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Negative refraction in the light of the plasma-optical effect
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 705–706 www.elsevier.de/ijleo
SHORT NOTE
Negative refraction in the light of the plasma-optical effect M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/ Julio Palacios 11, 9-B, 28029 Madrid, Spain Received 5 January 2007; accepted 10 April 2007
Abstract A mathematical expression for the refractive index of a semiconducting material with negative refraction is presented within the context of the plasma-optical effect. The basic ingredients for writing down this expression are the fact that the index of refraction of the above material is negative and the fact that free-carrier effective mass depends upon carrier spatial density. In particular, negative refraction for higher optical frequencies is discussed. r 2007 Elsevier GmbH. All rights reserved. Keywords: Negative index of refraction; Plasma-optical effect; Higher optical frequencies
The plasma-optical effect is based upon the fact that, in a semiconductor, the electrons of the conduction band and the holes of the valence band behave as very free carriers (see, for instance, Refs. [1,2]). Although the above-mentioned fact is apparently idealized, it provides a useful framework for elucidating the fundamental problems related to the plasma-optical effect and for understanding the most significant applications. The aim of this paper lies on the mathematical description of negative refraction in the light of the plasma-optical effect; in this description, carrier-concentration dependence of the effective masses of electrons and holes will be taken into account. As a relevant particular case, negative refraction will be examined for higher optical frequencies. Our starting point is the following relationship for the local refractive index of a semiconductor under the freecarrier approach (see, for example, Refs. [1,2]): " ZðoÞ ¼ Z1
e2 n p 1 2 þ o mnn mnp
!#1=2 ,
where o denotes optical angular frequency, ZN is the refractive index at very high frequencies (o-N), e is the electron charge, e is the dielectric permittivity, n and p are the electron and hole spatial densities, respectively, and mnn ; mnp are the electron and hole effective masses, respectively. Now we are interested in formulating an expression for a negative refractive index from formula (1). To this end, given that one has eo0 in the context of negative refraction and by taking only the minus sign in the square root of Eq. (1), it follows for a metamaterial of negative refraction (see Ref. [3]): " ZðoÞ ¼ Z1
e2 n p 1þ þ jjo2 mnn mnp
!#1=2 .
(2)
From Eq. (2) we state the following relationship for the refractive-index perturbation, assuming that free-carrier effective mass depends on carrier concentration:
(1)
Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.04.012
DZðoÞ Z1 ZðoÞ 8 " !#1=2 9 < = e2 an n ap p , þ ¼ Z1 1 þ 1 þ : ; jjo2 mnn mnp
ð3Þ
ARTICLE IN PRESS 706
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 119 (2008) 705–706
where "
1 an n ap p e þ n jj mnn mp
!#1=2 o0
stands for plasma resonant angular frequency, an and ap being corrective factors corresponding, respectively, to electrons and holes; these factors give a certain measure of the carrier-concentration dependence of free-carrier effective mass. Now we are interested in deriving from formula (3) an approximate relationship valid for the visible and ultraviolet regions. To this end, by expanding the right-hand side of relation (3) in a first-order Taylor series of 1/o2, we arrive at the following expression: o2 DZðoÞ Z1 2 þ 02 . (4) 2o Formula (4) is suitable to evaluate refractive-index perturbation for higher frequencies of the optical spectrum. Of course, from this formula, it can be appreciated that Z(o) is negative.
In summary, we have developed new ideas about negative refraction within the context of the plasmaoptical effect in a quantitative way. In fact, we have derived an expression for the (negative) refractive index of optical semiconducting metamaterials by taking into consideration that the corresponding dielectric permittivity is negative and the fact that free-carrier effective mass depends upon carrier concentration. In particular, from the above-mentioned expression, a formula that gives the refractive-index perturbation for higher frequencies has been obtained; this formula is valid for both the visible and ultraviolet ranges.
References [1] F. Wooten, Optical Properties of Solids, Academic Press, New York, NY, 1972, pp. 52–55. [2] M.A. Grado-Caffaro, M. Grado-Caffaro, Plasma-optical effect in GaAs PIN photodiodes, Act. Pass. Electron. Comp. 15 (1993) 63–66 and references therein. [3] M.A. Grado-Caffaro, M. Grado-Caffaro, Negative refractive index and anomalous dispersion, Optik 117 (2006) 141–142 and references therein.
