CHAPTER - VII

SIMULATION STUDY ON

CYLINDRICAL MICROLENS USING

ZEMAX® AND EFFICIENT LIGHT

TRAPPING SCHEME FOR

V-GROOVE THIN SILICON SOLAR

CELLS USING MICROLENSES

CHAPTER - VII

SIMULATION STUDY ON CYLINDRICAL MICROLENS USING ZEMAX® AND

EFFICIENT LIGHT TRAPPING SCHEME FOR V-GROOVE THIN SILICON SOLAR CELLS

USING MICROLENSES

7.1 Introduction

Theoretical study on focal performance of cylindrical microlenses

under closed boundary system using BEM yields the good results for

efficient light trapping in thin silicon solar cells were reported and

thoroughly analyzed in chapter 6. Based on the obtained data from the

previous chapter VI, simulation study was carried out for the designing[1,2]

of cylindrical microlens and its focal characteristics[3] using a ray tracing

method[4] through ZEMAX® software[5] tool. In the subsequent sections,

the simulated cylindrical microlens was coupled with V-Groove thin silicon

solar cells as photovoltaic concentrators to enhance the light trapping

capability[6]. The reflectivity is one of the key factor which significantly

reduces the efficiency of solar cells were taken into consideration. The

various factors such as quantum efficiency, throughput of microlens etc.,

150

were reported and its values were compared with simulated results using

ZEMAX®.

7.2 Cylindrical microlens designing and its focal

characteristics through ZEIVIAX® software

7.2.1 Introduction

Practical design of lens systems in accordance with given

specifications is essentially a problem of nonlinear optimization in a

constrained multivariate hyperspace. Since the advent of digital computers,

optical designers have experimented with different optimization techniques

that can yield an optimum solution, given a starting point in the multivarient

hyperspace[7]. These experiments have been mostly successful, and they

have yielded the results sought: a large number of in-house and commercial

lens design software packages are in routine use around the globe[8].

Nevertheless, it is well known that these solutions represent optimum

solutions in the neighborhood of the given starting point[9-11]. Often

different starting point lead to different local optima, the designer making

use of the best among them. Practical success in any venture of lens design

depends to a large extent on choice of the starting point. Several interesting

investigations are currently being undertaken to circumvent this

problem[12]. Some are also trying to adopt stochastic approaches for

151

seeking a global optimum[13,14]. However, except for the trivial cases, the

curse of dimensionality associated with the large number of variables in

lens design problems seems to posses a fundamental limitation in seeking a

true global optimum from an arbitrary starting point in a reasonable time

frame.

7.2.2 Design of cylindrical rnicrolens using ZEMAX®

The objective in lens design is to find appropriate system parameters

of lens system such as the curvatures of surfaces, gaps between surfaces and

glasses of lens elements determined by two optical properties, the refractive

indices, so that the lens system has good image forming capabilities. The

lens design problem is known as a very difficult problem. This is because it

has many parameters to be determined including continuous variables and

discrete ones, strong epitasis among parameters and many local optima, and

also because it has many criteria such as well-known seidel aberration,

chromatic aberration, size and cost.

Most programs for lens design employ the ABCD method[15] as the

core of the optimization process today. Several global optimization methods

for lens design have been proposed.

There are many parameters that a lens designer must determine to get

a lens with good image forming capabilities; the curvature of surfaces, gaps

152

between two surfaces, glasses of lens elements, the number of lens elements

etc. It can improve the blur and the distortion of an image at a certain

wavelength by changing the curvatures of surfaces and gaps between

surfaces because it control the refraction of rays with one wavelength at

each surface by doing them[16]. However, the way a ray is refracted at a

surface, which is given by index of refraction, depends on its wavelength.

As a result, two rays with different wavelengths entering the lens system

from the same direction and at the same position make image points at

different positions on the image surface, which causes the chromatic blur or

chromatic aberration.

ZEMAX® is a lens design program that uses ray tracing to model

refractive, reflective, and diffractive sequential and non-sequential optical

systems. Most optical systems, and virtually all imaging systems, are well

described by the sequential surface model[17]. It is fast, efficient, and lends

itself to optimization and detailed analysis. In sequential ray tracing a ray

starts at the object surface (always surface number 0). The ray is traced to

surface 1, then to surface 2, and so on. Sequential ray tracing uses a

"surface" model: each transition from one optical space to another requires

a surface.

There are several "editors" in ZEMAX® that are essentially

spreadsheets customized[18] for the needs of a lens design program in

153

which lens data editor holds the basic information about the lens data

which describes the basic lens data, including surface types, radius,

thickness, glass, etc. given in Figure (7.1). The design a cylindrical

microlens made of BK7 with a 5 um focal length, for use on-axis in the near

infra-red spectrum. The data entered into ZEMAX®, set the system aperture,

lens units, and wavelength range and then optimize the design given in

Figure (7.2). Different field angles of both on-axis and off-axis had tried

and different field angle for off-axis of 2, 4, 6 and 8 degrees on either side

of optical axis were attempted. For wavelengths, the near infra-red spectrum

was used and it was entered into the wavelengths, editor. It has various

values from 1 ,um to 1.6 gm. The solid model and wire frame model of

designed cylindrical microlens using ZEMAX® were given in Figures (7.3)

and (7.4).

System data summery retrieves most system specific data, such as

effective focal length, working F I# , apodization factors, and other data not

associated with any particular surface. The data is placed in one of the

vector array variables (either VEC1, VEC2, VEC3, or VEC4). The data

which is stored in the specified VECn array variable will be reported as text

file tabulated in Table (7.1).

154

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Figure (7.1). Lens data editor with cylindrical microlens parameters with ZEMAX® main window

General

?

Files I Non-Sequential I Polarization I Ray Aiming I Misc. Aperture I Title/Notes I Units I Glass Catalogs I Environment

Aperture Type: bject Space NA L7.1

Aperture Value: 10 2

Apodization Type: I Gaussian

Ii

Apodization Factor:

E Telecentric Object Space

r Afocal Image Space

r Iterate Solves When Updating

OK Cancel I Help

Figure (7.2). General data editor for cylindrical lens system

•••

155

Cylindric al microlen.s

Ray trajectory

Focusing plane

Figure (7.3). Solid model of Cylindrical microlens with figed angles 0, 2, 4, 6, 8, 10 and 12 (deg) Tram principal axis

156

Cylindric al tnicrolens

Ray tr eCto

Focusing plane

Figure (7.4). Wire frame model of Cylindrical microlens with fkled angles 0, 2, 4, 6, 8, 10 and 12 (deg) from principal axis

157

Table (7.1). Summary of system data to study the focal characteristics of cylindrical microlens

Practical Focal Length : 5 gm

Radius of Curvature : 15 gm

Aperture Shape : Cylindrical

Geometric Aperture : 0.2

Diffraction Limited Optical Aperture : 0.2

Geometric Numerical Aperture : 0.2

Numerical Aperture : 0.2

Pitch : 250 gm

(150mm of dia)

Lens Density : 40x40x3 (mm3)

Fill Factor : 79%

Surface Modulation Depth : 80 gm

Physical Thickness : 120 gm

Substrate : 150 mm in dia

Common Focal Plane : 250 gm

Peak-to-Valley : 0.86

Sag : 12.7 gm

System Aperture : 0.2

Glass Catalogs : SCHOTT

Conic Constant : -2.205

Strehl : 0.455

158

7.2.3 Focal characteristics of cylindrical microlens

Zemax® and many of the current design methods utilize ray tracing.

