Distortions of Gaussian pulses transmitted through

a transparent isotropic layer

Distorsi

´

on de pulsos gaussianos al atravesar una capa is

´

otropa

Eduardo O. Acosta

∗1

, Mar

´

ıa T. Garea

∗2

, Natalia C.

´

Alvarez

∗†3

and Liliana I. Perez

∗‡4

∗

Grupo de L

´

aser,

´

Optica de Materiales y Aplicaciones Electromagn

´

eticas

Facultad de Ingenier

´

ıa, Universidad de Buenos Aires

Paseo Col

´

on 850, C1063ACV, Buenos Aires, Argentina

1

eacosta@fi.uba.ar

2

mgarea@fi.uba.ar

†

Consejo Nacional de Investigaciones Cient

´

ıﬁcas y T

´

ecnicas (CONICET)

Buenos Aires, Argentina

3

nalvarez@fi.uba.ar

‡

Universidad de Buenos Aires. Consejo Nacional de Investigaciones Cient

´

ıﬁcas y T

´

ecnicas

Instituto de Tecnolog

´

ıas y Ciencias de la Ingenier

´

ıa ”Hilario Fern

´

andez Long”

Paseo Col

´

on 850, C1063ACV, Buenos Aires, Argentina

4

lperez@fi.uba.ar

Abstract—Lasers can operate in two regimes: continuous-

wave mode or pulsed mode. In the simplest case, the former

mode corresponds to monochromatic beams with Gaussian

distribution of amplitudes (beam limited in space); whereas

the latter mode corresponds to polychromatic beams with

Gaussian distribution of frequencies (pulse limited in time).

When the pulsed beams are reﬂected and refracted in different

types of interfaces, they undergo peculiar distortions that

bear some parallelism with those found for beams limited in

space. These effects, as shown in a previous work, correspond

to time delay (ﬁrst order) and change of pulse width (second

order). The distortions are clearly limited by the principle of

causality and their interpretation, while not straightforward,

emerges clearly when the associated ﬁelds are expressed in

magnitude and phase. Since the analytical expressions are

not simple even for the case where the pulse is transmitted

through a single layer of linear, homogeneous, isotropic and

transparent material, it makes it difﬁcult to solve the inverse

problem. In this work, we present an alternative analytical

development that makes it possible to explicitly determine

these distortion effects when a pulse impinges normally

on a transparent isotropic layer immersed in a medium of

analogous characteristics.

Keywords: Gaussian pulses; Geometric Optics; Phase

shift

Resumen— Los l

´

aseres pueden operar en dos reg

´

ımenes:

modo continuo

´

o pulsado. En los casos m

´

as simples, el

modo continuo se corresponde a haces monocrom

´

aticos con

distribuci

´

on de amplitudes gaussiana (es un haz limitado en

el espacio); mientras que el modo pulsado corresponde a

haces policrom

´

aticos con distribuci

´

on gaussiana de frecuencias

(pulso limitado en el tiempo). Cuando los haces pulsados se

reﬂejan y refractan en diferentes tipos de interfaces, sufren

distorsiones peculiares que tienen cierto paralelismo con los

encontrados para haces limitados en el espacio. Estos efectos,

como se muestra en un trabajo anterior, corresponden al

retardo de tiempo (primer orden) y al cambio de ancho de

pulso (segundo orden). Las distorsiones est

´

an claramente

limitadas por el principio de causalidad y su interpretaci

´

on,

aunque no es directa, emerge claramente cuando los campos

asociados se expresan en magnitud y fase. Pero como las

expresiones anal

´

ıticas no son simples (incluso en el caso de que

el pulso se transmita a trav

´

es de una capa

´

unica de material

lineal, homog

´

eneo, isotr

´

opo y transparente) se hace dif

´

ıcil

resolver el problema inverso. En este trabajo, presentamos un

desarrollo anal

´

ıtico alternativo que hace posible determinar

expl

´

ıcitamente estos efectos de distorsi

´

on cuando un pulso

incide normalmente en una capa isotr

´

opica transparente

sumergida en un medio de caracter

´

ısticas an

´

alogas.

