The asymmetric Goos-H\" anchen effect

14 Pages·2013·0.61 MB·English

¨
THE ASYMMETRIC GOOS-HA¨NCHEN EFFECT
THE ASYMMETRIC GOOS-HANCHEN EFFECT
•• JJoouurrnnaallooffOOppttiiccss1166,,001155770022--77 ((22001144)) ••
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Abstract. We show in which conditions opti- Manoel P. Araujo
o
cal gaussian beams, propagating throughout an Gleb Wataghin Physics Institute
N
homogeneous dielectric right angle prism, present State University of Campinas (Brazil)

9 an asymmetric Goos-H¨anchen (GH) effect. This mparaujo@ifi.unicamp.br
1 asymmetric behavior is seen for incidence at criti-
Silvˆania A. Carvalho

calanglesandhappensinthepropagationdirection Department of Applied Mathematics
]
s of the outgoing beam. The asymmetric GH effect State University of Campinas (Brazil)
c canbealsoseenasanamplificationofthestandard
i [email protected]
t GH shift. Due to the fact that it only depends on
p Stefano De Leo
o the ratio between the wavelengthand the minimal
Department of Applied Mathematics
. waist size of the incoming gaussian beam, it can
s State University of Campinas (Brazil)
c be also usedto determine one of these parameters.
[email protected]
i Multiple peaks interference is an additional phe-
s
y nomenon seen in the presence of such asymmetric
h effects.
p
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I. INTRODUCTION
The behavior of laser gaussian beams in the presence of symmetric and asymmetric wave number
distributions is the subject matter of this paper. The importance of the difference among such kinds
of distributions can be understood if one takes into account the analogy between optics [1,2] and
non relativistic quantum mechanics [3,4] and discuss the behavior between stationary and dynamical
maxima.
Anon-relativisticfree(gaussian)particleinitsrestframeisdescribedbythefollowingwavepacket
d +∞ +∞ (k2 +k2)d2 k2 +k2
Ψ(x,y,t) = Ψ dk dk exp x y + i k x+k y ~ x y t
0 4π x y "− 4 x y − 2m !#
Z−∞ Z−∞
x ~t y ~t
= Ψ , , , (1)
0 G d md2 G d md2
(cid:18) (cid:19) (cid:18) (cid:19)
where
α2 /
(α, β)=exp 1+2iβ .
G −1+2iβ
(cid:20) (cid:21)
p
This wave convolution is solution of the two-dimensional Schr¨odinger equation [3],
∂ +∂ +2im∂ Ψ(x,y,t)=0 . (2)
xx yy ~ t
(cid:2) 2 (cid:3)
Thegaussianprobabilitydensity, Ψ(x,y,t) ,growswiththebeamdiameterasafunctionoftime. Its
| |
maximum, whichdecreasesfor increasingvalues oftime, is always locatedatx=y =0. It represents
a stationary maximum. It is obvious that the previous analysis is a consequence of the choice of a
symmetric momentum distribution centered in k =k =0.
x y
To illustrate the idea behind our study, let us consider a gaussian momentum distribution with
only positive momentum values for k , i.e.
y
x ~t d +∞ k2d2 k2
Φ(x,y,t) = Φ , dk exp y + ik y i~ y t
0 G(cid:18)d md2(cid:19) √π Z0 y "− 4 y − 2m #
x ~t y ~t y / 2~t
= Φ , , 1+ erf i 1+ i . (3)
0 G(cid:18)d md2(cid:19) G(cid:18)d md2(cid:19) " d r md2 !#
The maximum of this distribution can be estimated by using a basic principle of asymptotic analysis
[5]. For oscillatoryintegrals,rapidoscillationsoverthe range ofintegrationmeans that the integrand
averagesto zero. To avoidthis cancelation rule, the phase has to be calculated when it is stationary,
i.e.
∂ k2 ~ k
0= k y ~ y t y = h yi t .
y
(∂ky " − 2m #) ⇒ m
ky=hkyi
The breaking of symmetry in the gaussian momentum distribution implies now an expected value of
k different from zero,
y
+∞ k2d2 / +∞ k2d2 2
y y
k = dk k exp dk exp = , (4)
y y y y
h i "− 4 # "− 4 # d√π
Z0 Z0
and, consequently, a dynamical maximum at
2~
y = t . (5)
max
md√π
For non-relativistic quantum particles, the difference between stationary and dynamical maxima can
be roughly represented as the difference between symmetric and asymmetric wave number distribu-
tions. Theaimofthispaperistoinvestigateinwhichconditionswecanreproducedynamicalmaxima
for laser gaussian beams propagating throughout an homogeneous dielectric right angle prism.