Ray tracing is certainly a powerful tool for designing and evaluating beam

line optics[19]. However, it is difficult by means of ray tracing to foresee

the relations between the parameters of individual optical components and

the resultant aberrations in the final spectral image plane. Furthermore, ray

tracing does not permit analysis of individual aberrations in the image plane

in a similar manner to that of seidel aberrations of lens system[20]. In this

present section simulation of cylindrical microlens system is developed by

analytically following a ray-tracing formalism. This simulation takes an

extended source into account and gives focal characteristics such as spot

diagram, chromatic aberration, point spread function and encircle energy to

determine the modulation transfer function for different field angle

variations.

a) Spot Diagram : The spot diagram, as the most complete expression of

the geometrical image of an object point, has all the information necessary

to evaluate optical systems in which the diffraction effects may not be taken

into account. [21]. Attempt made to evident the influence that the choice of

the sampling may have on the spot diagram and on any others functions that

might derive from it[22-24]

159

Much can be learned from the general shape of the image of a point

source. For that reason, a useful method of displaying aberrations is the spot

diagram. To produce a spot diagram a number of rays are traced through the

optical system to the image plane. The intersection of these rays with the

image plane is represented by small circles or squares. A spot diagram of

the optical system by ZEMAX® simulation software is shown in

Figure (7.6). In this some of the spot produced by cylindrical microlens is

not so simple off-axis spot diagram. This is due to the fact that in addition

to coma the other off-axis aberrations are present. Geometric optics was

used to generate the spot diagram, so there is no indication of the structure

that diffraction effect produce in the actual image, but the general shape is

as we expect. The modern view of the optical system utilizes the response

of an optical system to a point object as the mean of characterizing the

system. For this reason, the spot diagram and images of the point source

produced by the optical system are a useful representation of the

performance of the optical system.

The RMS radius of the spot size is also calculated by ray tracing

method theoretically by considering that the ray initially incident parallel to

the axis are focused at Z = H (2p + 1) where p is an integer. The

dispersion of the ray position at the focal point is expressed in terms of a

spot diagram which can be obtained by using

160

x = xicosilz + gsinaz ...(7.1)

where is the ray slope at z = 0.

Figure (7.5). Theoretical model for image configuration of cylindrical microlens

Let us consider that the spot that comes from point source point

source P and this makes a real image with uniform magnification as shown

in Figure (7.5). The notation J as measure of height x0 given by

X = (1-5) • A ...(7.2)

Where A is the radius of cylindrical microlens and x0 = Nr2A corresponds

to the highest object position included in the NA of the lens.

Using input data in this study, theoretically calculated results of

RMS spot radius for different field angles 0, 2, 4, 6, 8 and 10 degree with

various incident wavelength 2. (pm) for the range 1.0 to 1.6 ,um were

161

tabulated in Table (7.2). The simulation curve is generated for the same

input data using ZEMAX®, represented in the comparative graph in

Figure (7.6).

Table (7.2). Theoretical and simulation RMS value of radius of spot size `r.' (gm) for Varying field angle 0 (deg) from 0 to 10 (deg) with incident wavelength 4 (pm) varying from 1.0 to 1.6 pm

Field Angle

0 (Deg)

RMS radius of spot size 'r' (pm)

Incident Wavelength ko (im)

1.0 1.1 1.2 1.3 1.4 1.5 1.6 S 11.318 9.8110 8.346 6.903 5.461 3.987 2.454 T 11.118 9.6210 8.146 6.703 5.261 3.847 2.284 S 11.435 9.9280 8.461 7.017 5.572 4.092 2.542 T 11.325 9.7280 8.301 6.817 5.372 3.892 2.342 S 11.781 10.277 8.812 7.371 5.927 4.449 2.897 T 11.651 10.147 8.702 7.251 5.807 4.379 2.727 S 12.372 10.887 9.446 8.035 6.633 5.218 3.779 T 11.172 9.587 9.006 7.915 6.533 5.018 3.659 S 13.302 11.879 10.514 9.196 7.917 6.673 5.484 T 12.502 10.979 9.514 8.896 7.817 6.553 5.354 S 14.781 13.490 12.276 11.136 10.068 9.078 8.189 T 13.781 12.790 11.676 10.036 9.668 8.948 8.089

0

2

4

6

8

10

162

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Focal spot size variation from left to right, field angle variation from 0, 2, 4, 6, 8 and 12 Degree (b) from top to bottom, wavelength (kJ: 1, 1.1, 1.2, 1.3,1.4, 1.5 and 1.6um

163

RM

S R

adiu

s o

f sp

ot s

ize

r (p

in)

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Deg (Simulation) —I' 6 8 Deg (Simulation)

—4— 10 Deg (Simulation) 0 Deg (Theoretical) 2 Deg (Theoretical)

—44— 4 Deg (Theoretical)

• 6 Deg (Theoretical)

• 8 Deg (Theoretical) - 10,4Deg (Theoretical)

2 -

09 1.0

1 2 1 3 1.4 1.5 1.6 1.7

Incident Wavelength ko (.im)

Figure (7.7). RMS Radius of spot size r (pm) versus incident wavelength (um) for both simulation and theoretical calculation.

164

b) Point Spread Function : The point spread function (PSF) describes

the response of an imaging system a point source or point object. A related

but more general term for the PSF is an impulse response. The degree of

spreading (blurring) of the point object is measure for the quality of an

imaging system[25]. Ideally, the focus of the plane wave should be

infinitely small being the image of the point source located at infinity. In

our example the lens is supposed to have an ideal geometrical shape, e.g., to

focus an incoming collimated plane wave at a distance 1' from the lens.