Palabras clave: Pulsos gaussianos;

´

Optica geom

´

etrica;

Cambio de fase

I. INTRODUCTION

When a 2D ﬁnite-width monochromatic beam undergoes

reﬂection and refraction at an planar interface between two

isotropic dielectric media, it is deformed. This deformation

is due to the fact that a ﬁnite beam of light is composed

of a superposition of a large number of plane waves that

propagate at slightly different angles of incidence with

respect to the central plane wave.

Under the paraxial condition, this leads to a small correc-

tion in the Fresnel equations, producing small effects that

modify the beam. The ﬁrst and second order effects are lat-

eral displacement or Goos-H

¨

anchen effect and angular shift

(ﬁrst order effects) and change of focal point and change

of width (second order effects). These have been studied

through different approaches as energy considerations [1]

[2], superposition of two or more waves with different

directions of propagation [3] [4] [5] [6] [7] or moment

theory [8] [9]. Each of these methods have contributed

to the knowledge and determination of the different non-

geometrical effects.

Nevertheless, Gaussian or quasi-Gaussian beams have

been the most studied space-limited beams because they are

the lowest solution of the paraxial wave equation. In order

to obtain analytical expressions of these effects, usually the

Tamir’s generalized method (TGM) is used. This method,

an extension of the Tamir’s method [5], refers to the classic

Revista elektron, Vol. 2, No. 2, pp. 67-70 (2018)

ISSN 2525-0159

67

Recibido: 30/10/18; Aceptado: 30/11/18

model of multiple reﬂections on the interfaces of an isotropic

dielectric layer. The amplitudes of the successive terms of

the expansion rapidly decrease, i.e., it is a rapid convergence

method.

Chromatic dispersion is present in practically all optical

materials and is manifested as a dependence of the phase

velocity on the frequency or wavelength of light inclusive

in transparent media. As known, the group velocity is the

velocity that characterizes optical pulses either in dispersive

or non-dispersive media as well. An optical pulse propagates

at the group velocity

In the propagation, reﬂection and transmission of the real

Gaussian pulses there can be two types of deformations:

temporal and spatial. The latter corresponds to the distortion

that occurs with Gaussian beams (i.e. just spatially limited)

[10].

The propagation and transformation of laser pulses are

fundamental problems in the ﬁelds of laser technique, optical

communication, optical information processing, etc. Many

works have focused on propagation through dispersive and

anisotropic media [11] [12] [13]. However, there are no

studies about the distortion that occurs when a pulse (only

limited in time)is reﬂected or transmitted through non-

dispersive dielectric interfaces.

In this work we study the distorsions that occur on a

Gaussian pulse just limited in time when it is reﬂected

or transmitted on a single layer of non-dispersive material

using an analytical development alternative to TGM. The

alternative method applied to these type of pulses is a

generalization of the so call moment theory of light beam

propagation [14] [15] [16]. Finally the results obtained with

both methods are compared.

II. INCIDENT ELECTRIC FIELD OF A GAUSSIAN PULSE

Consider a Gaussian pulse, centered on a mean frequency

ω

0

and spectral width σ, which propagates in vacuum that

impinges normally on a parallel plate of refractive index n

and thickness d as shown in the ﬁgure 1. The electric ﬁelds

corresponding to the incident pulse

~

E

1

are given by

~

E

1

(x, t) =

~

E

0

√

2πσ

Z

+∞

−∞

e

−

(ω−ω

0

)

2

2σ

2

e

iω( x/c−t)

dω (1)

Fig. 1. Diagram of the system under analysis: incident (IP), reﬂected (RP)

and transmitted pulses (TP)

where ω

0

corresponds to the mean frequency. Without loss

of generality we assume that the vector

~

E has a ˆy direction.

The integral of Eq. (1) is solved by taking Ω = (ω−ω

0

)/σ

and using the series expansion of e

iz

=

P

∞

j=0

(−i)

j

z

j

/j!,

replacing in Eq. (1) we obtain:

~

E

1

(x, t) =

~

E

0

e

iω

0

(x/c−t)

∞

X

k=0

(−i)

j

j!

