Angular spectrum of quantized light beams.

Abstract

We introduce a generalized angular spectrum representation for quantized light beams. By using our formalism, we are able to derive simple expressions for the electromagnetic vector potential operator in the case of (a) time-independent paraxial fields, (b) time-dependent paraxial fields, and (c) nonparaxial fields. For the first case the well-known paraxial results are fully recovered.

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arXiv:quant-ph/0509082v2 20 Feb 2006
Angular spectrum of quantized light beams
A. Aiello, J. Visser, G. Nienhuis, and J. P. Woerdman
Huygens Laboratory, Leiden University
P.O. Box 9504, 2300 RA Leiden, The Netherlands
Compiled February 1, 2008
We introduce a generalized angular spectrum representation for quantized light beams. By using our formalism,
we are able to derive simple expressions for the electromagnetic vector potential operator in the case of: a)
time-independent paraxial fields, b) time-dependent paraxial fields, and c) non-paraxial fields. For the first
case, the well known paraxial results are fully recovered.
OCIS codes: 000.1600, 270.0270.
c ? 2008 Optical Society of America
Propagation of nonclassical states of the electromag-
netic field is an issue of growing interest in quantum op-
tics for both fundamental and technological purposes.1
Consider, for instance, the relevance of propagation of
entangled photons to quantum cryptographic systems.2
The purpose of the present Letter is to provide a
novel perfectly general formalism for the representation
of quantized light beams which can be used in any regime
of propagation. Such objective is achieved by using a
dispersion relation that some of us have recently in-
troduced in Ref.,3and a generalized angular spectrum
representation for the field operators.4The usefulness
of our approach5becomes manifest whenever one deals
with quantum systems for which both paraxial and non-
paraxial regimes of propagation may be relevant as, e.g.,
down-converted photon pairs.6
Consider the plane-wave expansion of the positive-
frequency part of the electromagnetic vector potential
operatorˆA(r,t) = ˆA(+)(r,t) + H.c. in the Coulomb
gauge7
ˆA(+)(r,t)=
?
×
d3k
?
ǫ(λ)(k)ˆ aλ(k)exp(ik · r − ic|k|t).
?
16π3ε0c|k|
?1/2
2
?
λ=1
(1)
Since we want to describe fields propagating mainly
along the z axis, we find it convenient to define kz= sζ,
where s ≡ sign(kz) = ±1, and ζ ≥ 0. Then we can
rewrite Eq. (1) as
ˆA(+)(r,t) =
?
2
?
s=±1
?
dkxdky
?∞
0
dζ
?
?
16π3ε0c|ks|
?1/2
×
λ=1
ǫ(λ)(ks)ˆ aλ(ks)exp(iks· r − ic|ks|t),
(2)
where we have defined ks = (kx,ky,sζ). The field an-
nihilation and creation operators satisfy the canonical
commutation relations7
[ˆ aλ(ks),ˆ a†
λ′(k′
s′)] = δλλ′δss′δ(2)(q − q′)δ(ζ − ζ′).
Let ω ≥ 0 be an arbitrary frequency; at a later point
(3)
in this Letter we shall identify ω with the carrier fre-
quency of a paraxial field. We perform a change of vari-
ables {kx,ky,ζ} → {qx,qy,ω} such that
kx= qx,ky= qy,ζ = f(q,ω),(4)
where f(q,ω) ≥ 0 is an (almost) arbitrary function to
be determined. For reasons that will be soon clear, we
require f(q,ω) to increase monotonically for increasing
ω in the domain
Iω(f,q) =?ω ∈ R+: f(q,ω) ≥ 0?.
This condition implies that df(q,ω)/dω > 0 for ω ∈
Iω(f,q). Therefore, in such domain, we can write
δ(2)(q − q′)δ(ζ − ζ′) = δ(2)(q − q′)δ [f(q,ω) − f(q,ω′)]
(5)
= δ(2)(q − q′)
δ(ω − ω′)
df(q,ω)/dω.
