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2320 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Cossmann et al.

Transient reflection of TE-polarized plane wavesfrom a Lorentz-medium half-space

Steven M. Cossmann, Edward J. Rothwell, and Leo C. Kempel

Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824

Received February 3, 2006; accepted March 16, 2006; posted March 24, 2006 (Doc. ID 67722)

The time-domain reflection coefficient for a plane wave obliquely incident on a Lorentz-medium half-space isdetermined analytically by inversion of the frequency-domain reflection coefficient. The resulting expressioncontains only simple functions and a single convolution of these functions. Owing to its simplicity, this form ofthe reflection coefficient provides insight into its temporal behavior, specifically how the relationship betweenthe damping coefficient and the oscillation frequency determines the shape of the response. The simple form ofthe reflection coefficient is validated numerically through comparison with the inverse fast Fourier transformof the frequency-domain reflection coefficient. © 2006 Optical Society of America

OCIS codes: 260.2030, 260.2110, 320.2250, 320.5550, 350.4010, 350.5500.

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. INTRODUCTIONhe propagation of short duration pulses in dispersiveaterials has generated much interest, and the effects of

ispersion on the shape of the propagating wave, particu-arly in the Sommerfeld and Brillouin precursors, haveeen extensively studied.1–4 Recently, the use of shortulses to probe materials has prompted the investigationf the reflection of transient waves from material half-paces of various types.5–8 Of particular interest is theorentz material, which is a good model for many mate-ials encountered in optics and engineering.9,10

The reflection of a short pulse by a Lorentz medium haseen considered for TE polarization by Gray11 and for TMolarization by Stanic et al.12 In each of these cases, theuthors find the impulse response of the reflected field byomputing the inverse transform of the frequency-domaineflection coefficient as an infinite series of fractional-rder Bessel functions. While this gives a convenient ana-ytic result, the series form provides little insight into theehavior of the reflected field waveform. In this paper weresent a simple, compact form for the TE reflection coef-cient that provides useful intuition about the response ofhe half-space. The formula is evaluated and comparedith the inverse fast Fourier transform (FFT) of the

requency-domain reflection coefficient to validate the ex-ression.

. FORMULATIONonsider a sinusoidal steady-state plane wave of fre-uency � incident on an interface separating free spaceregion 1) from a homogeneous Lorentz medium (region). The angle of incidence measured from the normal tohe interface is �, and the electric field is polarized per-endicular to the plane of incidence (TE polarization). Re-ion 1 is described by the permittivity �0 and permeability0, while region 2 is described by the complex permittiv-

ty ����=� ���� and the permeability � . The reflection

r 0 0

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oefficient, describing the ratio of the tangential incidento reflected electric fields, is given by13

���� =Z��� − Z0

Z��� + Z0, �1�

here the wave impedance of the incident wave is Z0�0 /cos � and the wave impedance of the transmittedave is

Z��� =����k���

kz���. �2�

ere �0= ��0 /�0�1/2, �= ��0 /��1/2, kz= �k2−k02 sin2 ��1/2, k0

���0�0�1/2, and k=���0��1/2. The time conventionxp�j�t� is assumed.

The relative permittivity of a single-resonance Lorentzedium takes the form9

�r��� = �� −�0

2��s − ���

�2 − 2j�� − �02 . �3�

ere �0 is the resonance frequency, and � is the dampingoefficient. The static relative permittivity, �s, is the valuef �r at zero frequency, while the optical relative permit-ivity, ��, is the value of �r as �→�. In optical problems,he case of ��=1 is of most interest. Then Eq. (3) can beritten as3

�r��� = 1 +b2

�02 − �2 + 2j��

. �4�

ere b is the plasma frequency of the medium.The time-domain reflection coefficient (impulse re-

ponse of the medium) is the inverse Fourier transform ofq. (1). This may be computed in terms of tabulatedaplace transform pairs. Let the Laplace transform vari-ble be s= j�. Then, substituting Eq. (4), the Laplace-omain reflection coefficient may be written in the form

006 Optical Society of America

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Cossmann et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2321

��s� =�s2 + 2�s + �0

2�1/2 − �s2 + 2�s + �02 + B2�1/2

�s2 + 2�s + �02�1/2 + �s2 + 2�s + �0

2 + B2�1/2, �5�

here B=b / cos �. Factoring the quadratic forms underhe radicals gives the alternative form