1
The Maxwell equations
∂
2 tt E(r,t)=0 , (6)
∇ − c2
(cid:20) (cid:21)
fortimeharmonicelectricfields(exp[ iωt])andforplanewaves,modulatedbyacomplexamplitude
−
A(r), which travel along the z-direction (exp[ikz] with k =ω/c=2π/λ),
E(r,t)=E ei(kz−ωt)A(r) ,
0
reduce to [6]
ω2
0 = E ei(kz−ωt) ∂ +∂ ∂ +2ik ∂ k2 + A(r)
0 xx yy zz z − c2
(cid:20) (cid:21)
= E ei(kz−ωt)[∂ +∂ +∂ +2ik ∂ ] A(r) . (7)
0 xx yy zz z
In the paraxial approximation [1], A(r) is a slowly varying function of z and the previous equation
becomes
[∂ +∂ +2ik ∂ ] A(x,y,z)=0 . (8)
xx yy z
The analogy between the paraxial approximation of the Maxwell equations, Eq.(8), and the non-
relativistic Scho¨dinger equation, Eq.(2), is then clear if we consider the following correspondence
rules [7,8]
z t and k m/~ .
←→ ←→
It is interesting to ask in which circumstances, by using optical paraxialbeams, it is possible to have
an asymmetrical wave number distribution and, consequently, produce a dynamical shift. The study
presented in this paper aims to give a satisfactory answer to this intriguing question.
Gaussian beams are the simplest type of paraxial beams provided by a laser source. The electric
field amplitude of the incident paraxial gaussian beam is given by [9,10]
w2 +∞ +∞ (k2 +k2)w2 k2 +k2
E(r,t) = E ei(kz−ωt) 0 dk dk exp x y 0 + i k x+k y x y z
0 4π x y − 4 x y − 2k
" !#
Z−∞ Z−∞
= E ei(kz−ωt)A(r) , (9)
0
where w is the minimal waist size of the beam. After performing the k and k integrations, we
0 x y
obtain
x z y z
A(r)= , , . (10)
G w kw2 G w kw2
(cid:18) 0 0(cid:19) (cid:18) 0 0(cid:19)
The density probability distribution of the gaussian electric field,
2
E(r,t) 2 = E 2 w0 exp 2 x2+y2 , (11)
| | | 0| w(z) − w2(z)
(cid:20) (cid:21) (cid:20) (cid:21)
growths in beam diameter as a function of the z-distance from the beam waist w ,
0
2
2z
w(z)=w 1+ .
0s kw
(cid:18) 0 (cid:19)
The maximum, which decreases for increasing values of z, is always located at x = y = 0. This
maximum plays the role of the stationary maximum for the quantum non-relativistic particle in its
rest frame.
In this paper, we investigate the behavior of optical gaussian beams which propagates through a
right angle prism, see Fig.1. For incidence angle θ > θ , the beam is totally reflected at the second
c
interface, its wave number distribution is symmetric and centered at k = k = 0. Consequently,
x y
the Goos-H¨anchen shift [11] is stationary in the direction of the beam propagation. The optical
phenomenon in which linearly polarized light undergoes a small phase shift, δ λ, when totally
≈
2
internally reflected is widely investigated in litterature [12–19]. For incidence at and near critical
angles [20–22], we find a frequency crossover in the GH shift which leads to an amplification effect,
δ √kw λ . Inthis paper,we shallpresenta new effectfor incidence atcriticalangles. Depending
c ≈ 0
on the magnitude of kw , only the positive values of k , in the wave number distribution, contribute
0 y
to reflection and this asymmetry produces a dynamical Goos-H¨anchen shift. It is thus the breaking
of the symmetry in the wave number distribution which opens the door to a dynamical maximum.
A detailed analysis of this new phenomenon will be discussed in section III. Before of our numerical
study, in section II, we introduce our notation and the geometry of the dielectric system used in this
paper. In such a section, we also give, for s and p polarized waves, the reflection and transmission
coefficients at each interface. Our final considerations and proposals are drawn in the last section.