(6)
If we substitute Eq. (6) into Eq. (3), we obtain
[ˆ aλ(ks),ˆ a†
λ′(k′
s′)] = δλλ′δss′δ(2)(q − q′)
δ(ω − ω′)
df(q,ω)/dω.
(7)
Equation (7) suggests the introduction of the “angular-
spectrum” field operators ˆ aλs(q,ω) defined as4
ˆ aλs(q,ω) ≡ ˆ aλ(ks)
?
df(q,ω)/dω.(8)
By using Eq. (8) and Eq. (7) it is easy to see that
[ˆ aλs(q,ω),ˆ a†
λ′s′(q′,ω′)] = δλλ′δss′δ(2)(q − q′)δ(ω − ω′).
(9)
Equation (9) is the first main result of this Letter; it
worthwhile to note that it is exact. No approximations
were made to obtain it.
The condition f(q,ω) ≥ 0 defines a volume Vq,ω(f)
in the half-space R2× R+spanned by {qx,qy,ω}.
This volume is bounded by the surface Sq,ω(f) =
∂Vq,ω(f) defined by the equation f(q,ω)
If we define the 2-dimensional domain Cq(f,ω)
?(qx,qy) ∈ R2: f(q,ω) ≥ 0?then, from Eq. (5) it read-
?
=0.
=
ily follows
R2d2q
?
Iω(f,q)
dω =
?
R+dω
?
Cq(f,ω)
d2q.(10)
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We use this equality to rewrite Eq. (2) immediately in
the new variables {qx,qy,ω} as
ˆA(+)(r,t) =
?
2
?
s=±1
?∞
0
dω
?
Cq(f,ω)
d2q
?
?/?q2+ f2
16π3ε0c
?1/2
×
λ=1
ǫ(λ)(q,sf)ˆ aλs(q,ω)
?
?
df/dω
×exp
?
iq · x + isfz − itc
q2+ f2?
,
(11)
where q2= q2
that the square root of the Jacobian J = df/dω was
used to pass from the original operators ˆ aλ(ks) to the
angular-spectrum operators ˆ aλs(q,ω). Since we want to
develop a formalism suitable for both non-paraxial and
paraxial light beams, we rewrite Eq. (11) as
x+ q2
y, f = f(q,ω), and x = (x,y). Note
ˆA(+)(r,t) =
?
s=±1
?∞
0
dωe−iω(t−sz/c)ˆΨs(r,t;ω), (12)
so that ω determines the plane carrier wave, and
ˆΨs(r,t;ω) is the envelope field which, at this stage, is
not required to be spatially and temporally slowly vary-
ing:
ˆΨs(r,t;ω)=
?
Cq(f,ω)
2
?
×exp[iq · x + izs(f − ω/c)]
×exp
d2q
?
?df/dω
16π3ε0c?q2+ f2
s (q,ω)ˆ aλs(q,ω)
?1/2
×
λ=1
ǫ(λ)
?
−it
?
c
?
q2+ f2− ω
??
.
(13)
Moreover, since ks
(qˆ q + sfˆ z)/?q2+ f2, and ˆ q = q/q; we have defined
=
ˆks
?q2+ f2, where ˆks
s (q,ω) = sˆ z × ˆ q, and
?
=
ǫ(λ)
s (q,ω) ≡ ǫ(λ)(q,sf), where ǫ(2)
ǫ(1)
s(q,ω) = (fˆ q − sqˆ z)/
Until now, we furnished expressions for the field opera-
tors in the angular spectrum representation, but not for
the energy, the momentum, etc. However, closed expres-
sions for these physical quantities can be easily found by
noting that the product
q2+ f2. (14)
ˆ a†
λ(ks)ˆ aλ(ks)dζ = ˆ a†
λs(q,ω)ˆ aλs(q,ω)dω,(15)
is invariant with respect to the change of variables Eq.
