1. INTRODUCTIONIt is well documented that electromagnetic fields within thegrooves of diffraction grating can be largely increased whenilluminated by light [1–4]. This phenomenon was recently pro-posed as a way to enhance the absorption of light in PV cellsand optoelectronic devices [5–8]. In fact, for the purpose ofincreasing the efficiency of photo-cells where sunlight maynot be absorbed so easily, the field enhancement appears tobe a useful approach. In this realm, however, rather than dif-ferential, one needs integrated figures. This is so becauselarge field enhancements, as those previously reported, arenot enough if not accompanied by a net increase of the lightenergy integrated over a representative volume of the cell,along the pertinent range of wavelengths, incidence anglesand states of polarization.
In a previous paper of the authors [9], the field enhance-ment was analyzed for the more general case of beams ofmonochromatic light arriving to the grating along several di-rections and different states of polarization. There, an en-hancement coefficient, namely η, is introduced, so thatη > 1 means that the amount of electromagnetic energy pre-sent within the grooves is larger than that one will have, overthe same volume, if the diffraction grating is replaced by a per-fectly reflecting mirror. In this paper, however, the previousstudy is extended to the case of illuminating the grating withsolar light. The results of these calculations show that, whenusing monochromatic, polarized, and well-collimated beamsof light, η exhibits a series of peaks at certain wavelengths,where the enhancement coefficient can be as large as severaldecades, or even greater. However, after taking an averageover incidence angles and polarization states, η is significantlyreduced as it may reach values that can hardly be larger thanapproximately 3. If the previous results are also averaged over
the solar spectrum, the enhancement coefficient is furtherreduced to approximately 1.20.
Although these results are to some extent discouraging, itdoes not mean that the diffraction grating structure cannot beadvantageously used in designing photo-electronic devices.This is so because, depending on the case, the aforementioned20% gain may suffice. In this regard, the present study could beinteresting to those who may need to increase the absorptionof light within a solar cell and might possibly be planning toincorporate a diffraction grating for such a purpose. This pa-per is organized as follows: the basic equations necessary toobtain the field within the grooves of a diffraction grating arederived in Section 2. The results of numerically calculating theenhancement coefficient as a function of the various para-meters in the model are shown and discussed in Section 3and, finally, Section 4 contains a summary and concludingremarks.
2. BASIC EQUATIONSThe calculation model is sketched in Fig. 1. There, one canobserve the incident light, a linearly polarized plane-wave, ar-riving to the grating from above at an angle θi with respect tothe y axis and azimuthal angle φi with respect to the x axis.The light has a wavevector ki ¼ ðkix; kiy; kizÞ, which is relatedto the wavelength in the upper medium (λ) as jkij ¼ ki ¼ 2π=λ.The electric field amplitude Ei is assumed to be perpendicularto ki and ϕ is the angle formed by Ei and the (y ¼ 0) plane.
The diffraction grating is assumed to be an infinite set ofequally spaced, rectangular-shaped grooves made on the sur-face of a perfect conductor. The spatial period of the groovesis d, whereas the width and depth are b and h, respectively.Finally, the upper medium is assumed to be a nonlossy dielec-tric material that fills the upper part of the grating, up to a
2650 J. Opt. Soc. Am. B / Vol. 28, No. 11 / November 2011 M. M. Jakas and F. Llopis
front surface which will not be taken into account in thiscalculations.
The way the electromagnetic fields within the groovesare calculated was already shown in [7,9]; however, for thepurpose of the present analysis, it is important to re-do it oncemore.
In the first place, the Rayleigh expansion is assumed forupper space, where qn is the wave vector of the nth orderreflected beam and Rn ¼ ðRnx; Rny; RnzÞ is the correspondingelectric field vector. Similarly, qn ¼ ðqnx; qny; kizÞ, whereqxn ¼ kix þ 2πn=d, n ¼ 0;�1;�2;…, and qny ¼þ
square root must be taken. Similarly, xN denotes the x coor-dinate of the right wall in the Nth groove, i.e., xN ¼ Ndfor N ¼ 0;�1;�2;….