Although in this ideal case no other aberrations are introduced, the focus

will have a fmite extension. Diffraction at the lens aperture(D) causes a blur

of the focus. The light distribution in the focus is determined by the Fourier

Transform of the pupil function of the lens. The 1D pupil function j54(v)

of this ideal lens is described by a rect-function [rect (-Alcv )]. The frequency

coordinate vx is related to the physical coordinate x in the Fourier domain •

by, 12, = —x

and Ay = —D where A denoted the wavelength of the

Af Af

illuminating light beam. The point spread function PSF p(x) i.e., the image

of the point source generated by the lens is calculated as the Fourier

Transform of the pupil function r(,) p (x)a J rect ( 12-1) e-2i1 xxdvxasinc (x. j ) —D ) ... (7.3)

Av

Here the following definitions were used for rect(x) and sinc(x):

165

rect (x) = .11:1x11 0:else

sin(rrx) sinc(x) = ...(7.4)

7rx

For the lenses with circular aperture the ideal PSF is calculated as the

Fourier Transform of the circ D) function. This yields the so called airy

(r. 14) pattern, D Where Ji (x) is the first order Bessel Function.

r.

The PSF corresponds to the shape of the point image formed by the

lens. In the absence of aberrations a lens is called ideal or diffraction

limited. This means that the psf is determined by the sinc-function resulting

from the diffraction at the lens pupil. When the lens aperture D decreases,

the extension of psf increases proportionally. This has important effects on

the scaling behavior of lenses. With the reduction of lens diameter it

becomes easier to achieve diffraction limited performance. Since, the

extension of the diffraction limited PSF becomes of larger the constraints of

the shape of the phase profile become less stringent. Phase error due to

different optical path length of the rays passing through the pupil vanishes,

because of the small extension of pinhole diameter. The simulation graph of

psf was given in Figure (7.8). The calculated psf in terms of normalized

intensity for field angle 0, 2, 4, 6, 8 and 10 degree varying incident

wavelength A. (gm) reported in Table (7.3). and also simulation graph

generated using ZEMAX® represented in Figure (7.9).

166

Table (7.3). Normalized intensity variation with incident wavelength k (pm) for varying filed angles 0 (deg).

Field Angle

0 (Deg)

Simulation/ Theoretical

Normalized Intensity Incident Wavelength 1.(lam)

1.0 1.1 1.2 1.3 1.4 1.5 1.6

0 S T

0.592 0.492

0.652 0.552

0.668 0.648

0.748 0.648

0.842 0.742

0.942 0.842

0.981 0.881

2 S T

0.586 0.486

0.631 0.531

0.653 0.533

0.731 0.582

0.816 0.806

0.928 0.828

0.976 0.876

4 S T

0.576 0.456

0.590 0.456

0.612 0.542

0.685 0.585

0.745 0.715

0.846 0.746

0.936 0.836

6 S T

0.553 0.453

0.578 0.448

0.591 0.451

0.643 0.543

0.679 0.529

0.718 0.618

0.845 0.745

8 S T

0.541 0.411

0.562 0.432

0.584 0.424

0.633 0.503

0.671 0.501

0.694 0.594

0.718 0.618

10 S T

0.531 0.390

0.550 0.415

0.562 0.392

0.610 0.441

0.665 0.495

0.686 0.556

0.698 0.568

4., 0.9- (.4 a)

.5 «18-

vi -8

Am 1

Auxia•

Point spread function for different incident wavelength A (pm) varying with field angles 0 (deg)

Figure (7.8). Simulation representation of normalized intensity distribution for different field angle 0 (deg) varying with incident wavelength A, (ptm)

167

c) Modulation Transfer Function :

Encircled Energy : Encircled energy is typically plotted as a graph

relating the proportion of the incident light which falls within a circle of a

given radius to that radius. The amount of energy contained in a square or

circle of given dimension to determine best focus. This method is

particularly useful for non-imaging applications in which the purpose of the

lens is to focus energy onto a detector for maximum signal. It is also useful

in maximizing the amount of energy falling onto single pixels in an array.

In addition the measurement takes a lot of care in taking up if stray

reflections are to be avoided and all ambient light must be eliminated. On

balance, as the encircled energy can also be derived from wavefront

aberration and PSF results[26] obtained from simulation software has been

found to be the best measurement method. The theoretical computation

method had been done to obtain the normalized encircle energy for different

field angle 0 derived from aberration results for which input data taken from

Table (7.1). Calculated results were tabulated in Table (7.4) and these

results were compared with simulation findings were represented in

Figure (7.11).

Modulation Transfer Function (MTF) : The maximum spatial

frequency which can be resolved in the image plane of the cylindrical

microlens can be tested using a simple USAF resolution chart[27-29]. More

168

information is gained, however, by measuring the modulation transfer

function (MTF) of the cylindrical microlens. The MTF is a measure of the

ability of a cylindrical microlens to form a clear image of an extended

object. It is a plot as a function of spatial frequency of the contrast of the

image of perfectly modulated microlens. It is the function which probably

best summarizes the performance of an optical system in an imaging

application and takes account of the stray light and the energy in the wings

of the point spread function. For this reason it has been widely used by

researchers developing microlenses for various practical applications.

[30-32]. In order to measure the MTF directly, an object in the form of fully

modulated sinusoidal variation of intensity is set beneath the lens array.

The real image formed by the cylindrical microlens under test is then

scanned using line intensity on a computer using image analysis software.

The spatial frequency of the sine wave image formed is u per mm. The

contrast of the image for a particular spatial frequency as obtained by

measuring the maximum and the minimum intensities Lax and Imin and

calculating the contrast or modulation of the fringes as

MT F (U) = Imax-1min

Inta-r+Imin

the MTF curve is a versatile measurement has the curve changes with field

angle and with the conjugate ratio. If the cylindrical microlens under test as

astigmatism or coma, different MTF curves are obtained with different

169

azimuths in the image plane through a single image point. For a perfect

aberration free cylindrical microlens at an arbitrary conjugate ratio, MTF is

given by,

2 r MTF(u) = — [arc cos(x) — x V1— x2,1 ...(7.5)

Where, the radian more is indented for the arc cosine function and x is the

normalized spatial frequency

...(7.6)

where u is the absolute spatial frequency and Uk is the incoherent

diffraction cut-off spatial frequency. A number of formulae can be used to

calculate (Tic including,

2 nsin (u) = a ...(7.7)

Where, A is the wavelength and nsin(u) is the image space numerical

aperture.

At infinite conjugate ratio, Uic = n ...(7.8)

Where, n is the image space refractive index, D is the diameter of the

cylindrical microlens and f i§ the focal length of the cylindrical microlens.

The Variation of normalized optical transfer function with spatial frequency

(mm) for different field angle (0) is represented in Figure (7.12).

