
(8)
93
and eq. (6) written in oper_orial form becomes
exp(- t) +c.c.]
J
where A 0 is a constant.
(9)
t
1.. t/
_ft
Rg. 1 A planar waveguide
We would like to remark that in the case of dispersionless material the time
evolution of the field operators( Heisenberg equation) is the same as the one in
space , i.e. the Hamiltonian operator and the momentum operator
approach provide the same results. This remark is particularly important when
we consider the quantum treatmen of a guided mode inside a guiding
structure, due to the fact that in general we propagate different spatial modes
of equal frequency and we are not obliged to take into account dispersion of the
medium if weassume a c.w. propagation.
The same kind of considerelion is still valid if we study the propagation of single
or more modes inside a nonlinear planar waveguide with third order nonlinearity.
These cases have been extensively studied in the paper [5].
In the following we analyze the case of propagation in a directional coupler,
which is one of the most interesting guiding devices , very important from the
point of view of its switch properties.
04
4. Behaviour of a linear dlrecUonal coupler when nonclassical staWs are
involved (squeezing)
The directional linear coupler consists in two adjacent and parallel
waveguides [ 5,6 ](channels ).When radiation goes through the
structure,exchange of power between the channels is possible because of the
evanescent field which is present in the region between them.
In the frame of classical theory the coupler is studied by using the
[5,6 ],in which a perturbation polarization responsible for the
coupling contains the refractive index of the guides. Complete power transfer
occurs in a distance L - ( x/2)K , where K is the coupling constant
determined by the refractive indices of the structure: if the detuning parameter 5
is zero,that is in the case of complete phase matching [6 ] ,being
I
s= (13a- (lO)
where _a and 13o are the wavevectors of two modes of equal frequency
propagating in channel a and b respectively .If 8 is not zero the maximum
fraction of power that can be trasferred is proportional to
 +82"
From the classical equations for the complex amplitude for the directional
coupler, in the frame of the couplad-mode theory,with obvious generalization we
have the following Heisenberg equations for the operators,
f da/dz - -iKbexp(i2 8z),
db / dz - -il_ exp(-N 6 z), (11)
95
where _ and _ are the field annihilation operators in channel a and b
respectively .For the sake of simplicity we neglect damping terms because we
are interested in the coupling effect only,this is a good approximation in the
region of low temperature and optical frequencies. Wfthout taking into account
dispersion the set of eqs. ( 11 ) is the same that we can write startling from the
momentum operator with the substitutions t -> z/c and _./-> c _;.
In this way we get the following solutions of eqs.(11)
t,= t,o. (12)
where a0 and _x) are the input annihilation operators and
C== eBJ[cos(yz)- i5ITsin(¥z)]
Cu= e- BZ[cos(yz) + i 8/-f sin(yz)]
Ga = - i _ sin(yz) • _z
G_ = - i K/-fsin(yz) e- ,6z
where y2 = /_2 + 82.
To study the propagation of nonclassical field through the structure we use the
following characteristic function
CN(13) = Tr{pexp[_a + ] e_o[-13"_] = exp[- MII31 2+ _(S'132 + Sl3°2)+13 Wr-13" W]
(13)
which is able to describe a field which is not a pure coherent or squeezed
state,but has simultaneously squeezed ,coherent and chaotic features [ 7 ].In
eq.( 13 ) W ,, W exp(i 0 ) is the coherent signal,and M and S are related to the
96
noncoherentpart of the field.So for the vacuum state we have M-S,, W-0
;in the pure coherent state M=S=0, and for the chaotic field W-S-0.
The state is a pure squeezed state if
M- O= 0.5((4 S + 1) -1)
S,= exp(i _ ) cosh(r) sinh(r) (1.4)
r being the squeezing parameter [13 ].
A mixed state is given from a superposition of a pure squeezed state with
coherent signal W and a cahotic field described by the normally ordered
characteristic function given by eq.( 13 ) if
M= Q+ N,
where N is the noise photon number.