II. DIELECTRIC SYSTEM GEOMETRY AND OUTGOING BEAM
The incidentgaussianbeam(9) propagatesalongthe z-axisandformsanangle θ with z ,normalto
in
the first air/dielectric interface (see Fig.1a),
y cosθ sinθ y y
in = =R(θ) . (12)
z sinθ cosθ z z
(cid:18) in (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
Observing that the spatial phase of the incoming beam is
k r = k r , (13)
in · in ·
with k =k (k2 +k2)/2k, we obtain, for the beam propagating within the dielectric after the first
z − x y
air/dielectric interface, the following phase
qin · rin = kxx+kyin yin + n2k2 −kx2 −ky2in zin . (14)
q
Inordertofollowthebeammotionwithinthedielectric,wehavetointroducetwonewaxesrotations,
see Fig.1a,
y 3π y π y
out =R ∗ =R in , (15)
z − 4 z −2 z
(cid:18) out (cid:19) (cid:18) (cid:19)(cid:18) ∗ (cid:19) (cid:16) (cid:17)(cid:18) in (cid:19)
withz andz respectivelynormaltothesecondandthirddielectric/airinterface. Thespatialphase
∗ out
of the beam moving within the dielectric in the direction of the last dielectric/air discontinuity can
be given in terms of the outgoing axes,
q r = k x+q y +q z , (16)
out · out x yout out zout out
where
qyout =R 3π qy∗ = −kyin .
qz − 4 qz qz
(cid:18) out (cid:19) (cid:18) (cid:19)(cid:18) − ∗ (cid:19) (cid:18) in (cid:19)
Observe that the spatialphase of the reflectedbeam at the second dielectric/airinterface is obtained
replacing q by q . Finally,
z∗ − z∗
k r = k x+q y + k2 k2 q2 z
out · out x yout out − x− yout out
= k x+k z +kq y
x yin in zin in
= k x + [k cos(2θ) k sin(2θ)] y + [k sin(2θ)+k cos(2θ)] z , (17)
x z y z y
−
As expected from the Snell law [1,2],
(k r ) =[0, k cos(2θ), k sin(2θ)] . (18)
∇ out · out (kx=0,ky=0)
(cid:2) (cid:3)
The amplitude of the outgoing beam is given by [9,10]
w2 +∞ +∞ (k2 +k2)w2
A[s,p](r,θ)= 0 dk dk T[s,p](k ,k ) exp x y 0 + iϕ (k ,k ;r,θ) , (19)
out 4π x y θ x y "− 4 out x y #
Z−∞ Z−∞
3
where
k2 +k2
ϕ (k ,k ;r,θ)=k x+k [cos(2θ)z sin(2θ)y] x y [cos(2θ)y+sin(2θ)z]
out x y x y − − 2k
and T[s,p](k ,k ) are obtained by calculating the reflection and transmission coefficients at each in-
x y
θ
terface. For s-polarized waves, this means
2kz qz kz 2qz
in ∗ − ∗ exp[2iq a ] out exp[i(q k )a ] .
kz + qz × qz + kz z∗ ∗ × qz + kz zout − zout out
in in ∗ ∗ out out
By using the geometry of the dielectric system, see Fig1a, and Eqs.(16,17), we have
a =a/√2 , a =b a , q =q and k =k .
∗ out − zout zin zout zin
Consequently, the transmission coefficient becomes
T[s](k ,k )= 4kzin qzin qz∗ −kz∗ exp i[q a√2+(q k )(b a)] . (20)
θ x y k + q 2 qz + kz { z∗ zin − zin − }
z z ∗ ∗
in in
For p-polarized waves,w(cid:0)e find (cid:1)
T[p](k ,k )= 4n2kzin qzin qz∗ −n2kz∗ exp i[q a√2+(q k )(b a)] . (21)
θ x y n2kz + qz 2 qz∗ + n2kz∗ { z∗ zin − zin − }
in in
Due to the fact tha(cid:0)t the motion i(cid:1)s on the y-z plane, only second order k contributions appear in
x
the transmission coefficient, T[s,p](k ,k ). Thus, without lost of generality, to calculate the complex
x y
θ
amplitude A[s,p](r,θ) we can take the following approximation T[s,p](k ,k ) T[s,p](0,k ). Conse-
x y y
out θ ≈ θ
quently,
w2 +∞ +∞ (k2 +k2)w2
A[s,p](r,θ) 0 dk dk T[s,p](0,k ) exp x y 0 + iϕ (k ,k ;r,θ)
out ≈ 4π x y θ y "− 4 out x y #
Z−∞ Z−∞
x cos(2θ)y+sin(2θ)z
[s,p]
= , (y,z) , (22)
G w z Fθ
(cid:20) 0 R (cid:21)
where
w +∞ k2w2
[s,p](y,z)= 0 dk T[s,p](0,k ) exp y 0 + iϕ (0,k ;r,θ) .