170

Table (7.4). Normalized encircled energy variation with incident wavelength), (gm) for change in field angle 0 (deg)

Field Angle

0 (Deg)

Simulation/ Theoretical

Normalized encircled energy Incident Wavelength A(jim)

1.0 1.1 1.2 1.3 1.4 1.5 1.6

0 S T

0.61 0.56

0.64 0.59

0.65 0.60

0.73 0.68

0.82 0.77

0.93 0.88

0.97 0.92

2 S T

0.59 0.54

0.62 0.57

0.64 0.59

0.72 0.67

0.82 0.77

0.93 0.88

0.98 0.93

4 S T

0.58 0.53

0.60 0.55

0.62 0.57

0.69 0.64

0.75 0.71

0.85 0.85

0.95 0.89

6 S T

0.56 0.51

0.58 0.53

0.62 0.55

0.65 0.61

0.68 0.63

0.72 0.67

0.85 0.81

8 S T

0.55 0.50

0.57 0.52

0.59 0.54

0.64 0.59

0.68 0.63

0.70 0.68

0.72 0.70

10 S T

0.54 0.49

0.56 0.51

0.57 0.52

0.61 0.55

0.67 0.62

0.69 0.66

0.73 0.69

171

I GC,

El

G

1..00 DEC

10.00 DSC 0 821 OEM

8.148 DUG

q 00 OGG ..00 OMO

.0. DEC

0,00 008

DEC; .0

6

7

0.00 DIM 2.00 0110

00 DEC 6.0m COG S.00 DEC

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IC

a) A= 1.3 ttni

/4/ A = 1.4 lun

7 / -1_00 DEC 0.00 DEC

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00 DEG

A = 1.5 lun ,

z

06 0.0

Radius from the centroid (pm)

Figure (7.10). Simulation representation normalized encircled energy with radius r from the centroid (pm) for various field angle 0 (deg)

172

7.2.4 Results and Discussion

The propagation of moments of an optical beam focused by a

cylindrical microlens has been considered. These can be done both

theoretical and simulation using ZEMAX® method. The theoretical

functions such as spot diagram, point spread function and modulation

transfer function has been derived and verified with ZEMAX® simulation.

These functions establish the basic for quickly and simply calculating

parameters of RMS radius of spot size `r', normalized intensity and

modulation transfer function which particularly measures chromatic

aberration, spherical aberration, astigmatism and coma etc. ray tracing is

powerful tool for evaluating and designing an optical system and provides

spot diagram that contains information on resultant aberration of the

system. The data from Table (7.1) is given as input parameters to simulation

study and the resultant were compared with theoretical findings.

Most of the lens design employed the Damped Least Square Method

(DLS) as the core of the optimization process today. The DLS is a local

search method based on differential information of an evaluation function.

This causes two important issues such as, (a) an appropriate trade-off ratio

among multiple criteria is chosen by trial and error because the DLS can

handle many evolution function, (b) virtual glass whose refractive indices

and Abbe numbers are allowed to vary continuously within a specified

173

boundary must be used during the optimization process because the DLS

requires that the evaluation function can be differentiated. As a result the

glasses found by the DLS will be easily available commercially.

In present study the care has been taken for the optimization run

must be completed with the specified material in order to have physically

viable, cheaper and better lens. Here, it is necessary to find materials of lens

elements to reduce chromatic aberration. Hence, for a given focal length f,

an F/# number, and a field size as a specifications, it is considered to search

the curvature of the surface and Abbe number to obtain a lens that has good

image forming capabilities. 190 glasses available in the Schott catalog has

been tried. The BK7 glasses, which are well known to be effective in

eliminating the chromatic blur are used because they are easily available

and less cost. Given a focal length, the number determines the brightness

of the image. The curvature of the lens surface are modified to meet the

required focal length f, by applying the geometric optics. In ray tracing

mechanism, there are six wavelengths A. (1.0, 1.1, 1.2, 1.3, 1.4, 1.5 and 1.6

pm). For each wavelength 10 bundle of rays whose entrance angle is 0, 2, 4,

6, 8 and 10 degrees respectively are traced. Each bundle of rays consists of

a principle ray and other rays surrounding and parallel to the principal ray.

A cylindrical microlens is evaluated by using simulated spot diagram

as shown in Figure (7.6). Spot diagram shows image forming status on the

174

image surface. The pictorial representation reveals the absence of chromatic

aberration but astigmatism and coma were predominant in higher field

angles. The RMS radii get decreases when we switch over from wavelength

of 1.0 gm to 1.6 gm. This indicates that for lower wavelengths the rays are

missing the image plane which fails to concentrate to the centre spot which

clearly evident from Figure (7.6) [A. = 1.0, 6 = 0 deg]. A slower increase

in the RMS value of radii of spot size for increasing field angle of 0, 2, 4, 6,

8 and 10 degrees is attributed to a slower increase of astigmatic and coma

image length. The numerical values of RMS radii of spot size for simulation

and theoretical findings were reported in Table (7.2). All these results when

compared as graphical representation in Figure (7.7) shows the theoretical

results well agrees with simulation findings.

The theoretical expression to analyze the Point Spread Function

(PSF) by using pupils function from Equation (7.3) in the presence of

defocus aberration is derived for cylindrical microlens system. The PSF

characteristics infer the information of normalized intensity distribution in

image plane focused by lens system. There are two factors to be drawn from

the Figure (7.8) of simulated PSF representations. The primary factor is that

the maximum intensity drawn for a specific lens system and the width of

the central maximum peak.

175

The pictorial representation of PSF image generated from the

ZEMAX® simulation contains a collection PSF images for one single

wavelength of different field angles of 0, 2, 4, 6, 8 and 10 degree

respectively were arranged horizontally from left to right side in the

increasing wavelengths of 1.0, 1.1, 1.2, 1.3, 1.4, 1.5 and 1.6 gm

respectively. As expected, for higher wavelength of zero field angle a sharp

high intense single peak is observed. For example, for a wavelength of 1.6

gm the normalized intensity is observed to be maximum of 0.981 for zero

field angle and a minimum of 0.698 is drawn for greater field angle of 10

degrees. This may be due to steady increase in astigmatism makes the rays

to get defocus at the central maximum.

176

SzA

• simulation o deg) - simulation 2 deg)

—A-- simulation 4 deg) —v-- simulation 6 deg) --•— simulation 8 deg)

- simulation 10 deg) Theoretical (0 deg) Theoretical (2 deg)

—le— Theoretical (4 deg) —I,— Theoretical (6 deg) --co— Theoretical (8 deg)

I Theoretical (10 de

Aolt

1 0 -

0.9 -

Norm

aliz

ed in

ten

sity

0.8 -

0.7 -

0.6 -

0.5 -

0.4 -

• 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

Incident wavelength ko

Figure (7.9). Variation of normalized intensity with incident wavelength A (ium)

for varying filed angles H (deg).

177

The similar trend is observed for other wavelengths also but the

normalized intensity gets reduced when we go from higher wavelength to

lower wavelength. The intensity of peak for zero and two degrees are very

nearer but when field angle further increased the intensity drop is higher.

So, the incident wavelength incident on to the lens surface of field angle

near to the optical axis of up to approximately 2 degree will be having good

focusing ability.