σ

j

(t − x/c)

j

Q

j

(2)

with

Q

j

=

1

√

2π

Z

+∞

−∞

e

−

Ω

2

2

Ω

j

dΩ

The term Q

j

of the Eq. (2) is the central moment of

the standard normal distribution E(Ω

n

). By the symmetry

of the normal function, the terms of odd j are canceled and

the moments of j pairs are calculated following the equation

E(Ω

2k

) = (2k)!/2

k

k!. Replacing in Eq. (2), we obtain the

electric ﬁeld for the incident Gaussian pulse:

~

E

1

(x, t) =

~

E

0

e

iω

0

(x/c−t)

e

−

σ

2

2

(x/c−t)

2

(3)

III. REFLECTED AND TRANSMITTED ELECTRIC FIELD OF

A GAUSSIAN PULSE

Assuming that the incident pulse impinges the surface

normally, the electric ﬁelds corresponding to the reﬂected

~

E

2

and transmitted

~

E

3

pulses are given by

~

E

2

(x, t) =

~

E

0

√

2πσ

Z

+∞

−∞

R(ω)e

−

(ω−ω

0

)

2

2σ

2

e

−iω(x/c+t)

dω

(4)

~

E

3

(x, t) =

~

E

0

√

2πσ

Z

+ inf

−∞

T (ω)e

−

(ω−ω

0

)

2

2σ

2

e

iω(x/c−t)

dω

(5)

where R(ω) and T (ω) are the reﬂection and transmission

coefﬁcients. An analytical expression of the ﬁelds can be ob-

tained by approximating these coefﬁcients by a polynomial

series around the average frequency ω

0

:

S(ω) = S(ω

0

) +

∞

X

k=0

1

k!

∂

k

S

∂ω

k

ω

0

∆ω

k

(6)

where S is R or T and ∆ω = ω − ω

0

. Replacing Eq. (6)

in Eq. (4) and using the same calculation procedure as for

the incident ﬁeld, we obtain the reﬂected electric ﬁeld

~

E

2

(x, t) =

~

E

0

e

−iω

0

(x/c+t)

R(ω

0

)e

−

σ

2

2

(x/c+t)

2

+ A

2

(7)

with

A

2

=

∞

X

k=1

i

k

k!

∂

k

R

∂ω

k

ω

0

∞

X

j=0

(−i)

j

(x/c + t)

j

j!

σ

k+j+1

E(Ω

j+k

)

using the expression of the moments we can obtain a

simpler expression for A

2

A

2

=

∞

X

k=1

i

k

k!

∂

k

R

∂ω

k

ω

0

∂

k

∂t

k

e

−

σ

2

2

(x/c+t)

2

Revista elektron, Vol. 2, No. 2, pp. 67-70 (2018)

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In the same way, replacing Eq. (6) in Eq. (5) and using

the same procedure the transmitted ﬁeld is written as

~

E

3

(x, t) =

~

E

0

e

iω

0

((x−d)/c−t)

T (ω

0

)e

−

σ

2

2

((x−d)/c+t)

2

+ A

3

(8)

with

A

3

=

∞

X

k=1

i

k

k!

∂

k

R

∂ω

k

ω

0

∂

k

∂t

k

e

−

σ

2

2

((x−d)/c−t)

2

IV. COMPARISON WITH THE RESULTS OBTAINED BY

MEANS OF TAMIR’S GENERALIZED METHOD

In the Tamir’s generalized Method (TGM) the logarithm

of the reﬂection and transmission coefﬁcients can be re-

placed by their second order approximations around the

mean frequency [17]

ln S(ω) = ln S(ω

0

)+

∂ ln S

∂ω

ω

0

∆ω +

1

2

∂

2

ln S

∂ω

2

ω

0

∆ω

2

(9)

Introducing Eq. (9) in Eq. (4) and integrating we obtain

~

E

T GM

2

(x, t) =

~

E

0

R(ω)

σ/σ

2

e

−iω

0

(x/c+t)

e

−

σ

2

2

2

(−x/c−t+τ

2

)

2

(10)

where we deﬁne

τ

2

= −i

∂ ln R

∂ω

ω

0

and σ

2

2

=

σ

2

1 − σ

2

∂

2

ln R

∂ω

2

ω

0

(11)

Eq. (10) show that the pulse suffers a ﬁrst order effect,

corresponding to a complex time delay τ

2

and a second order

effect, corresponding to a complex change of half-width σ

2

.

To compare both methods we truncate the series from Eq.

