Optical Design for Increase of the Photon Density in Matter

The photon density in matter and likewise the degree of photons available for exploitation is governed not only by incomplete absorption A — 1 — exp (a (ha) d) < 1 but equivalently by incomplete coupling of photons to the absorber resulting from reflection. As a consequence of the contrast in refractive indices between two media, e. g., air and absorber, a substantial fraction of photons may be reflected unless anti-reflection strategies are applied.

The reflection coefficient for the amplitude (rampl) of an electromagnetic wave is found by considering the electric field strength propagating across an interface between materials with (complex) refractive indices n 1 and n2 (see Appendix A.2 and see footnote 3 in Sect. 6.1.3.2). This leads for perpendicular incidence on the absorber to

Подпись: rampl —П1 — n2 П1 + П2

Optical Design for Increase of the Photon Density in Matter Подпись: n 1 — n 2 Подпись: 2

whence the reflection factor for the photon flux will be the square of this magnitude, viz.,

For a typical absorber like c-Si with real part of the refractive index in the near IR – range n2 ^ 3.5, one gets a reflection factor of the photon flux for normal incident illumination of ry — (2.5/4.5)2 ^ 0.31, implying that only 69% of the light incident on the absorber would be coupled in, an entirely unacceptable situation. For this reason, anti-reflective (AR) films are generally used to coat the absorber. These exhibit a particular combination of spectral refractive indices and thicknesses which allows most solar photons with energies above the band gap to be wave-optically fed to the absorber medium.

In addition to AR coatings, specific surface contours are prepared, such as comparatively large scale (mm) 2D regular or random sized pyramids, or inverted pyramids, 1D grooves, etc. to ‘scatter’ that part of the solar light that is reflected

absorber with absorber with

flat surface rough surface

Optical Design for Increase of the Photon Density in Matter

at first contact with the device back towards the surface again in order to provide another opportunity to be coupled in (see Fig. 6.6).

Another artificial increase in photon propagation length at the front and/or rear surface can be obtained by ideally randomized photon scattering into the solid angle of 2n, facilitated by small scale rough surface or interface contours. Due to the absorption of photons, the flux Г(х) and the total rate of photoexcited species g(d), be they electron-hole pairs or excitons, only asymptote towards the maximum achievable values of initial flux Г0 and total possible accumulated generation rate go

Г(х)

! 0 , or g(d) ! go,

Г 0

with a strong dependence on the absorption coefficient a(rn), and on the length d of the path the photons travel in the absorber. The photon path length d is usually an unambiguous quantity, since we consider only one direction of light propagation in the absorbing medium. However, when front or rear surfaces of the absorber are strongly scattering, photon propagation becomes randomly oriented and the average path length increases, whereupon the accumulated generation rate g(d) also rises (see Fig. 6.7).

From geometrical optics in conjunction with statistics, the upper limit for the increase in effective absorption lengths to be achieved by ideal scattering into the 2n-solid angle (Lambertian type of scattering) at the front as well as at the side amounts to a factor of 4n2 (see [10] and Appendix C). In Fig. 6.8, the

Fig. 6.8 Normalized absorbed photon flux versus absorption length in absorbers with flat and with ideally rough surface (optimum scattering). The increase in effective absorption length is governed by the absorber refractive index n

Optical Design for Increase of the Photon Density in MatterOptical Design for Increase of the Photon Density in MatterFig. 6.9 Ratio of absorption of photons in matter with ideally scattering surfaces with FScatt =

Log[exp(—n2ad)/exp(—ad)] (Lambertian scattering) as compared with a flat surface plotted against absorber thickness for different refractive indices n

accumulated generation rate g(d)—equivalent to the absorbed photon flux—for maximum random scattering versus absorber thickness is sketched for different refractive indices n = 2-4. For comparison, the absorbed flux is displayed for perpendicular propagation through a flat surface without scattering.

The factor Fscat for the increase in total generation by scattering surfaces versus absorber thickness d as compared with a flat surface is shown in Fig. 6.9.

An even greater light-trapping effect has been proposed by combining random­ized scattering surfaces and spectrally selective filters [11].