(15)
We shall suppose that the input statistics of light in both modes can be
described by the normally ordered characteristic function ( 13 ).Putting solutions
(12) into eq.(13 ) we can see that the truncated normally ordered output
characteri- stic functions will have the same functional form as the input ones with
new terms
s,.  0)Gt
W a = I_ O) Ca + H_oO) Ga (16)
where the superscript (o) labels the input quantities,and similar expressions can
be found for the b-mode,by interchanging the subscripts a and b.
9?
<(
We are interested in finding expressions for the variances < ( A Q ) 2 • and
A A
A p)2 • ,(where _= _+a+ and J>= -i(a-_ +)).lnmf.(14)it
is shown that
=I+2M+S+S'
=1+2M- S- S' (17)
Several interesting cases can be considered which depend on the way the
coupler is feeded.
Let us suppose first that a pure squeezed state enters channel b and a
coherent state (or vacuum) channel a. It can be shown in this case that for
L- x /2K and 8- 0 we have
< (A_)2> a = < (A_ z>_o. < (A_')2>a =
< (Ab)2>, = < (A_ e.. 1.
(18)
This means that at the output of channel a we have an opposite squeezing than
at the input of channel b,while the output in channel b shows no squeezing.
A relaIed situation is obtained when two opposite squeezed fields enter the
two channels in the same conditions as in the previous case.In this case
squeezing is preserved in both channels because the field entedng channel a
comes out of channel b with opposite squeezing and the same happens with
field entedng channel bat intermediate lengths of the coupler the squeezing is
not completely preserved.
A very interesting role is played by the detuning parameter. In general [14]
if 8 _, 0 some noise is added to both channels and squeezing is reduced, and
for some special values of 8, noise is absent. Let us consider for example
the case in which 8 = _ K.. In this case ¥ - 2k and for the same coupler
isngth I.-Tl'/2k if a squeezed field entered channel b and a coherent or vacuum
field channel a then we have at the output
98
< (Ab)2>a = < (A_)2>a= 1.
< (Ab)2> b = < (_% eo.
< (A_')2>_ = < (_)% bo. (19)
We can see that changing the detuning parameter from zero to '_/'3"K
switches from one channel to another.This result is rather interesting for the
purpose of measurement.The squeezed state is detected by interfering it with a
coherent reference light and looking at fluctuations.The switching behaviour just
described allow to preserve both the squeezed state and its reference beam.
5 Dlrectiorml and contradlrectiorml coupler with modes with small
different frequency propagating Inside
We have studied the problem of propagation of radiation in a coupler
assuming two different frequencies inside the channels, with the hypothesis that
each channel can support one only guided mode : this is possible if the two
frequencies am quite similar. In general for a coupler the more realistic
description of the field propageting inside all the structure is the one which takes
into account the superposition of the single modes propagating in each channel
( so called supermodes); in the case of different frequencies this approach is
particularly convenient and it is the one that we have adopted but in its
quantum analogous, i.e. introducing this concept in the statistical dependence of
the modes supported by the structure.
99
The motion equation of the operator describing the
structure is given by ( Heisenberg form)
a__ i ^
az - ,
I:)ropagation inside the
(20)
where j is the index corresponding to the mode ( j,, 1,2) and this equation is
related to the momentum operator G, that for this case is given by
where k)= _/= -_, is the mode wave vector, 7. is the coupling constant, which
depends on the refractive index n((o/) distributJon inside the coupler. It is very
interesting fo observe that the G operator looks like the one of a second order
nonlinearity for a bulk material.
Using the approach of the supermodes we can descdbe the two fields of
different frequency supported by the strucure as
I1> = 2-_(111n >+ Ull Ilou¢> + _112ou¢> )
12> = 2-_(12in >+ u_211ou_> + ¢¢z212ou¢> ) (22)
being Uj,k a function related to the transformation law of the coupler ,
containing all the informations about the structure , such as the coupling
constant, the datuning parameter, etc. ( see functions C a,b and G a,b of the
provious paragraph ).