(7) to second order (k = 2)

~

E

2

(x, t) =

~

E

0

e

−iω

0

(x/c+t)

e

−

σ

2

2

(x/c+t)

2

P

2

(u

2

) (12)

with u

2

= σ(x/c + t) and

P

2

(u) = R(ω

0

) − iσ

∂R

∂ω

ω

0

u −

1

2

σ

2

∂

2

R

∂ω

2

ω

0

(u

2

− 1)

When comparing expressions Eq. (7) and Eq. (10) it is

not possible to determine if both expressions are similar.

Figure 2 shows incident pulse and reﬂected pulse calcu-

lated using equations (7) and (10) for ω

0

= 1.77×10

15

s

−1

;

σ = ω

0

/500 assuming a plate of thickness d = 22πc/ω

0

and

refractive index n = 1.33; calculated for x = 0 as a function

of the dimensionless time ω

0

t. The curves calculated using

both approximations show a similar shift, for the case of

the TGM approximation the shift of ω

0

τ ≈ 91, while in the

presented development it is ω

0

τ ≈ 98. However, the half-

width of the half-height pulse (∆) in both cases coincides

with ω

0

∆ = 607 being 1.03 times the width of the initial

pulse.

The electric ﬁeld of the transmitted pulse calculated using

the TGM approximation is calculated in a similar way by

replacing equation Eq. (9) in Eq. (5)

Fig. 2. Incident pulse (black), reﬂected pulse using the method of moments

(red) and TGM (◦) calculated at x = 0

~

E

T GM

3

(x, t) =

~

E

0

S(ω)

σ/σ

3

e

−iω

0

((x−d)/c−t)

e

−B(t)

(13)

with

B(t) =

σ

2

3

2

(−(x − d)/c + t + τ

3

)

2

where we deﬁne

τ

3

= −i

∂ ln T

∂ω

ω

0

and σ

2

3

=

σ

2

1 − σ

2

∂

2

ln T

∂ω

2

ω

0

(14)

In order to compare both methods we developed the Eq.

(8) to second order

~

E

3

(x, t) =

~

E

0

e

−iω

0

((x−d)/c−t)

e

−

σ

2

2

((x−d)/c−t)

2

P

3

(u

3

)

(15)

with u

3

= σ((x − d)/c − t) and

P

3

(u) = T (ω

0

) − iσ

∂T

∂ω

ω

0

u −

1

2

σ

2

∂

2

T

∂ω

2

ω

0

(u

2

− 1)

In Fig. 3 shows the incident pulse and the transmitted

pulse calculated using equations (15) and (13) calculated

for x = d. In this case the curves calculated using both

approximations show a similar shift ω

0

τ ≈ 93, however the

pulse half-width in both cases coincides ω

0

∆ = 588 being

the same as that of the initial pulse.

V. CONCLUSION

We show that Gaussian pulses, when reﬂected or trans-

mitted on a single layer, exhibit peculiar distortions. We

determine expressions for the ﬁrst and second order effects:

time delay and pulse width. The method used (method

of moments) is an alternative to the generalized Tamir’s

method and extended to pulses limited in time. Although

the distortions are very low at optical frequencies and begin

to be noticeable at frequencies of microwaves or less we

can see that the results obtained do not differ numerically

from each other. Nevertheless, from the method of moments

it is not simple to extract explicit expressions for such

Revista elektron, Vol. 2, No. 2, pp. 67-70 (2018)

ISSN 2525-0159

69

http://elektron.fi.uba.ar

Fig. 3. Incident pulse (black), transmittes pulse using the method of

moments (red) and TGM (◦) calculated at x = 0

deformations; but instead the TGM provides explicit expres-

sions, simple to calculate. In all cases the major analytical

difﬁculty is to calculate explicit expressions of the real and

imaginary parts of the ﬁelds (or of their module and phases)

when dealing with complex interfaces (several interfaces).

Although Tamir’s approximation is likely to be better than

the second order and can easily been extended up to fourth

order, the latter is easily extended to higher orders since a

recurrence equation has been obtained

ACKNOWLEDGMENT

This work was supported by the Universidad de

Buenos Aires under Grants UBACyT (2014-2017)

20020130100346BA, (2017- 2020) 20020160100042BA

and (2017-2020) 20020160100052BA. Also, a postdoctoral

grant from CONICET for one of the authors is gratefully

acknowledged.

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