Both, the continuity of the x and z component of the elec-tric field along the groove aperture, and the boundary condi-tions of electromagnetic fields on the surface of a perfectconductor (see [10,11]), allow one to write
Eixeikixx þXþ∞
n¼−∞
Rnxeiqnxx ¼�Pþ∞
m¼0 Ex;m cosðKmxxÞ sinðKmyhÞ 0 < x < b0 b ≤ x < d
ð3Þ
and
Eizeikixx þXþ∞
n¼−∞
Rnzeiqnxx ¼�iPþ∞
m¼0 Ez;m sinðKmxxÞ sinðKmyhÞ 0 < x < b0 b ≤ x < d
: ð4Þ
By taking the Fourier transform of these equations in the xvariable and after rearranging terms, one obtains
Rnx ¼ −Eixδn;0 þ1d
Xþ∞
m¼0
Ex;m~Cm;n sinðKmyhÞ ð5Þ
Rnz ¼ −Eizδn;0 þid
Xþ∞
m¼0
Ez;m~Sm;n sinðKmyhÞ; ð6Þ
where
~Sm;n ¼Z
b
0dxe−iqnxx sinðKmxxÞ and
~Cm;n ¼Z
b
0dxe−iqnxx cosðKmxxÞ: ð7Þ
Similarly, the continuity of the x and y components of themagnetic field along the grooves aperture yields
2ðk2iy þ k2izÞEiz þ kixkizEix
kiyexpðikixxÞ
¼Xþ∞
m¼0
��KmxkizKmy
sinðKmxxÞ cosðKmyhÞ
− sinðKmyhÞU ðxxÞm ðxÞ
�Ex;m
þ�K2
my þ k2izKmy
sinðKmxxÞ cosðKmyhÞ
− sinðKmyhÞU ðxzÞm ðxÞ
�Ez;m
�; ð8Þ
and
2ðk2iy þ k2ixÞEix þ kixkizEiz
kiyexpðikixxÞ
¼Xþ∞
m¼0
��K2
mx þ K2my
KmycosðKmxxÞ cosðKmyhÞ
− sinðKmyhÞU ðzxÞm ðxÞ
�Ex;m
þ�KmxkizKmy
cosðKmxxÞ cosðKmyhÞ
þ sinðKmyhÞU ðzzÞm ðxÞ
�Ez;m
�; ð9Þ
M. M. Jakas and F. Llopis Vol. 28, No. 11 / November 2011 / J. Opt. Soc. Am. B 2651
where
U ðxxÞm ðxÞ ¼ 1
d
Xþ∞
n¼−∞
qnxkizqny
~Cm;neiqnxx;
U ðxzÞm ðxÞ ¼ i
d
Xþ∞
n¼−∞
q2ny þ k2izqny
~Sm;neiqnxx
U ðzxÞm ðxÞ ¼ i
d
Xþ∞
n¼−∞
q2nx þ q2nyqny
~Cm;neiqnxx;
U ðzzÞm ðxÞ ¼ 1
d
Xþ∞
n¼−∞
qnxkizqny
~Sm;neiqnxx: ð10Þ
It must be noticed that no equations for the y componentsof the field are necessary since the null divergence condition,i.e., ∇ · E ¼ 0, implies that only two out of the three compo-nents of both Eðy>0Þ and Eðy<0Þ are independent. Finally, thevalues of Ex;m and Ez;m can be found by evaluating Eqs. (8)and (9) over a finite set of equally spaced x’s along the grooveaperture, and the resulting system of linear algebraic equa-tions is then solved by resorting to the Gauss elimination al-gorithm in [12]. Once Ex;m and Ez;m are obtained, the reflectionamplitudes, i.e., Rnx and Rnz, can readily be calculated fromEqs. (5) and (6).