Fθ 2√π y θ y "− 4 out y #
Z−∞
The detailed analysis of [s,p](y,z) will be the subject matter of the next section.
Fθ
III. ASYMMETRIC GH EFFECT AND MULTIPLE PEAKS INTERFERENCE
Let us consider the momentum distribution
(k2 +k2)w2
g[s,p](k ,k )=T[s,p](0,k ) exp x y 0 (23)
x y y
T θ "− 4 #
responsible for the shape of the transmitted beam. The contour plots of g (k ,k )[s,p] clearly show
T x y
that for decreasingvalue ofkw , see Fig.2,andfor incidence angles approachingto the criticalangle,
0
seeFig.3,thesymmetrybetweenk andk inthewavenumberdistributionisbroken. Asanticipated
x y
in the Introduction, this symmetry breaking is responsible for the creation of a dynamical maximum.
To examine in detail this phenomenon, let us first consider incidence at θ =π/4,
w +∞ k2w2 k2
[s,p](y,z)= 0 dk T[s,p](0,k ) exp y 0 ik y i y z . (24)
Fπ4 2√π Z−∞ y π4 y "− 4 − y − 2k #
4
To estimate the maximum, we can apply the stationary phase method [5],
∂ k2
[s,p] y
0= phase T (0,k ) k y z .
(∂ky " π4 y − y − 2k #)
h i ky=hkyi
Due to the fact that the phase of the transmission coefficient T[s,p](0,k ) is not dependent on the
y
θ
spatial coordinates, we can immediately find an analytical expression for the shift in y between two
maxima, i.e.
k
y
∆y = h i∆z . (25)
− k
For θ = π/4, n = √2, and kw 10, the wave number distribution is a symmetric distribution
0 ≥
centered at k = 0. Consequently, k = 0 and the maximum does not change its position. We
y y
h i
thus recognize a stationary maximum. The numerical analysis confirms this theoretical prediction,
see Fig.4.
Let us now consider incidence at critical angle,
sinθc+ n2 sin2θc =√2 .
−
For n=√2 the critical angle is θ =0 and tphe transversal y-z profile is determined by
c
w +∞ k2w2 k2
[s,p](y,z)= 0 dk T[s,p](0,k ) exp y 0 + ik z i y y . (26)
y y y
F0 2√π 0 "− 4 − 2k #
Z−∞
In this case, the z-shift of the maximum in terms of the y location of the detector is given by
k
y
∆z = h i∆y . (27)
k
For kw =103, the wavenumber distributionis notcompletely symmetric ink andthis produce the
0 y
first modifications on the transmitted beam, see Fig.5. Such a little modification is more evident for
p-polarized waves. By decreasing the value of kw up to 10, we lose the symmetry (see Fig.2) and
0
we clearly find a dynamical maximum. To estimate this dynamical shift, we observe that, as seen in
section I, for kw =10 only positive values of k contribute to the mean value, this implies
0 y
2
k = (28)
y
h i w √π
0
and, consequently, the shift in z of the transmitted optical beam is given by
2
∆z = ∆y . (29)
kw √π
0
The numericalanalysis,showninFig.6,confirmssucha prediction. Insucha plot,it isalsoclearthe
asymmetric interference which appears in the presence of dynamical maxima.
IV. CONCLUSIONS
Optics sure represents a very stimulating field to reproduce quantum mechanical phenomena. For
example, the well known Goos-H¨anchen shift [11] is the optical analogous of the delay time in non-
relativistic quantum mechanics [3,13]. These optical and quantum effects are due to the fact that
evanescent waves exist in the classical forbidden region. This intriguing shift which is always matter
of scientific investigation [16–19] is, in general, a stationary shift. In this paper, we have analyzed in
which situations this stationary shift becomes a dynamical shift.