(4). Then, for example, starting from the well known
expression for the Hamiltonian operator of the electro-
magnetic field (see, e.g., Ref.7), after a straightforward
calculation one obtain
?∞
2
?
ˆH =1
2
?
?
s=±1
0
dω
?
Cq(f,ω)
d2q?c
?
q2+ f2(q,ω)
×
λ=1
ˆ a†
λs(q,ω)ˆ aλs(q,ω) + ˆ aλs(q,ω)ˆ a†
λs(q,ω)
?
.
(16)
Similar calculations can be easily done for the other
relevant quantities. Equation (16) shows that, as ex-
pected for an arbitrary field, the frequency ω of the
carrier plane wave is not equal to the frequency c|k| =
c?q2+ f2(q,ω) of the plane-wave mode exp(ik·r). How-
ial limit.
At this point the function f(q,ω) is still undeter-
mined, therefore we can exploit this freedom by imposing
some constraints on the envelope fieldˆΨs(r,t;ω) which
is, until now, perfectly general. In particular, we want
to find an expression for the envelope field in which the
Fresnel propagator6plays a role even beyond the parax-
ial regime. To this end, we proceed as in Ref.3and we
requireˆΨs(r,t = 0;ω) ≡ˆΨs(r;ω) to satisfy the time-
independent paraxial equation:
ever, as we shall see later, c|k| reduces to ω in the parax-
∂2ˆΨs(r;ω)
∂x2
+∂2ˆΨs(r;ω)
∂y2
+ 2isω
c
∂ˆΨs(r;ω)
∂z
= 0. (17)
In this way we obtain an expression for ˆA(+)(r,t)
which is an exact solution of the full d’Alembert equa-
tion for any time t > 0 and its corresponding enve-
lope fieldˆΨs(r,t = 0;ω) satisfies the time-independent
paraxial wave equation at t = 0, as initial condi-
tion. If we substitute from Eq. (13) the plane wave
exp[iq · x + izs(f − ω/c)] into Eq. (17), we easily find
f(q,ω) =ω
c
?
1 −q2c2
2ω2
?
.(18)
For q ∈ Cq(f,ω) this function satisfies all our require-
ments: it is positive and df(q,ω)/dω =?1 + ϑ2?/c > 0
that the plane-wave frequency c|k| = ω(1 + ϑ4)1/2re-
duces to ω in the paraxial limit ϑ ≪ 1. Finally, a closed
expression for the field operatorˆA(+)(r,t) can be given:
where we have defined8ϑ ≡ qc/(√2ω) . It is easy to see
ˆA(+)(r,t) =
?
s=±1
?∞
0
dω e−iω(t−sz/c)
×
?
?
Cq(f,ω)
2
ǫ(λ)
d2q
?
?(1 + ϑ2)
16π3ε0cω√1 + ϑ4
?1/2
×
λ=1
s (q,ω)ˆ aλs(q,ω)exp
?
.
iq · x − isq2c
2ωz
?
×exp
?
−iωt
??
1 + ϑ4− 1
??
(19)
Equation (19) is the second main result of this Let-
ter. It is easy to recognize in the exponential func-
tion in the third row, the sought Fresnel propagator
in momentum space. The spatial behavior of the enve-
lope field is entirely governed by this term. It worth
to note that Eq. (19) is exact, that is it has been
obtained without any approximation and, therefore, it
holds for both non-paraxial (ϑ ? 1) and paraxial (ϑ ≪
1) beams. In the latter case, the slowly varying term
exp?−iωt?√1 + ϑ4− 1??
shows that the envelope field
2

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ˆΨs(r,t;ω) cannot be strictly monochromatic for any
t > 0.