Having arrived at this point, the enhancement coefficient(η) is introduced, namely,
η ¼ 1
2AjEij2Z
b
0dx
Z0
−hdyjEðx; yÞj2; ð11Þ
where jEij is the electric field amplitude of the incoming light,Eðx; yÞ is the electric field within the groove, and A is the crosssectional area of the groove, i.e., A ¼ hb. Firstly one has tosolve Eqs. (5) and (6); then the electric field in Eq. (11) is ob-tained from Eq. (2). It must also be observed that no integra-tion over the z coordinate is necessary, since for 1D gratingsjEðx; yÞj2 does not depend on z.
As was previously mentioned [9], such a coefficient allowsone not only to quantify the field enhancement but also, andmost importantly, it tells when there is a net gain of electro-magnetic energy within the grooves. Actually, one can readilysee that η ¼ 1 denotes the case for which, on an average, theelectrical energy density within the grooves is the same as thatpresent above, far away from the diffraction grating.
It must be stressed that the main goal in this paper is thecalculation of η, as well as analyzing its sensitivity upon thevarious parameters of the model. Before going into the results,however, it must be noted that for the purpose of exploitingthe trapping of light within the grooves, one can reasonablyassume that d ¼ b. By doing so, the number of variables is re-duced by one and b can be used as the unit of length. Besides,even in the worst case in this paper, most metals exhibit skindepths approximately 2 orders of magnitude smaller than thevalues of d. Accordingly, apart from θi, φi, jEij, and ϕ, the so-lutions to Eqs. (8) and (9) must be functions of bki (or λ=b) andh=b. The results of numerically calculating these equations un-der the aforementioned assumptions are produced in the fol-lowing section. It must be mentioned, however, that all alongthe following sections, when referring to wavelength it is as-sumed to be that in vacuum. Only within the numerical code,
the wavelength is translated into that of silicon using therefractive index from [13].
3. RESULTS AND DISCUSSIONSThe enhancement coefficient is calculated for a 30° conicalbeam of nonpolarized light. The results are plotted in Fig. 1,where η appears as a function of the wavelength and forgrooves of depths h=b ¼ 1, 2, 4, and 8. These data are obtainedby calculating the mean-value of η in Eq. (11) over all polar-ization states, incidence angles within the aforementionedcone, and an interval of wavelengths, whose width equals thedistance between the points in the graph, namely Δλ ¼ 0:1d.To this end, a Monte-Carlo integration scheme is used, whererelative uncertainties of the order of 10% or less are normallyachieved.
As one can readily see in Fig. 1, the enhancement coeffi-cient exhibits a single large peak around λ=b ≈ 1:8, becomingas large as approximately 3 for the four groove depths anal-yzed in this paper. For λ=b ≤ 1:8, η decreases and becomesslightly greater than unity for λ=b ≤ 1:2. However, one cansee that η falls below unity as soon as λ=b > 2. Curiously en-ough, the enhancement coefficient does not exhibit a strongdependence with the groove depth h and, as a matter of fact,differences between the results corresponding to the differentgroove depths are masked by the noisy aspect of the curves.This noise stems in part from the statistical fluctuations in theMonte-Carlo calculations, and also from the narrow reso-nances occurring within the groove which, as will be seenbelow, have been considerably reduced as a result of takingaveraged values.
In order to analyze the results in Fig. 1, η is calculated forlight arriving to the grating along the normal direction and fortwo limiting cases of polarization states, namely, those withthe electric field perpendicular and parallel to the grooves, re-spectively. The results are plotted in Fig. 2, where, in order toavoid a busy plot, only the results corresponding to h=b ¼ 1and 8 appear.