It is interesting to observe that the width of the peaks obtained for all

the field angles for any particular wavelength doesn't have much variation

reveals the very important result that the variation of angles may cause the

aberration to some extent but it doesn't fails to deviate to focus into

specified area in the image plane. The width of the peak is observed to be

approximately over the diameter of 10 p.m for an incident wavelength of

1.6 gm and it found to a maximum of 15 gm at an incident wavelength

1.0 p.m. hence, over wavelength region taken in this present study is

suitable to get trapped by the designed lens. All the above discussion was

pictorially represented both theoretical and simulation results in Figure

(7.9).

The focal shift can be considered in terms of the Optical Transfer

Function (OTF). It is interesting to note that the second moment of intensity

has a special significance in terms of the Fourier transform of the intensity,

178

which is formally identical with the OTF. The Fourier integral is applied on

to the OTF to get Modulation Transfer Function (MTF) which is studied

interms of spatial frequency. The MTF study on our microlens system

begins with characterizing the lens by encircled energy distribution which

gives the quantum intensity of light beam from which OTF is obtained.

The simulated encircled energy curve for present system of study is

given in Figure (7.10). The normalized encircled energy is obtained a

maximum value of 0.97 for an incident wavelength of 1.61.im and field

angle of 0 degrees. This shows that maximum light energy passes across the

lens. When field angle get increased it drops to 0.73 shows that some part of

energy is lost during this transmission as given in Table (7.4). The

decreasing trends is observed when calculation was done for lower

wavelength and the theoretical results were quit agrees with simulation

findings reported in Figure (7.11). There is large variation in normalized

encircled energy values were observed for increasing field angle at lower

wavelength may due to diffraction effect. The maximum and minimum

encircled energy are taken from which MTF was calculated theoretically

and the results were compared with the simulation findings.

179

1.00 -

0.95 -

0.90 - •

0.85 -

0.80 -

0.75 -

0.70 -

0.65 -

0.60 -

0.55 -

0.50 -

Nor

ma

lized

enc

ircle

d e

nerg

y

• Simulation (0 deg) Simulation (2 deg) Simulation (4 deg)

—v— Simulation (6 deg) Simulation (8 deg)

—1— Simulation (10 deg) Theoretical (0 deg)

-4- Theoretical (2 deg) ---*— Theoretical (4 deg) —0— Theoretical (6 deg) —0— Theoretical (8 deg)

Theoretical (10 deg)

1 ' 1 ' 1 ' 1 ' 1

1.0 1.1 1.2 1.3 1.4

1.5 1.6

Incident wavelength X (pm)

Figure (7.11). Variation of normalized encircle energy with incident wavelength k (um) for varying field angle 0 (deg)

180

The MTF is the critical parameter of aberrated optical system.

Except in restricted cases, such as when the well known Fourier transform

relationship are valid, the numerical methods of wave-optics analysis can

obscure the underlying physical process of MTF formation. By simulation

method the MTF curves were generated and the results were plotted for

modulus OTF against spatial frequency in mm, given in Figure (7.12) in

which polychromatic light of wavelength ranges from 1.0 to 1.6pm is

considered. The cut-off frequency is represented for each field angle for

both transverse plane and sagittal plane is represented in Figure (7.12) and

its values were plotted as graph and compared with theoretical fmdings

given in Figure (7.13). When the field angle of 0 and 2, degrees the

normalized optical transfer function maximum for lower cut-off frequency

due to existence of minimum aberrations which is also gets agrees with PSF

calculations. Lower the cut-off frequency give aberration free image quality

which can be achieved by sending the incident ray to the angle very close to

the optical axis of the optical system. the field angles of 4, 6 and 8 degrees

has still higher cut-off frequency and further 10 degree has very high cut-off

frequency confirms larger aberration coefficient which is also reported in

spot diagram section 7.2.3.1. The theoretically determined normalized OTF

values quite aggress with simulation results.

181

MODULUS OF THE OTF

I.0

.7

,6

TS L1.00 DEC S 6.00 DEC IS DEC

IS 10.00 DEC IS 0.00 DEC TS 2.00 DEC

0.00 200.00

400.00 SPATIAL FREQUENCY IN CYCLES PER MM

Figure (7.12). Variation of normalized Optical Transfer Function with spatial frequency (mm) for different field angle (A)

182

-2 0 110 8 I ■ I I

2 4 6

Field Angle 0 (deg)

• FST FSS

—A— FTT FTS

}-- OST "-A 0 S S

-6-- OTT OTS S

pa

tial

Fre

qu

en

cy (m

m)

Norm

alize

d O

ptica

l Tra

nsfe

r Fu

nctio

n

FST = Simulation - Cut-off frequency for transverse plane

FSS = Simulation - Cut-off frequency for sagittal plane

FTT = Theoretical - Cut-off frequency for transverse plane

FTS = Theoretical - Cut-off frequency for sagittal plane

OST = Simulation – Optical transverse function for transverse plane

OSS = Simulation – Optical transverse function for sagittal plane

OTT = Theoretical – Optical transverse function for transverse plane

OTS = I heoretical – Optical transverse function for sagittal plane

Figure (7.13). Variation of cut-off spatial frequency (mm) and normalized Optical Transfer Function for different field angle 0 (deg)

183

7.3 V-groove thin silicon solar cells

7.3.1 Introduction

The light trapping properties of textured optical sheets have become

of recent interest in photovoltaically energy conversion since light trapping

allows a significant reduction in the thickness of active solar cell material.

There are many reasons for keeping the thickness of photovoltaically active

material in solar cells to its minimum possible value. In the case of

crystalline silicon cells, it has been shown that this is necessary to minimise

recombination rates due to intrinsic Auger recombination processes and

hence obtain maximum possible open-circuit voltage[33,34]. To

simultaneously obtain high short-circuit current, some form of light

trapping is required to boost the light absorbing properties of such thin

layers.

Two types of schemes have been proposed to achieve light trapping

in solar cells. One of type is based on randomizing the direction of light

within the cell substrate. Once so randomized, only a small fraction of the

light will lie within the escape cone for small fraction out of substrate

surface from within. The rest is totally internally reflected giving rise to

very effective light trapping[35]. The second type of scheme is based on

regular geometrical structures [36]. The aim is to control the direction of the

light within the substrate so that it is kept away from the escape cone

184

associated with each surface for the maximum number of internal passes

within the substrate.

One geometrical feature widely incorporated into commercial

crystalline silicon cells is the square based pyramid formed by intersecting

(111) orientated crystallographic planes exposed by anisotropically etching

silicon surface originally of (100) orientation. Silicon solar cells with high

efficiency have been achieved through the process of texturing the silicon

surface so that light is coupled into the cell obliquely[37] thus making it

possible for the photogeneration to take place near the junction. Wet

chemical etching[38,39} laser grooving[40] and mechanical sawing[41] are

now being used to texture the silicon solar cell surface.