In the remaining part of this Letter, we give two differ-
ent examples of the application of our theory in order to
illustrate its generality. As a first example, let us gener-
alize the previous case and requireˆΨs(r,t;ω) to satisfy
the time-dependent paraxial wave equation, for any t:9
∂2ˆΨs
∂x2+∂2ˆΨs
∂y2+ 2isω
c
∂ˆΨs
∂z
+ 2iω
c2
∂ˆΨs
∂t
= 0, (20)
whereˆΨs≡ˆΨs(r,t;ω) for short. If we substitute from
Eq. (13) the relevant term exp[iq · x + izs(f − ω/c)] ×
exp[−it(c?q2+ f2− ω)] into Eq. (20), we obtain a new
f(q,ω) =ω
c
dispersion relation
?
1 −q2c2
4ω2
?
.(21)
Once again, for q ∈ Cq(f,ω) this function satisfies
all our requirements: it is positive and df(q,ω)/dω =
?1 + η2?/c > 0, where η ≡ qc/(2ω). It is easy to see that
Also for this case a closed expression for the field oper-
atorˆA(+)(r,t) can be given:
?∞
?
2
?
the plane-wave frequency c|k| becomes c|k| = ω(1+η2).
ˆA(+)(r,t) =
?
d2q
s=±1
0
dω e−iω(t−sz/c)
×
Cq(f,ω)
?
?
16π3ε0cω
?1/2
×
λ=1
ǫ(λ)
s (q,ω)ˆ aλs(q,ω)exp
?
iq · x − iq2c
4ω(sz + ct)
?
.
(22)
This expression is quite simpler than Eq. (19). However,
its exponential part (in the last row), differ by a factor of
1/2 from the Fresnel propagator expression. Once again,
we stress that Eq. (22) is exact, no approximation were
made.
As a last example of application of our formalism, we
choose to determine the function f(q,ω) by requiring ω
to coincide with the plane-wave frequency4c|k| : ω =
c|k|. It is easy to see that in this case we have
f(q,ω) =ω
cω2
?
1 −q2c2
?1/2
≃ω
c
?
1 −q2c2
2ω2
?
,(23)
where the last approximate equality holds in the parax-
ial limit qc/ω ≪ 1. For q ∈ Cq(f,ω) this function is
positive and df(q,ω)/dω = 1/(c?1 − (qc/ω)2) > 0,
in the paraxial limit Eq. (23) coincides with Eq. (18).
Since by definition ζ = |kz| = (ω/c)|cosθ|, it follows
that |cosθ| = f(q,ω)c/ω, and we can write
ˆA(+)(r,t) =
?
×
Cq(f,ω)
16π3ε0cω
×exp[iq · x − iszω(1 − |cosθ|)/c].
therefore all our requirements are fulfilled. As expected,
s=±1
?∞
??/|cosθ|
0
dω e−iω(t−sz/c)
?
d2q
?1/2
2
?
λ=1
ǫ(λ)
s (q,ω)ˆ aλs(q,ω)
(24)
Equation (24) is our last result. It gives an exact expres-
sion for the electromagnetic potential vector operator of
a generic, non-paraxial light beam, in the angular spec-
trum representation. By expanding in Taylor series the
|cosθ| term around θ = 0, it is easy to see that Eq.
(24) reduces to the well know classical paraxial expres-
sion (with the quantum operators ˆ aλs(q,ω) substituted
by the corresponding classical amplitudes). Moreover, at
the lowest order in θ, it coincides with Eq. (19) calcu-
lated at the lowest order in ϑ.
In conclusion, in this Letter we presented a novel for-
malism for the representationof arbitraryquantized light
beams. First, we introduced an angular spectrum rep-
resentation for the field annihilation and creation op-
erators. Then, we used our formalism to derive an ex-
act expression for the “paraxial-like” envelope field of
a light beam. Finally, we illustrated the generality of
our theory, by applying it to the description of time-
dependent, paraxial and non-paraxial, light beams. It
worth to note that, although our formalism is fully quan-
tum, all the previous results can be straightforwardly ex-
tended to classical fields just by replacing the quantum
operators ˆ aλs(q,ω) with the corresponding classical am-
plitudes aλs(q,ω).
We acknowledge support from the EU under the IST-
ATESIT contract. This project is also supported by
FOM.
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