��s� =��s − s1��s − s2��1/2 − ��s − s3��s − s4��1/2

��s − s1��s − s2��1/2 + ��s − s3��s − s4��1/2 , �6�

here

s1,2 = − � ± 1, 1 = ��2 − �02�1/2, �7�

s3,4 = − � ± 3, 3 = ��2 − �02 − B2�1/2. �8�

ote that 1 and 3 may be either real or imaginary, de-ending on the values of � ,�0, and B.The reflection coefficient, Eq. (6), may be put into a

orm amenable to inversion via a look-up table by ratio-alizing the denominator, leading to

��s� = −�F�s��2

B2 , �9�

here

F�s� = ��s − s1��s − s2��1/2 − ��s − s3��s − s4��1/2. �10�

hrough simple manipulation, F�s� can be written as

F�s� = s��� s − s2

s − s1�1/2

− 1 − �� s − s4

s − s3�1/2

− 1− s1�� s − s2

s − s2�1/2

− 1 + s3�� s − s4

s − s3�1/2

− 1− �s1 − s3�. �11�

ow, define the transform pairs

� s − s2

s − s1�1/2

− 1 ↔ f1�t�, �12�

� s − s4

s − s3�1/2

− 1 ↔ f2�t�. �13�

hen, with the differentiation theorem of Laplaceransforms,14 the inversion of F�s� can be written as

−1�F�s�� = f�t� �14�

=d

dt�f1�t� − f2�t�� − s1f1�t� + s3f2�t� − �s1 − s3���t�.

�15�

sing the tabulated transform pair15

s −

s − ��1/2

− 1 ↔1

2�− + ��exp�−

1

2�− − ��t

��I1�1

2�− + ��t + I0�1

2�− + ��tu�t� �16�

nd substituting from Eqs. (7) and (8) then give

f1�t� = 1 exp�− �t��I1�1t� + I0�1t��u�t�, �17�

f2�t� = 3 exp�− �t��I1�3t� + I0�3t��u�t�. �18�

ere u�t� is the unit step function, and I0�x� and I1�x� arehe modified Bessel functions of orders 0 and 1, respec-ively.

The derivatives required to specify f�t� may be foundhrough direct differentiation:

df1�t�

dt= 1 exp�− �t��1I1��1t� + �1 − ��I1�1t�

− �I0�1t��u�t� + 1��t�, �19�

df2�t�

dt= 3 exp�− �t��3I1��3t� + �3 − ��I1�3t�

− �I0�3t��u�t� + 3��t�. �20�

ubstituting these into Eq. (15) then gives

f�t� = 12 exp�− �t��I1��1t� − I0�1t��u�t�

− 32 exp�− �t��I1��3t� − I0�3t��u�t�. �21�

sing the derivative identity16

1

xI1�x� = − I1��x� + I0�x� �22�

llows Eq. (21) to be simplified to

f�t� = �− 12 exp�− �t�

I1�1t�

1t+ 3

2 exp�− �t�I1�3t�

3t u�t�.

�23�

he time-domain reflection coefficient, ��t�, can now beound by computing the inverse transform of Eq. (9) andsing the convolution formula.14 Substituting from Eq.

23) for f�t� leads to

B2��t� = f�t� * f�t� �24�

=exp�− �t����− 12I1�1t�

1t+ 3

2I1�3t�

3t u�t�* ��− 1

2I1�1t�

1t+ 3

2I1�3t�

3t u�t�� �25�

=14 exp�− �t��� I1�1t�

1tu�t� * � I1�1t�

1tu�t�

− 2123

2 exp�− �t�

��� I1�1t�

1tu�t� * � I1�3t�

3tu�t�

+ 34 exp�− �t��� I1�3t�

3tu�t� * � I1�3t�

3tu�t�

�26�

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2322 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Cossmann et al.

=14 exp�− �t�fA�t� − 21

232 exp�− �t�fB�t�

+ 34 exp�− �t�fC�t�. �27�

he convolutions comprising fA�t� and fC�t� can be writtenxplicitly. Writing fA�t� in integral form gives

fA�t� = u�t� 0

t I1�1t��

1t�

I1�1�t − t���

1�t − t��dt�. �28�

sing the substitution u=1�t− t�� then gives

fA�t� = u�t� 1t

0 I1�1t − u�

1t − u

I1�u�

u �−1

1du� . �29�

efining =1t allows the integral to be evaluated asollows17:

fA�t� = −u�t�

1

0

I1� − u�

− u

I1�u�

udu �30�

=u�t�

1�2

I2� � . �31�

herefore fA�t� and fC�t� can be written as

fA�t� =2

1I2�1t�, fC�t� =

2

3I2�3t�, �32�

here

In�x� =In�x�

xu�x�. �33�

ith these, the time-domain reflection coefficient be-omes

��t� = −2

B2 exp�− �t��13I2�1t� + 3

3I2�3t�

− 123

2I1�1t� * I1�3t��. �34�

his is the final form for the reflection coefficient. If aransient plane wave with waveform g�t� is incident onhe interface, then the reflected field will be a transientlane wave with waveform g�t�*��t�.The form Eq. (34) is generally valid. However, when 1

r 3 is imaginary, the modified Bessel functions may beeplaced by ordinary Bessel functions according to threeossible cases. Case 1 occurs when �0