This chapter organized as follows: theoretical design[42] geometry

on microlens coupled solar cells was explained schematically in section

7.3.2 and pitch[43] of the groove is given in section 7.3.3. The

reflectivity[44] of V-groove thin silicon solar cells theoretically

analyzed[45] and its coefficient values were compared with simulation

results in section 7.3.4. In the subsequent section 7.3.5 discuss on numerical

aperture(NA) and intensity distribution in solar cells and a comparison was

made with and without microlens array. The next two successive sections

have a theoretical approach to calculate high throughput microlens and

calculating the quantum efficiency of solar cell with cylindrical microlens

185

array. Theoretical attempt was made to calculate the numerical aperture

under normalized intensity distribution to obtain throughput of the

microlens. The section 7.3.8 discusses the overall results obtained through

theoretical and simulation work, and it is compared with reported values.

7.3.2 Theoretical design of microlens coupled with solar cell

The aim of the present study to improve the light trapping efficiency

of V-groove thin silicon solar cell by using cylindrical microlens as

concentrators. In the cell V-grooves are positioned such that the coupled

light passes through a V-groove misses the adjacent one as it travel to the

bottom of the cell. One face of the V-groove is coated with aluminium so

that the light reflected off this face would be coupled into the cell through

the opposite face of V-groove as illustrated in Figure (7.14).

7.3.3. Pitch of the V-groove silicon solar cells

It is proposed to focus light onto beam steering mirrors in V-grooves of the

cell surface in trapping light. The specific value for the silicon thickness,

the distance between microlens and silicon, and the separation shown

between the two microlenses are determined by the twice of the optical path

length[46]. The pitch of the V-grooves determines the pitch of the

cylindrical microlenses. For manual alignment of two wafers on to top of

the other, a jig alignment tolerance of +20 pm is possible. On the basis of

this it can be deduced that the minimum width of the V-groove is 80Am and

186

Figure (7.14). Schematic diagram of V-grooved solar cell with microlens array.

187

has a depth of (40 tan 54.7°) = 56.5 pm. To determine the pitch of the V-

grooves, a model has been taken as a worse case alignment[47] in which

focused as shown in Figure (7.15). This figure shows a principal axis (PA)

ray that has normal incidence to the cell surface and an extreme ray to the

left of PA that makes an angle a with PA. This extreme ray passes through

EB and finally F where it just misses the vertex of the next V-groove. It

makes an angle of about (30.8 - a/n)° with the horizontal, where n=1.5 is

the refractive index of silicon. The horizontal distance L a, between the

points D and F represents the pitch of the grooves. A derived relationship is

given by

FD sin(94.5+) 3.5 From the triangle FBD — = BD sin(30.8-51--)

,

3.5

from the triangle BDE,

BD sin(35.3+a) — =.- On combining these two triangle, —FD x BD = FD

ED sin(74.1—a)' BD ED ED

FD sin(35.3+a) sin(30.8 + (A) — = a ), ED sin(74.1—a) sin(30.8 — Draw a normal along the E, From the

EO Triangle DEO„ tan 54.7 = , EO = 0.5 al tan 54.7Therefore the

relationship between La, a and al is given as,

FD L = x EO L a if) a

= 0.5a1 tan 54.7° ( sin(35.3 + a)\ sin (94.5° +

sin(74.1 — a)) sin (30.8 —

...(7.9)

188

when a=16.07, = 0.5a1 tan 54. 7. Cin(35.3+16.07°)) sin(94.50 + 16.07)

si n(74.1-16.07°)) sin(30.8 1 63. 057 \

3.5 )

= 0.5a1 (1.412350)(0.920946)(2.230475),

La + 0.5a1 Ped (1.95058 al) ...(7.10)

For small a Equation (7.9) gives an approximate value of La as

1.45286al . Taking the case where a is small then minimum pitch becomes

La + 0.5a1 1.953a1. For al = 80 gm the minimum pitch would be

about 160gm.

In the design reported here the pitch was made 165#7n. For a given

cell thickness W the optical path length for a PA ray on first transit to the

back reflector is given by

Optical path length — [ .] = 2W sm30.8

...(7.11)

If the cell had a plain front surface the optical path length for a

normal incidence ray that travels to the back reflector would be 2W. Hence,

using this design geometry, the cell thickness can at least be halved while

maintaining the optical path length. The advantage of reducing the cell

thickness is that photo generation takes place closer to the junction on the

average, especially for the low energy photons and thus there is an increase

in current collected. In thick cells, the minority carriers that created at

189

011408/ 8A7,5 (PA)

' , •

BDE= 70.6a

Ch. D--30.8a

Ea0=74.

Ckg=35.

al -4 /,'

`v•

Figure (7.15). Schematic diagram for determining the V-groove pitch used aluminium coated onto right-hand side face of V-groove.

190

distances greater than the diffusion length from the junction are likely to

recombine in the bulk or surface, hence resulting in less photocurrent.

The second advantage of this design is the light trapping capability

that is made possible by covering most of the cell surface with aluminium

except in optical coupling areas. A photon enters the cell obliquely and can

be absorbed on first transit or during subsequent passes as it bounces off the

internal walls. By comparison, in a plain surface solar cell the photons that

are reflected from the back surface may be coupled out without contributing

to photocurrent generation. Cell series resistance is reduced by the large

area of aluminium contact and hence a minimized power loss in the cell.

7.3.4 Theoretical approach on reflectivity in V-groove cells

To fabricate solar cells with maximum efficiency, it is necessary to

reduce the reflectivity of the surface to zero, if possible. Most traditional

solar cells designs have used polished planar surfaces, and the reflectivity

for normally incident light (which is essentially independent of wavelength

for 1.0 p.m < Xo < 1.6 Jim) is approximately 35% for silicon. The use of

antireflection coatings can reduce the reflectivity to approximately 10%

(single layer cells) or 7% (multiple layer cells) in silicon cells[48]. Further

reductions are possible only with modification in the geometry of the cell.

Recently, several reports have shown that reduced reflectivity can be

achieved in solar cells by preferentially etching the silicon surfaces [49-51].

191

The resulting surfaces is no longer planar and can be said to be "serrated" if

the etching produces regularly spaced grooves or to be "textured" if the

etching produces randomly spaced and sized etch features. Without

antireflection coatings, the serrated silicon surface has an average reduced

reflectivity of roughly 22% (for 35% coverage of the surface with grooves)

and the textured silicon surface approximately 12%[52]. The application of

antireflection coatings reduces these values even further. Solar cells

fabricated form these etched substrates demonstrate an enhanced short-

circuit current and spectral response[53]. Clearly, the addition of one

relatively simple etching step to the fabrication process can yield solar cells

with dramatically improved performance.