The results in Fig. 3(a) clearly show that when the electricfield is perpendicular to the grooves, the field enhancementsare oscillating functions of λ=b, becoming scarcely larger than
Fig. 1. A linearly polarized electromagnetic wave, with wave vectorki, electric field Ei, incidence angle θi and azimuthal angle φi, arrivesfrom the upper half-space upon a rectangular-shaped diffraction grat-ing. Grooves have a spatial period d, width b, and depth h, and ϕdenotes the polarization angle with respect to the (x-z) plane.
2652 J. Opt. Soc. Am. B / Vol. 28, No. 11 / November 2011 M. M. Jakas and F. Llopis
unity over nearly the entire range of wavelengths calculated inthe present paper. On the contrary, when the electric field isdirected along the z axis, as shown in Fig. 3(b), η exhibits aseries of large peaks, regularly spaced, with an amplitude thatappears to increase with increasing wavelength. This is so,however, for λ=b ≤ 2, because for λ=b > 2 the field enhance-ment falls below unity, and it does so at a rate that seemsto be an increasing function of the groove depth. This howeveris not at all unexpected, since Ez must be zero all over thesurface of the grating; therefore, a Ez-polarized light cannotpenetrate within the groove as soon as its wavelength be-comes comparable to, or larger than, the groove width.
Figure 4 shows four limiting cases of beams arriving to thegrating with 30° incidence angle (θi). These comprise the TE-and TM-polarization state and 0 and 90° azimuthal angles (φi).The first conclusion one may extract from these curves is thatif the electric field of the light lies on a plane that is parallel tothe grooves, such as the (b) and (c) cases, then there appearsa cut-off wavelength above which the field enhancement fallsdown below unity quite rapidly. However, if the electric fieldis perpendicular to the grooves, η remains greater the unityover the entire range of wavelengths analyzed in this paper.This is obvious, since no large electric fields can be developedwithin the grooves if they are parallel to the lateral and bottomsurfaces.
As was already observed elsewhere [1,2,7,9], the results inFigs. 4(a)–4(d) show a number of peaks that can reach valuesas large as several decades. These peaks are observed to oc-cur at wavelengths that depend on the groove depth h, theincidence and azimuth angles, and the state of polarization.Curiously though, such peaks are often so narrow that manyof them nearly disappear after taking an average, as seenin Fig. 2.
Finally, one may find the enhancement coefficient of thegrating exposed to sunlight. In this case, one must take anaverage of η over all polarization states and wavelengths inthe solar spectrum,
ηsun ¼R λ0m0 dλ0ΦEðλ0Þλ0ηR λ0m0 dλ0ΦEðλ0Þλ0
; ð12Þ
where λ0 is the wavelength of the light in vacuum, ΦEðλ0Þ isthe spectral flux density of the sun light, η is the enhancementcoefficient for a given wavelength and after averaging over allpolarization states, and λ0m is the largest wavelength of thelight that can promote electrons from the valence to the con-duction band. Notice that the factor λ0 appearing in both in-tegrals is required in order to obtain the photon densityspectrum from ΦEðλ0Þ, which is obtained from the so-calledReference AM 1.5 spectra in [14]. It must be also noticed that,since the wavelength in vacuum is used, the dimensions of thegrating must be scaled using the refractive index n of the med-ium that fills the groove. For the sake of simplicity, such anindex is assumed to be constant along the solar spectrum andabsorption is ignored; as a consequence, ηsun will be a functionof both the optical spacing of the grooves, i.e., dn and theaspect ratio h=b.
Equation (12) is calculated using the Monte-Carlo method,and the results are plotted in Fig. 5. There, one can see that theenhancement coefficient becomes larger than unity for dn ap-proximately greater than 500nm, whereas for nd smaller thanthis value, ηsun drops down fairly fast. For optical spacing be-tween the grooves greater than 500nm the enhancement co-efficient becomes as large as 1.2 and stays around this valueup to dn ¼ 3600nm, which is the largest dn value calculated inthis paper. Remarkable though, in a similar fashion as was pre-viously observed in Fig. 1, ηsun appears to nearly not dependon the aspect ratio h=d.
