An electromagnetic plane wave incident obliquely from a transparent medium onto the surface of an absorbing medium can be reflected with 0 or π phase shift if (i) the wave is p (TM) polarized, and (ii) the complex relative dielectric function ε is such that 0 ≤|ε| 2/2Re(ε) ≤ 1. Furthermore, the locus of ε such that the reflection coefficient for the p polarization is real at the same angle of incidence, is a circle, and that of ε½ (the complex relative refractive index) is Bernoulli’s lemniscate

Azzam, R. M.A.

English

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University of New Orleans 
ScholarWorks@UNO 
Electrical Engineering Faculty Publications Department of Electrical Engineering 
3-1979 
Reflection of an electromagnetic plane wave with 0 or π phase 
shift at the surface of an absorbing medium 
R. M.A. Azzam 
University of New Orleans, razzam@uno.edu 
Follow this and additional works at: https://scholarworks.uno.edu/ee_facpubs 
 Part of the Electrical and Electronics Commons, and the Physics Commons 
Recommended Citation 
R. M. A. Azzam, "Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an 
absorbing medium," J. Opt. Soc. Am. 69, 487-488 (1979) 
This Article is brought to you for free and open access by the Department of Electrical Engineering at 
ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an 
authorized administrator of ScholarWorks@UNO. For more information, please contact scholarworks@uno.edu. 
Reflection of an electromagnetic plane wave with 0 or π phase shift 
at the surface of an absorbing medium 
R. M. A. Azzam 
Hematology Division and Department of Internal Medicine, The University of Nebraska Medical Center, Omaha, Nebraska 68105 
(Received 24 July 1978) 
An electromagnetic plane wave incident obliquely from a transparent medium onto the surface of 
an absorbing medium can be reflected with 0 or π phase shift if (i) the wave is p (TM) polarized, and 
(ii) the complex relative dielectric function e is such that 0 < | e\ 2/2Re(e) < 1. Furthermore, the locus 
of ε such that the reflection coefficient for the p polarization is real at the same angle of incidence, is 
a circle, and that of ε1/2 (the complex relative refractive index) is Bernoulli's lemniscate. 
Except under conditions of total internal reflection, a plane 
electromagnetic wave experiences a "trivial" phase shift of 
either 0 or π upon reflection at the interface between two 
transparent (homogeneous and isotropic) media at all angles 
of incidence and for both the s (TE) and p (TM) polarizations. 
However, when the medium of refraction (or incidence) be­
comes absorbing, reflection induces a phase shift δ that is 
generally neither 0 or π. In fact, if surface layers (films) can 
be ruled out, δ ≠ 0 or π may be considered as a manifestation 
of the existence of absorption. The question arises: Can δ 
revert to one of its trivial values of 0 or -π even though ab­
sorption is present? The answer is yes. In this Letter we 
determine when this happens. 
The interface Fresnel reflection coefficient for the p po­
larization (with electric vibration parallel to the plane of in­
cidence) can be written as1 
where φ is the angle of incidence and e is the relative dielectric 
function of the two media, that is, the ratio of the dielectric 
487 J. Opt. Soc. Am., Vol. 69, No. 3, March 1979 0030-3941/79/030487-02$00.50 © 1979 Optical Society of America 487 
FIG. 1. (a) Domain of the relative dielectric function ε (hatched) in which 
the reflection coefficient for the p polarization rp can be real. The locus 
of e such that rp is real at the same fixed angle of incidence φ is a circle with 
center (sin2φ, 0) and radius sin2φ. (b) Domain of the complex relative re­
fractive index v (hatched) in which rp can be real. The locus of v such that 
rp is real at the same fixed angle of incidence φ is Bernoulli's lemnis-
cate. 
function of the medium of refraction to that of the medium 