The major difference between a planar surface and a grooved surface

is that there exists a possibility for multiple reflections of the incident light

rays with the serrated surface. Multiple reflections decrease the total

reflectivity of the cell and thus permit a greater amount of photoabsorption.

Although the V-grooves described in Section 7.3.2 are formed by (1 1 1). A

planes and have a fixed groove angle given by[54]

a, = 1800 — cos- r(111).(111)1 ...(7.12)

= 70.5°

192

a) Calculation of reflection coefficient : In this section, it shall calculate

the reflection coefficient for a serrated surface as a function of the groove

angle a. Using these results, the total reflectivity at a = ay.

When a =1800, the surface is planar, and incident rays can strike the

surface only once. In fact, for groove angles between 1800 and 120°, there

can be only one reflection, for with specular reflection and vertically

incident light the outgoing rays cannot intercept the opposite side of the V-

groove. The incident angle is given by

81 = 90° — 2 ...(7.13)

When a = 120 °, 0, =30° , and the outgoing ray is parallel to the

opposite groove side. This is depicted in Figure (7.16(a)). When a is

reduced from 120 0, say to 95 ° as shown in Figure (7.16(b)), a fraction of

the rays striking the left side of the groove will be involved in a second

reflection from the right side of the groove. Rays incident near the top will

not intercept the opposite side.

Using the law of sines it can be shown that the fraction of rays that can be

reflected twice is

fCOSO2 1

COSei ...(7.14)

where cos 02 is the second incident angle and is given by 02 = a — 01

= —3a

— 90° 2 ...(7.15)

193

At a groove angle a of 90 °, all rays will be reflected twice, and the

outgoing ray is vertical. Between the angles of 90° and 72 0, in fact, all rays

are reflected twice.

When a = 72 °, the outgoing ray is parallel to the left side of the groove.

This general scheme is repeated and can be summarized in the following

form. When

180° a = n = 1, 2, 3 ... (7.16)

the outgoing ray is vertical, having undergone 'n' reflections. When

1800 a = n = 1, 2, 3 ...(7.17) n+1/2

the outgoing ray, having undergone n reflections, is parallel to one of the

etched sides.

This ray is parallel to the first reflecting side if n is even, and parallel

to the opposing side if n is odd. When the groove angle is given by,

180 180 < a < n+1/2 n+1

...(7.l8)

a fraction of the incident rays have n + 1 reflections, and the remainder 'n'

reflections.

This fraction is,

fn = cosen+i cosei

where, On+iis the (n + 1)th incident angle and is found to be

1 180 - (2n+1)al en+1 2

...(7.19)

—(7.20)

194

180' It should be noted that fn = 0 and a = (

1801) and that fn = 1 for a

= (n+1)

Table (7.5) lists the critical angles for vertical and parallel final reflections

and the applicable number of reflections. Once a groove angle is specified,

the number of reflections can be specified using Equations (7.16-7.20). To

calculate the total reflectivity, we employ Fresnel's equations for the

oblique incidence of light [55]. The electric field vector for the incident

radiation can be written as,

E0 = (E0,p ft p E0,N ñ N)expf—i (coot — ko.r)) ...(7.21)

where, E0, p and EON are the parallel and normal components of the electric

field, ft pandiI N are the unit vectors along the two polarization directions,

(Do is the angular frequency and Icc, is the waventunber of the radiation. We

shall assume that the incident radiation is initially unpolarized, so that

(E0,P)time average = (EO,N)time average ...( 7.22 )

After one reflection, the electric field is then given by

= ELp ftp + ELN fiN exp{—i (coot — /cc,. r)} ... (7.23)

where ELF, and ELN are found from Fresnel's equations:

ELF, = (Rp(1))1/2

E0, p ELN = (D (1) )1/2

I LN la QN ...(7.24)

Rp(i) tan' (91 — 91) tan2 (O i +

RN(1) = sin2 (Oi — OD sin2 (01 + OD

195

(a)

(b)

Figure (7.16). Ray-tracing models for the V-groove structure: (a) groove angle 0, = 120 °, only one reflection with 01 = 300; (b) groove angle 0, = 95 °, a fraction of the rays are reflected twice with 01 = 42.5 0 and 02 =475 0.

196

n++ A deg Out going

Ray*

Number 'm' of the

reflection

180 V 1 120 P 1<m<2

90 V 2 2i 72 P 2<m<3

60 V 3 3+ 51.43 P 3<m<4

45 V 4 4 + 40 P 3<m<5

36 V 5 51 32.72 P 5<m<6

30 V

1

2

3

4

5

6

Rp(1) and RN(1)) are the first reflection components, and 01 and 01 ' are the

first incident angle and the first refracted angle respectively. They are

related by the law of refraction

sine' = no ...(7.25)

where no is the index of refraction for silicon. The total reflection

1E11 2 coefficient, designated by R(1), is then computed to be = 1E0 12

Rp(1) 1E0,1312+ RN(1) lEo,N12

lEo.P124- 1E0,N12 ...(7.26)

Table (7.5). List of Critical angle for V-Groove Reflection

* V-Vertical and P-Parallel

For unpolarized light the reflection coefficient becomes

R(1) = R p (1) ÷Riv (1)

2 ... (7.27)

If the light rays undergo a second reflection, the electric field vector is

197

len) = —2

(FIRp(i) + RN“)) 1

i=1 i=1

... (7.32)

written as

E2 = (E2,p flp) + (E2,N fl Ar)exP[—itoot — k.

r (2).0.12 r () 1/2 = tELpifip np + ELN

2\ fiN} eXp[—if-Oot — /co. 7]

...(7.28)

where, the second reflection components are given by

= tan2( 02-02)

R p (2) tan- (92+9'2)

...(7.29)

sin2(02-02) RN (2/

sitt 2 (0242) ...(7.30)

and, again, 02 and 02' are the second incident and second refracted angles.

The total reflection coefficient, after two reflections, becomes

1E2 1 2 R(2) =

1E112

Rp(2)Rp(1) RN(2)RN(1)

...(7.31) 2

for initially unpolarized light.

In general then, after m reflections, the total reflection coefficient is

where,

Rpm = tan2( At 0;)

tan2 (Ai + 0;)

198

RN (1) sin2 (0i — sin2 (0i +

and 0; and Oi' are the ith incident and refracted angles.

For any specified groove angle, the range into which a falls must be

determined: for range A,

180° 180° < a <

n + n/2 ... (7.33)

and, for range B,

180° 180° < a < n + 1/2 n + 1

If a lies in range A, the reflection coefficient is given by

R(a) = R(a)

If a fails into range B, the reflection coefficient becomes

R(a) R(a) f ro(n+i)(a) _ R(a))

... (7.34)

... (7.35)

... (7.35)

where fn, is the fraction given by Equation (7.19).