To follow the statistics of the field we start from the characteristic function ( the
antinormal one) from which It is possible to derive all the factorial moments and
the photon counting distnbution. As in the previous paragraph we suppose the
input staIe is a superposition of a coherent state and noise , including
squeezing.
I00
Due to to the hypothesisof the supermode we can write the charaIedstic
function for all the fields _1 and _ , CA (1_1,_), but we can follow also the
behaviour of each separate mode CA (_) [ 8 ] :
t..
c_. (_1._ = exp{Z, [- Bj. I _Jl2+).1
(noise) (squeezing)
+ (- B12_; _2 + C_2in _1_4Z + c.c.)]}.
1
#. c.c.)+ - c.c.)+
(coherent)
( interference of noise ) (23)
The output characteristic function is of the same form as the input, where all the
features of the coupler are inside the B and C coefficients of the eq.(23).
Several cases of inputs states have been studied, such as coherent, two-photon
coherent , two-mode squeezed states and all factorial moments have been
calculated [8] finding as the detuning pammetr plays a very important role on the
evolution of the flisds : it adds eddidional noise if it is non zero [8].
It is interesting also to follow the photon counting distribution which put into
evidence the switch properties of the structure 81ways starting from the hypotesis
of supermode supported by the coupler. An example is shown in Rg.2, where
the detuning parameter 8 is zero, the input state in the first channel is a two
photon coherent state in the tirst mode , with 8 " smaJl amount of
squeezing" ,and a coherent state in the other mode. The picture shows the
marginal photon number distribution in the channel 2 ; at a suitably distances the
sub-Poissonlan behaviour turns super-Poissonlan, which characterizes the field
in the squeezed vacuum state: this confirms the switching of ligth of certain
photon statistics from one mode to the other one.
101
O.g )
¢4
0.7
"_0.6
11.4
0a
0._
0.1
0.Q
o
G,m f
_,_p dP
Fig.2 - The marginal photon number distribution for 8 -0 and K -i for the
channel 2
Conclusions
The propagation of ligth in a linear directional coupler can be studied without
taking into account the dispersion of the dielectric constant until c.w. field
propagation is considered; of course dispersion must be taken into account in
non stationary cases and when the structure of the propag_ng device supports
different frequencies.
The ability of the coupler to switch from one channel to the other by introducing a
phase lag allows to change the squeezing directions, until the _ parameter
is of suitably values; in general a detuning different from zero reduces the switch
properties of the coupler and adds additional noise to the propagating fields. This
effect is in turn evident also on the photon counting distribution.
102
Roforoncos
. See,e.g. "Photon and Quantum Fluctuations', ed. by £1_ Pike and H.Waither
(Hilger, Bristol,1988)
2- I.Abram, Phys. Rev.A35, 4661 (1987)
3- B. Huttner,S. Semlnik,Y.Ben-Aryeh, Phys. Rev.A42, 5594 (1990)
4- D.Marcuse "Theory of Optical Dielectric Waveguides" (/Cad. Press,
N.Y.,1974)
5 - M,Bertolotti et.aJ in" Nonlinear Waves in Solids State Physics "- Ed by
A.D.Boardman et. ai.( Plenum Press ,N.Y.1990)
6- A.Yariv, P.Yeh - "Optical Waves in Crystals" ( Wiley Int. ,N.Y. 1984)
7- J.Janszky, C.Sibilia, M.Bertolotti ,Y.Yushin, J.Mod.Optics 35,1757 (1988)
8- V.Pednova,A.Luks,J.Kreplelka,C.Sibilia,M.Bertolotti, J..Mod.Optics 38, 2429
(1991)
103



Non classical effects in planar waveguides

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