4. SUMMARY AND CONCLUDINGREMARKSThe performance of a diffraction grating as a light trappingstructure for PV cells applications is analyzed. To this end,
Fig. 2. Enhancement coefficient for a h=b ¼ 1 (○), 2 (▵), 4 (▿), and8 (⋄) diffraction grating, illuminated by nonpolarized light, uniformlydistributed over a 30° wide, vertical cone.
Fig. 3. Enhancement coefficient for h=b ¼ 1 (full line) and 8(short-dashed line) diffraction gratings, illuminated by a normally in-cident light with electric field directed along the (a) x and (b) z axis,respectively.
M. M. Jakas and F. Llopis Vol. 28, No. 11 / November 2011 / J. Opt. Soc. Am. B 2653
the electric fields produced within the grooves of a perfectlyconducting diffraction grating by a well collimated, mono-chromatic, linearly polarized beam of light are calculated.The grating has a d period and is assumed to be made of in-finitely long rectangular-shaped, h deep and b wide grooves,although, for the purpose in this paper, it is assumed thatd ≈ b. Furthermore, in order to properly assess the grating,an enhancement coefficient (η) is introduced. η is definedas the average electromagnetic-energy within the volume ofthe grooves relative to that one will have if grating is replacedby a flat perfect reflector. In this regard, a η greater than unityimplies that light is being trapped within the grooves of thediffraction grating. Results in this paper agree with previouscalculations [1–4,7] that, for certain polarization state, wave-length and incidence angle, η can be substantially larger thanunity. When using conical, nonpolarized beams, such en-hancements, however, are reduced and η can hardly be larger
than 3. Moreover, these values are observed for wavelengthsin the propagating medium within the range b < λ < 2b. Forλ < b, η appears to be slightly greater than unity irrespectiveof the polarization state and incidence angle, provided thatthis later is not greater than 30 deg. For λ > 2b, however,the enhancement coefficient seems to be always smaller thenunity. This is particularly so when the electric field of the in-coming light is parallel to the groove direction. Finally, theresults of calculating the enhancement coefficient of the dif-fraction grating under the so-called Reference AM1.5 solarspectrum, namely ηsun, show that ηsun can be, in the best ofthe cases, of the order of 1.2. This maximum occurs for opticalspacing of the groove dn approximately equal to 500nm,whereas for dn ≤ 500nm ηsun becomes negligibly small. Inall the cases, ηsun does not seem to depend on the aspect ratioh=b. It is worth mentioning that, although the structure of theideal cell sketched in this paper is different to that in [8], thefigures reported by these authors are similar to those in thepresent calculations. In short, diffraction grating may concei-vably act as a light trapping structure and, consequently, in-crease the efficiency of a PV cell. Figures appear to bearound 20% for solar spectrum and groove spacing is expectedto be in the submicrometer scale.
ACKNOWLEDGMENTSWe are indebted to I. Tobias, A. Martí and A. Luque forencouraging the authors to work on this problem.
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Fig. 4. Enhancement coefficient for h=b ¼ 1 (full line) and 8 (short-dashed line) diffraction gratings. Light arrives at 30° incidence angle (θi) uponthe grating, whereas the azimuthal angle (φi) and the electric field direction angle (ϕ), both in degrees, are: (a) 0, 90; (b) 0, 0; (c) 90, 90; and (d) 90, 0.(see Fig. 1 for an explanation of these angles).
Fig. 5. Enhancement factor of the diffraction grating under AM 1.5solar irrandiance assuming normal incidence. Calculations are per-formed for several values of the optical spacing of the grooves, i.e.,dn, and two aspect ratios h=d ¼ 1 (○) and 8 (⋄).
2654 J. Opt. Soc. Am. B / Vol. 28, No. 11 / November 2011 M. M. Jakas and F. Llopis
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