Due to the fact that the dynamical shift is a direct consequence of the breaking of symmetry
in the wave number distribution, this new optical phenomenon can be also seen as an asymmetric
5
GH effect. In our study, we have seen that the more convenient circumstances to reproduce this new
phenomenonarethechoiceofincidenceatcriticalanglesandofbeamwaists,w 10/k 1.6λ,ofthe
0 ≈ ≈
orderofthewavelengthoftheincominggaussianbeam. Thisseemstobetoorestrictiveforapossible
experimental implementation of the theoretical analysis presented in this article. Nevertheless, this
difficulty is very similar to the difficulty found in detecting the standard Goos-H¨anchen sfhit which
is of the order of the wavelength of the incoming beam. Consequently, it can be overcome with the
same trick, i.e. amplifying the shift. For example, by preparing a dielectric structure which allows
2N +1 internal reflections, we obtain for the transmission coefficient the following expressions
2N+1
[s,2N+1] 4kz qz qz kz
T (k ,k ) = in in ∗ − ∗ (30)
θ x y 2 q + k
(cid:12) (cid:12) kzin + qzin (cid:12)(cid:12) z∗ z∗ (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
and (cid:12) (cid:12) (cid:0) (cid:1) (cid:12) (cid:12)
2N+1
[p,2N+1] 4n2kz qz qz n2kz
T (k ,k ) = in in ∗ − ∗ . (31)
At critical angles, we(cid:12)(cid:12)(cid:12)hθave x y (cid:12)(cid:12)(cid:12) (cid:0)n2kzin + qzin(cid:1)2 (cid:12)(cid:12)(cid:12)(cid:12)qz∗ + n2kz∗ (cid:12)(cid:12)(cid:12)(cid:12)
2 2
k >0 for k <0 and k <0 for k >0 .
z y z y
∗ ∗
Consequently, by increasing the number of internal reflections, we can select the positive k compo-
y
nents in the transmitted wave number distribution for value of the beam waist, w , greater than the
0
wavelength,λ, ofthe incoming laserbeam. The symmetry breakingin the wavenumber distribution,
responsible for recovering the second order k contribution to the phase which contributes to the
y
maximum with the term k y/k, can be thus optimized for experimental proposals by using the
y
h i
number of internal reflection N and the ratio w /λ.
0
In a forthcoming paper, we shall analyze the asymmetric GH effect for frustrated total internal
reflection [23,24] and resonant photonic tunneling [25]. Another interesting future investigation is
represented by the possibility to include in our calculation the focal shift [20]. This additional shift
representsasecondordercorrectiontotheGHshiftandconsequentlyactsasadelay inthe spreading
of the outgoing optical beam.
ACKNOWLEDGEMENTS
We gratefully thank the Capes (M.P.A.), Fapesp (S.A.C.), and CNPq (S.D.L.) for the financial
support and the referee for his useful suggestion on the new title and for drawing our attention to
references on the GH shift and, in particular, on the interesting second order correction which leads
to the focal shift [20].
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7
Geometric Layout
Axes Rotations
z
out
n
b
θ
b
a
z
z
z in
(a)
∗
Critical Incidence for n = √2
k x + k z + k y
x y z
k x + k y + k z a
x y z
a
(b)
Fig. 1
Figure 1: Geometric layout of the dielectric structure analyzed in this paper.
8
Contour plots of the transmitted
wave number distribution g[s,p](k , k )
x y
T
-2.4 -1.2 0 1.2 2.4 -2.4 -1.2 0 1.2 2.4
kw = 10 kw = 10 2.4
0 0.2 0 0.2
0.4 0.4
1.2
0.6 0.6
0 0.8 0.8
w
0
y
k
-1.2
-2.4
(a) (b)
s-polarization p-polarization
kw = 100 kw = 100 2.4
0 0.2 0 0.2
0.4 0.4
1.2
0.6 0.6
0 0.8 0.8
w
0
y
k
-1.2
-2.4
(c) (d)
k w k w
x 0 x 0
Fig. 2
n = √2 , θ = 0
Figure 2: Contour plots of the transmitted wave number distribution, g (k ,k ), at criticalangle for
T x y
increasing values of kw . The numericaldata show that the symmetry, which is brokenfor kw =10,
0 0
is recoveredby increasing the value of kw (total internal reflection).
0
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