The enhancement of the electromagnetic field within the grooves of a metallic diffraction grating is  analyzed. A perfect conductor, rectangular-shaped diffraction grating is assumed, therefore, the field enhancement  analyzed in this work is due to resonances caused by the structure itself, and not to the so-called surface-plasmon  resonances which are observed to be influenced by metal properties as well. The modal approach has been employed  for calculations. Basically, the squared amplitude of the electric field within the grooves is calculated as a function of  the wavelength of the incident light. Similarly, the behaviour of the structure when exposed to a conical bundle of  light is also analyzed. The present results show that a field intensity enhancement of the order of 100 is achievable.  However, the spectral width of the resonances decrease with increasing enhancement as is expecte

Jakas, Mario M.

Llopis, Francisco

Tobías Galicia, Ignacio

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

LIGHT TRAPPING BY MEANS OF ELECTRIC FIELD ENHANCEMENT IN 
METALLIC GRATINGS 
 
F.J. Llopis1*, M. M. Jakas1 and I. Tobías2 
1Departamento de Física Fundamental y Experimental, Electrónica y Sistemas. 
Universidad de La Laguna, c/Astrofísico Sánchez s/n, E-38025 La Laguna, S.C. de Tenerife, Spain. 
Corresponding autor: fllopis@ull.es 
2Instituto de Energía Solar. 
ETSI Telecomunicación, Universidad Politécnica de Madrid, E-28040 Madrid, Spain. 
 
ABSTRACT: The enhancement of the electromagnetic field within the grooves of a metallic diffraction grating is 
analyzed. A perfect conductor, rectangular-shaped diffraction grating is assumed, therefore, the field enhancement 
analyzed in this work is due to resonances caused by the structure itself, and not to the so-called surface-plasmon 
resonances which are observed to be influenced by metal properties as well. The modal approach has been employed 
for calculations. Basically, the squared amplitude of the electric field within the grooves is calculated as a function of 
the wavelength of the incident light. Similarly, the behaviour of the structure when exposed to a conical bundle of 
light is also analyzed. The present results show that a field intensity enhancement of the order of 100 is achievable. 
However, the spectral width of the resonances decrease with increasing enhancement as is expected. 
 
Keywords: Light Trapping; Optical Properties; Absorption 
 
 
1 INTRODUCTION 
 
Since the performance of silicon solar cells is 
affected by partial absorption of infrared wavelengths,    
textured surfaces have been commonly employed to 
confine light by means of total internal reflection                
[1] . This approach, however, is not always easily applied 
in thin film devices. On the other hand, recently proposed 
devices based on up-conversion schemes, or intermediate 
band solar cells (IBSCs), also exhibit low absorbance in 
the infrared range. In both cases, it has been suggested 
that the conversion efficiency relies on the achievement 
of high electromagnetic densities [2,3]. 
Light absorption increases with the squared 
amplitude of the electric field, which in turn can be 
boosted by means of resonant enhancement.  To this end, 
in this work we focus our attention on the behaviour of 
the electromagnetic field inside the grooves of 
rectangular-shaped metallic gratings. It has been shown 
previously that, at certain wavelengths, the intensity of 
the electric field can be enhanced up to two orders of 
magnitude inside the grooves if geometrical features are 
properly chosen The electric field intensity is reinforced 
by means of constructive interference, in a similar 
manner as the resonances supported by electromagnetic 
resonators [4, 5].  
The results of this work show that the enhancement 
increases with the groove depth to width ratio and that 
the spectral width of the resonances behaves in the 
opposite sense. The enhancement of the electric field is 
assumed to be linked to the resonances produced by the 
ideal structure itself. The so called surface plasmon 
resonances (the excitation of collective oscillations of the 
electronic surface density), are influenced by the metal 
properties [5], and are not considered in this work.  
 
2 OUTLINE OF THE THEORY 
 
As depicted in figure 1, we assume that a perfectly 
conducting grating with groove spacing d is exposed to a 
monochromatic TE polarized wave with a vacuum 
wavelength λ. The refractive index of the incidence 
medium is ν1. The groove depth and aperture are h and c, 
respectively. The incident electric field can be expressed 
as 
 ( ) ( )[ ]yxiyxEzi 00exp, χα −=  , (1) 
 
where the components of the incident wavevector are 
given by α0=κν1sinθ and χ0=κν1cosθ, with θ  being the 
incidence angle and κ=2π/λ the vacuum wavenumber. 
 
Figure 1: Grating illuminated by TE polarized light. 
 
 The interaction between the incoming wave and the 
grating results in [4]: 
(1) A reflected electric field that can be expressed in 
terms of the plane wave expansion 
 
( ) ( ) ( )[ ]∑∞
−∞=
++=
n
nnnziR yxiAyxEyxE χαexp,,  , (2) 
 
where An is the amplitude of the nth reflected plane 
wave, whose  wavevector componentes are αn= α0+2πn/d 
and χn=[(κν1)2− αn2]1/2.   
 