Optik
Optics
Optik 119 (2008) 705–706 www.elsevier.de/ijleo
SHORT NOTE
Negative refraction in the light of the plasma-optical effect M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/ Julio Palacios 11, 9-B, 28029 Madrid, Spain Received 5 January 2007; accepted 10 April 2007
Abstract A mathematical expression for the refractive index of a semiconducting material with negative refraction is presented within the context of the plasma-optical effect. The basic ingredients for writing down this expression are the fact that the index of refraction of the above material is negative and the fact that free-carrier effective mass depends upon carrier spatial density. In particular, negative refraction for higher optical frequencies is discussed. r 2007 Elsevier GmbH. All rights reserved. Keywords: Negative index of refraction; Plasma-optical effect; Higher optical frequencies
The plasma-optical effect is based upon the fact that, in a semiconductor, the electrons of the conduction band and the holes of the valence band behave as very free carriers (see, for instance, Refs. [1,2]). Although the above-mentioned fact is apparently idealized, it provides a useful framework for elucidating the fundamental problems related to the plasma-optical effect and for understanding the most significant applications. The aim of this paper lies on the mathematical description of negative refraction in the light of the plasma-optical effect; in this description, carrier-concentration dependence of the effective masses of electrons and holes will be taken into account. As a relevant particular case, negative refraction will be examined for higher optical frequencies. Our starting point is the following relationship for the local refractive index of a semiconductor under the freecarrier approach (see, for example, Refs. [1,2]): " ZðoÞ ¼ Z1
e2 n p 1 2 þ o mnn mnp
!#1=2 ,
where o denotes optical angular frequency, ZN is the refractive index at very high frequencies (o-N), e is the electron charge, e is the dielectric permittivity, n and p are the electron and hole spatial densities, respectively, and mnn ; mnp are the electron and hole effective masses, respectively. Now we are interested in formulating an expression for a negative refractive index from formula (1). To this end, given that one has eo0 in the context of negative refraction and by taking only the minus sign in the square root of Eq. (1), it follows for a metamaterial of negative refraction (see Ref. [3]): " ZðoÞ ¼ Z1
e2 n p 1þ þ jjo2 mnn mnp
!#1=2 .
(2)
From Eq. (2) we state the following relationship for the refractive-index perturbation, assuming that free-carrier effective mass depends on carrier concentration:
(1)
Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.04.012
DZðoÞ Z1 ZðoÞ 8 " !#1=2 9 < = e2 an n ap p , þ ¼ Z1 1 þ 1 þ : ; jjo2 mnn mnp
ð3Þ
ARTICLE IN PRESS 706
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 119 (2008) 705–706
where "
1 an n ap p e þ n jj mnn mp
!#1=2 o0
stands for plasma resonant angular frequency, an and ap being corrective factors corresponding, respectively, to electrons and holes; these factors give a certain measure of the carrier-concentration dependence of free-carrier effective mass. Now we are interested in deriving from formula (3) an approximate relationship valid for the visible and ultraviolet regions. To this end, by expanding the right-hand side of relation (3) in a first-order Taylor series of 1/o2, we arrive at the following expression: o2 DZðoÞ Z1 2 þ 02 . (4) 2o Formula (4) is suitable to evaluate refractive-index perturbation for higher frequencies of the optical spectrum. Of course, from this formula, it can be appreciated that Z(o) is negative.
In summary, we have developed new ideas about negative refraction within the context of the plasmaoptical effect in a quantitative way. In fact, we have derived an expression for the (negative) refractive index of optical semiconducting metamaterials by taking into consideration that the corresponding dielectric permittivity is negative and the fact that free-carrier effective mass depends upon carrier concentration. In particular, from the above-mentioned expression, a formula that gives the refractive-index perturbation for higher frequencies has been obtained; this formula is valid for both the visible and ultraviolet ranges.
References [1] F. Wooten, Optical Properties of Solids, Academic Press, New York, NY, 1972, pp. 52–55. [2] M.A. Grado-Caffaro, M. Grado-Caffaro, Plasma-optical effect in GaAs PIN photodiodes, Act. Pass. Electron. Comp. 15 (1993) 63–66 and references therein. [3] M.A. Grado-Caffaro, M. Grado-Caffaro, Negative refractive index and anomalous dispersion, Optik 117 (2006) 141–142 and references therein.