of incidence.2 For r to become real, Eq. (1) shows that x must 
be real. Consequently, x2 must be real and positive, or, from 
Eq. (2), 
where η is a positive real number. Equation (3) can be re­
written as a quadratic in ε whose roots are 
In Eq. (4) when 77 sin2 2φ ≤ 1, ε is a positive real number. This 
is the case of interfaces between transparent media. More 
interesting is the case when 77 sin2 2φ > 1. Equation (4) then 
indicates that ε becomes complex, ε = (εr,εi), with real and 
imaginary parts given by 
Elimination of 77 between Eqs. (5a) and (5b) gives 
Equation (6) indicates that the locus of ε = (εr,εi), such that 
the reflection coefficient for the p polarization at a given angle 
of incidence φ is real,3 is a circle in the complex ε plane with 
center on the real axis, ε = (sin2φ,0), and radius of sin2φ, Fig. 
1(a). Varying φ generates coaxial circles that touch the 
imaginary axis at the origin, with centers on the segment of 
the real axis 0 ≤ ε ≤ 1. φ = 0 (normal incidence) gives a null 
circle coincident with the origin (ε = 0), while φ = 90° (grazing 
incidence) yields a circle with unit radius with center at (1,0). 
The interior and boundary of the latter circle define the do­
main of the ε plane where the reflection coefficient for the p 
polarization is real at a specific angle of incidence φ for each ε. 
It is often convenient, particularly in the optical region of 
the electromagnetic spectrum, to work with refractive indices 
as well as dielectric functions. If we denote the relative re­
fractive index by v = (n,k), then ε = v2 and 
If we substitute Eqs. (7) into Eq. (6), we obtain 
which, interestingly enough, is the standard equation of the 
lemniscate of Bernoulli4 in the v plane, Fig. 1 (b). Varying φ 
between 0 and 90° generates a family of these curves whose 
symmetry axes are coincident with the n and k coordinate axes 
and with maximum dimension, n(k = 0) = √2 sinφ, that in­
creases from 0 to √2, respectively. The interior and 
boundary of the largest lemniscate (that corresponds to φ = 
90°) defines the domain of the v plane in which rp can become 
real. Notice that in this domain n ≤ √2 and \k\ <n. 
So far we have considered the p polarization only. For the 
s polarization, the reflection coefficient rs is also given by Eq. 
(1) but with x = (ε - sin2φ)1/2/ cosφ. In this case rs is real 
when x is real, which, in turn, is satisfied only if ε is real. This 
is the uninteresting case of interfaces between transparent 
media. 
To conclude, when the relative dielectric function ε, or 
relative refractive index v, is complex, only the reflection 
coefficient for the p polarization (rp) can become real. If f 
is defined by 
then rp is real when 
and the angle of incidence at which this happens is given by 
φ = sin_1ξ1/2∙ For a given φ, the locus of ε = (ε rε i) is a circle, 
and that of v = (n,k) is the lemniscate of Bernoulli. 
1See, for example, M. Born and E. Wolf, Principles of Optics (Per-
gamon, New York, 1975), 5th edition, p. 40. 
2For passive or absorbing media, ε (and the relative refractive index 
v) is limited to the first or fourth quadrant of the complex plane 
depending upon whether the e~jwt or ejwt time dependence of the 
harmonic fields is selected, respectively. By allowing ε (and v) to 
assume values in the entire right-half plane, both choices of the time 
dependence are simultaneously represented. 
3The reflection phase shift δ (arg r) is either 0 or π dependent upon 
the sign conventions used. If the ejut time dependence is chosen, 
and the p directions in the incident and reflected beams are selected 
such that they are antiparallel at normal incidence (in other words, 
if we assume the Nebraska (Muller) conventions), then we have δ 
= 7Г. 
4S. M. Selby, Editor, Standard Mathematical Tables (Chemical 
Rubber Company, Cleveland, 1972), 20th edition, p. 377. 
488 J. Opt. Soc. Am., Vol. 69, No. 3, March 1979 JOSA Letters 488 


Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an absorbing medium

University of New Orleans

University of New Orleans ScholarWorks@UNO Electrical Engineering Faculty Publications Department of Electrical Engineering 3-1979 Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an absorbing medium R. M.A. Azzam University of New Orleans, razzam@uno.edu Follow this and additional works at: https://scholarworks.uno.edu/ee_facpubs  Part of the Electrical and Electronics Commons, and the Physics Commons Recommended Citation R. M. A. Azzam, "Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an absorbing medium," J. Opt. Soc. Am. 69, 487-488 (1979) This Article is brought to you for free and open access by the Department of Electrical Engineering at ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact scholarworks@uno.edu. Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an absorbing medium R. M. A. Azzam Hematology Division and Department of Internal Medicine, The University of Nebraska Medical Center, Omaha, Nebraska 68105 (Received 24 July 1978) An electromagnetic plane wave incident obliquely from a transparent medium onto the surface of an absorbing medium can be reflected with 0 or π phase shift if (i) the wave is p (TM) polarized, and (ii) the complex relative dielectric function e is such that 0 < | e\ 2/2Re(e) < 1. Furthermore, the locus of ε such that the reflection coefficient for the p polarization is real at the same angle of incidence, is a circle, and that of ε1/2 (the complex relative refractive index) is Bernoulli's lemniscate. Except under conditions of total internal reflection, a plane electromagnetic wave experiences a "trivial" phase shift of either 0 or π upon reflection at the interface between two transparent (homogeneous and isotropic) media at all angles of incidence and for both the s (TE) and p (TM) polarizations. However, when the medium of refraction (or incidence) be­comes absorbing, reflection induces a phase shift δ that is generally neither 0 or π. In fact, if surface layers (films) can be ruled out, δ ≠ 0 or π may be considered as a manifestation of the existence of absorption. The question arises: Can δ revert to one of its trivial values of 0 or -π even though ab­sorption is present? The answer is yes. In this Letter we determine when this happens. The interface Fresnel reflection coefficient for the p po­larization (with electric vibration parallel to the plane of in­cidence) can be written as1 where φ is the angle of incidence and e is the relative dielectric function of the two media, that is, the ratio of the dielectric 487 J. Opt. Soc. Am., Vol. 69, No. 3, March 1979 0030-3941/79/030487-02$00.50 © 1979 Optical Society of America 487 FIG. 1. (a) Domain of the relative dielectric function ε (hatched) in which the reflection coefficient for the p polarization rp can be real. The locus of e such that rp is real at the same fixed angle of incidence φ is a circle with center (sin2φ, 0) and radius sin2φ. (b) Domain of the complex relative re­fractive index v (hatched) in which rp can be real. The locus of v such that rp is real at the same fixed angle of incidence φ is Bernoulli's lemnis-cate. function of the medium of refraction to that of the medium of incidence.2 For r to become real, Eq. (1) shows that x must be real. Consequently, x2 must be real and positive, or, from Eq. (2), where η is a positive real number. Equation (3) can be re­written as a quadratic in ε whose roots are In Eq. (4) when 77 sin2 2φ ≤ 1, ε is a positive real number. This is the case of interfaces between transparent media. More interesting is the case when 77 sin2 2φ > 1. Equation (4) then indicates that ε becomes complex, ε = (εr,εi), with real and imaginary parts given by Elimination of 77 between Eqs. (5a) and (5b) gives Equation (6) indicates that the locus of ε = (εr,εi), such that the reflection coefficient for the p polarization at a given angle of incidence φ is real,3 is a circle in the complex ε plane with center on the real axis, ε = (sin2φ,0), and radius of sin2φ, Fig. 1(a). Varying φ generates coaxial circles that touch the imaginary axis at the origin, with centers on the segment of the real axis 0 ≤ ε ≤ 1. φ = 0 (normal incidence) gives a null circle coincident with the origin (ε = 0), while φ = 90° (grazing incidence) yields a circle with unit radius with center at (1,0). The interior and boundary of the latter circle define the do­main of the ε plane where the reflection coefficient for the p polarization is real at a specific angle of incidence φ for each ε. It is often convenient, particularly in the optical region of the electromagnetic spectrum, to work with refractive indices as well as dielectric functions. If we denote the relative re­fractive index by v = (n,k), then ε = v2 and If we substitute Eqs. (7) into Eq. (6), we obtain which, interestingly enough, is the standard equation of the lemniscate of Bernoulli4 in the v plane, Fig. 1 (b). Varying φ between 0 and 90° generates a family of these curves whose symmetry axes are coincident with the n and k coordinate axes and with maximum dimension, n(k = 0) = √2 sinφ, that in­creases from 0 to √2, respectively. The interior and boundary of the largest lemniscate (that corresponds to φ = 90°) defines the domain of the v plane in which rp can become real. Notice that in this domain n ≤ √2 and \k\ <n. So far we have considered the p polarization only. For the s polarization, the reflection coefficient rs is also given by Eq. (1) but with x = (ε - sin2φ)1/2/ cosφ. In this case rs is real when x is real, which, in turn, is satisfied only if ε is real. This is the uninteresting case of interfaces between transparent media. To conclude, when the relative dielectric function ε, or relative refractive index v, is complex, only the reflection coefficient for the p polarization (rp) can become real. If f is defined by then rp is real when and the angle of incidence at which this happens is given by φ = sin_1ξ1/2∙ For a given φ, the locus of ε = (ε rε i) is a circle, and that of v = (n,k) is the lemniscate of Bernoulli. 1See, for example, M. Born and E. Wolf, Principles of Optics (Per-gamon, New York, 1975), 5th edition, p. 40. 2For passive or absorbing media, ε (and the relative refractive index v) is limited to the first or fourth quadrant of the complex plane depending upon whether the e~jwt or ejwt time dependence of the harmonic fields is selected, respectively. By allowing ε (and v) to assume values in the entire right-half plane, both choices of the time dependence are simultaneously represented. 3The reflection phase shift δ (arg r) is either 0 or π dependent upon the sign conventions used. If the ejut time dependence is chosen, and the p directions in the incident and reflected beams are selected such that they are antiparallel at normal incidence (in other words, if we assume the Nebraska (Muller) conventions), then we have δ = 7Г. 4S. M. Selby, Editor, Standard Mathematical Tables (Chemical Rubber Company, Cleveland, 1972), 20th edition, p. 377. 488 J. Opt. Soc. Am., Vol. 69, No. 3, March 1979 JOSA Letters 488 

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Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an absorbing medium

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