(2) An electric field inside the grooves that can be 
expressed by the modal expansion 
 
( ) ( )∑∞
=
=
1
,,
m
mmM yxayxE φ  , (3) 
where:  
 φm(x,y)=sin[μm(y+h)]sin(mπx/c) if m is even, 
 φm(x,y)=sin[μm(y+h)]cos(mπx/c) if m is odd, 
and μm = [(κν1)2 –(mπ/c)2]1/2. 
We have employed the procedure reported in [4] to 
25th European Photovoltaic Solar Energy Conference and Exhibition /
5th World Conference on Photovoltaic Energy Conversion, 6-10 September 2010, Valencia, Spain
404
calculate the modal amplitudes. The main results are 
discussed in the following section. 
 
3 RESULTS 
 
3.1 TE polarization 
For normal incidence, the behaviour of the internal 
TE electric fields can be more easily discussed if the 
conditions for operation of the monomode regime hold 
[4]. In this case, the only non-imaginary y-spatial 
frequency is that of the m=1 mode (fundamental mode). 
The remaining modes, i.e. m=2,3,…, are evanescent. The 
resonance wavelengths are close to the zeros of 
tan[μ1(λ)h], with μ1 being the y-spatial frequency of the 
fundamental mode. These quasi-resonance wavelengths 
are expressed as  
 
 λj = 2ν1(j2/h2+1/c2)-1/2, j=1,2,… 
 
As appointed in [4], the magnitude of the electric field 
will be larger the smaller is μ1(λj). This implies to select 
h as large as possible. In the limit ∞→h  we find λj = 
2ν1c. Thus, we have an estimate of the groove width c for 
a given resonance wavelength, provided that grooves are 
deep enough.  
 The electric field enhancement can be estimated in 
terms of the squared amplitude of the electric field at the 
centre of the grooves. The plots depicted in figure 2 have 
been obtained for ν1=3.6, a resonant wavelength λ=4.13 
μm (1.24/0.3 eV) and a ratio d/c=1.3. The enhancement 
increases with the h/c ratio and can be of the of the order 
of 100 (or even greater). The main peaks are also located 
near our objective for the greater h/c ratios, as expected. 
But the spectral width decreases with the h/c ratio. (For a 
more comprehensive discussion of the spectral behaviour 
of the electric field, see Ref. [6]).  
 
 
Figure 2: Plots of the squared magnitude of the 
normalized electric field for d=1.3c and different depth to 
width ratios. 
 
 It must be pointed out that prototype IBSCs operating 
with concentrated light can be exposed to a conical 
bundle of light of about 11º for concentration levels of  
1000× [3]. Thus, bearing in mind that the case of oblique 
incidence must not be disregarded, we have performed 
the calculation of the electric field enhancement for 
several incidence angles (figure 3). The same d/c ratio 
has been assumed. It can be noticed that the enhancement 
levels for θ =0º and θ =10º are nearly the same. The main 
peaks for θ =30º drop to about the half of normal 
incidence ones, and the enhancement effect is fairly 
small for angles above 50º. 
 
 
Figure 3: Influence of the incidence angle. 
 
3.2 TM polarization 
 When employing the aforementioned criterion for 
TM polarization, we cannot expect a noticeable 
enhancement of the electric field. This drawback cannot 
be overcome since the fundamental mode for TM 
polarization is enhanced for wavelengths that obey a 
quite different condition, i.e. cos(μ0h)=0 [5]. The electric 
field can be enhanced up to a factor of 2d2/c2, which 
implies a decrease of the grooves size. The simultaneous 
enhancement of the TE and TM electric fields can be 
attained by imposing a condition of surface resonance 
(d=λ/ν1) [5]. However, the results obtained so far show 
that the enhancement is very sensitive to changes of the 
width c and depth h of the grooves [6]. 
 
4 SUMMARY 
 
 We have investigated the possibility of increasing the 
absorption of solar cells by using a metallic grating.  
Results in this work show that a noticeable enhancement 
of the electric field can be achieved for TE polarization if 
grooves are deep enough for small incidence angles. 
Since the internal field enhancement obey quite different 
conditions for TE and TM modes, a condition that ensure 
the existence of surface resonances can be employed 
instead [5,6].  
 
5 ACKNOWLEDGEMENTS 
 
This work has been supported by the European Union 
(EU) Program IBPOWER (211640). 
 
 
6 REFERENCES 
 
[1] B. Sopori, in “Handbook of Photovoltaic Science and 
Technology”, A. Luque and S. Hegedus (eds.), Wiley 
(2004) 307. 
[2] H. Shpaisman, O. Niitsoo, I. Lubomirsky, and D. 
Cahen, Sol. Energy Mater. Sol. Cells 92 (2008) 1541. 
[3] A. Luque, A. Martí, Advanced Materials 21 (2010) 1. 
[4] A. Wirgin, A. A. Maradudin, Phys. Rev. B 31 (1985) 
5573. 
[5] F. J. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, 
E. Bustarret, T. López-Ríos, T. Fournier, and B. 
Pannetier, J. Lightwave Tech. 17 (1999) 2191. 
[6] F. Llopis, I. Tobías, and M. M. Jakas, paper accepted 
for publication in J. Opt. Soc. Am. B. 
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5th World Conference on Photovoltaic Energy Conversion, 6-10 September 2010, Valencia, Spain
405


Light trapping by means of electric field enhancement in metallic grantings

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Light trapping by means of electric field enhancement in metallic grantings

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