2��2. Then both 1nd 3 are purely imaginary. Defining

1 = ��02 − �2�1/2, 3 = ��0

2 + B2 − �2�1/2 �35�

nd using16

In�jx� = jnJn�x� �36�

ead to the expression

��t� = −2

B2 exp�− �t��13J2�1t� + 3

3J2�3t�

− 123

2J1�1t� * J1�3t��, �37�

here

Jn�x� =Jn�x�

xu�x�. �38�

ere Jn�x� is the ordinary Bessel function of order n.Case 2 occurs when �0

2+B2��2. In this case both 1 and3 are purely real, and the expression Eq. (34) may besed directly. Case 3 occurs when �0

2+B2��2 and �02��2.

n this case 1 is purely real, and 3 is purely imaginary.his leads to the expression

��t� = −2

B2 exp�− �t��13I2�1t� + 3

3J2�3t�

+ 123

2I1�1t� * J1�3t��. �39�

. NUMERICAL RESULTSo validate the expressions derived in Section 2, the time-omain reflection coefficient is evaluated numerically andhen compared with the inverse fast Fourier transformFFT) of the frequency-domain reflection coefficient, Eq.1). A set of parameters corresponding to each of the threeossible cases is used.The first set of parameters is the same as those chosen

y Brillouin18: �0=4.0�1016 s−1, b2=20.0�1032 s−2, �0.28�1016 s−1. This choice of parameters corresponds toase 1. With �=30�, Eq. (37) has been plotted in Fig. 1 andompared with the inverse FFT. The results show excel-ent agreement. Since ���0, the waveform is highly os-illatory and only lightly damped. This is reflected math-matically in the composition of ��t�, which involves onlyhe oscillatory ordinary Bessel functions.

The next choice of parameters is �0=2.0�1015 s−1, b2

20.0�1029 s−2, �=0.28�1016 s−1, which corresponds toase 2, Eq. (34). The results are shown in Fig. 2. Again,he closed-form expression and the inverse FFT compareell. For this choice of parameters �2��0

2+B2, and the re-ulting waveform is overdamped, showing no oscillatoryehavior and only a single negative peak. Since the ex-ression for case 2 involves only modified Bessel func-ions, which do not have the oscillatory behavior of ordi-

ig. 1. Time-domain reflection coefficient with incidence angle=30� and material parameters �0=4.0�1016 s−1, b2=20.01032 s−2, �=0.28�1016 s−1. This choice of parameters corre-

ponds to case 1, Eq. (37).

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Cossmann et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2323

ary Bessel functions, this observed behavior is easilyredicted from the mathematical form of the expression.The final choice of parameters, which corresponds to

ase 3, is �0=2.0�1015 s−1, b2=20.0�1032 s−2, �=0.281016 s−1. The analytic expression again matches the in-

erse FFT, as seen in Fig. 3. As expected, since ���0, but2��0

2+B2, we find more damping and less oscillationhan with case 1, but more oscillation than with case 2.ere the expression for the reflection coefficient has a

ombination of ordinary and modified Bessel functions.

. CONCLUSIONhe time-domain reflection coefficient for a plane wave in-ident on a Lorentz-medium half-space can be written as

compact expression involving a single convolution ofimple functions. This form allows the behavior of the

ig. 2. Time-domain reflection coefficient with incidence angle=30� and material parameters �0=2.0�1015 s−1, b2=20.01029 s−2, �=0.28�1016 s−1. The choice of parameters corre-

ponds to case 2, Eq. (34).

ig. 3. Time-domain reflection coefficient with incidence angle=30� and material parameters �0=2.0�1015 s−1, b2=20.01032 s−2, �=0.28�1016 s−1. The choice of parameters corre-

ponds to case 3, Eq. (39).

emporal response to be predicted on the basis of the ma-

erial parameters, according to three possible cases, eachf which is determined by a different relationship amonghe damping coefficient, the oscillation frequency, and thelasma frequency. The result is an exponentially dampedaveform that oscillates according to the conditions of the

pecific case.

Corresponding author E. J. Rothwell can be reached by-mail at rothwell@egr.msu.edu.

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