The reflectivity is plotted versus groove angle a (deg) for a serrated

silicon structure for an assumed refractive index of 'n' such as 1.5, 2.0 and

2.5 in Figure (7.17). The step-like nature of the curve is a consequence of

the abruptness of the grooves themselves; the number of reflections varies

in a nearly discontinuous fashion with the groove angle. It is apparent from

Figure (7.17) that, for groove angles less than 20°, the reflection coefficient

199

is reduced virtually to zero. For the angle (a, = 60 °) of the ( 111 ) -

defined groove, the reflection coefficient R(--v) is approximately 0.001, or

72% smaller than the reflection coefficient for a planar surface. On the

average, approximately 11% of the incident rays undergo three reflections,

and the remainder two, in a V-groove structure with a = a,.

The daily average reflectivity was obtained by multiplying the

reflectivity with the fraction of light, cos a and by integrating over the angle

of incidence 0°-90°. Figure (7.18) show the daily average reflectivity for tilt

angle a (deg) and Table (7.7). The reflectivity measured for various

incident angle 0 (deg) both theoretical and simulation method for various

groove angle of a (deg) at three different refractive index values were

represented in Figures (7.19 - 7.21) and its numerical results were given in

Table (7.8- 7.10).

200

020-% 1

0 15-

• n=1.5 Theoretical method -9- n=2.0 Theoretical method -A- n=2.5 Theoretical method

n=1.5 Simulation method n=2.0 Simulation method

• • n=2.5 Simulation method

0.05-

• t • 10 20 30 40 50 60 70 80

Tilt angle (1 (deg)

0.00

0 90

Table (7.6). Change in reflectivity with angle of incidence u (deg) for refractive index n 1=1.5, n2=2.0 and n3=2.5

SI.No Tilt

Angle Reflectivity

Theoretical method Simulation method a (deg) n 1 =1.5 n2=2.0 n3=2.5 n1 =1.5 n2=2.0 n3=2.5

1 0 0.040 0.120 0.190 0.043 0.123 0.195 2 10 0.040 0.120 0.190 0.043 0.123 0.195 3 20 0.040 0.120 0.190 0.043 0.123 0.195 4 30 0.040 0.120 0.190 0.043 0.123 0.195 5 33 0.035 0.100 0.150 0.035 0.100 0.155 6 36 0.022 0.080 0.120 0.023 0.080 0.125 7 39 0.010 0.060 0.090 0.008 0.063 0.095 8 42 0.002 0.040 0.070 0.002 0.043 0.073 9 45 1x10-c 0.020 0.050 1x10 0.023 0.051 10 48 1 x10-5 0.020 0.050 1 x10-c 0.023 0.051 11 51 1 x10-5 0.020 0.050 1x10-5 0.023 0.051 12 54 1x10 5 1 x10-s 0.005 1x10 5 1x10-s 0.011 13 57 1x10-'' 1x10-5 0.002 1x10-5 1x10-5 0.005 14 60 lx10-5 1)(10-5 iX10-5 IX10-5 iX10-5 IX10-5

Figure (7.17). Variation of reflectivity with angle of incidence u (deg) for

refractive index n 1=1.5, n2=2.0 and n3=2.5

201

30 40 50 60 70 80 90 20 0 10

• n 1 = 2.5 (Simulation)

--dr— n2 = 2.0 (Simulation) —A— n3 = 1.5 (Simulation)

ni = 2.5 (Theoretical)

n2 = 2.0 (Theoretical)

—4— n3 = 1 .5 (Theoretical)

•

0.150

0.000

0 125 4

0.100 —

a) 0 0 075 —

co

(7) 0.050 —I

0

0.025 —

Table (7.7). Variation of daily average reflectivity with various tilt angle for different refractive indices.

Sl . No .

Tilt Angle a (deg)

Daily average reflectivity Theoretical method Simulation method

n1 =1.5 n2=2.0 n3=2.5 n1=1.5 n2=2.0 n3=2.5 1 0 0.050 0.090 0.126 0.040 0.080 0.125 2 30 0.035 0.087 0.120 0.025 0.078 0.110 3 60 0.020 0.070 0.070 0.010 0.060 0.050 4 75 0.005 0.007 0.006 0.003 0.004 0.005

Tilt angle (I (deg)

i2,ore (7.18). Variation of daily average reflectivity with tilt angle a (deg) for refractive index n1=1.5, n2=2.0 and n3=2.5

202

Table (7.8). Variation of reflectivity for incident angle 0 (deg) against tilt angles when n1=1.5

SI. No.

Incident angle 0 (deg)

Reflectivity (n1 =1.5) Tilt angle a

(deg)(Theoretical) Tilt angle a (deg)

(Simulation) 0 30 60 75 0 30 60 75

1 0 0.04 0.04 1x10 3 1x10 3 0.05 0.05 1x10 3 1x10 3 2 10 0.04 0.04 1x10-3 1x10 3 0.05 0.05 1x10 3 1x10 3 3 15 0.04 0.04 1x10-3 1x10-3 0.05 0.05 1x10-3 1x10-3 4 20 0.04 0.04 1x10 3 1x10 3 0.05 0.05 1x10-3 1x10-3 5 25 0.04 0.04 1x10 3 1x10 3 0.05 0.05 1x10 3 1x10 3 6 30 0.04 0.03 1x10 3 1x10-3 0.05 0.04 1x10-3 1x10-3 7 35 0.04 0.03 1x10 3 1x10 3 0.05 0.04 1x10 3 1x10 3 8 40 0.04 0.03 1x10-3 1x10-3 0.05 0.04 1x10-3 1x10-3 9 45 0.05 0.03 0.005 1x10-3 0.05 0.04 0.00 1x10-3 10 50 0.07 0.03 0.02 1x10 3 0.07 0.04 0.02 1x10 3 11 55 0.07 0.04 0.03 1x10 3 0.07 0.04 0.03 1x10 3 12 60 0.08 0.04 0.04 1x10-3 0.09 0.04 0.06 1x10-3 13 65 0.12 0.04 0.04 1x10-3 0.15 0.04 0.06 1x10-3 14 70 0.17 0.04 0.04 0.01 0.18 0.04 0.06 0.01 15 75 0.25 0.04 0.04 0.02 0.35 0.04 0.06 0.03 16 80 0.40 0.04 0.05 0.05 0.44 0.04 0.06 0.06

0.50

0.45

--N-a = 0 (deg) (simulation method)

0.35 - ID- a = 30 (deg) (simulation method) -A- a = 60 (deg) (simulation method)

0.30 v a = 75 (deg) (simulation method) - • a = 0 (deg) (Theoretical method) 5

0.25 ---4- = 30 (deg) (Theoretical method) a = 60 (deg) (Theoretical method)

0.15

0.10

0.40

44:15 0.20 • a = 75 (deg) (Theoretical method)

0.05

0.00

Angle of Incidence o (deg)

Figure (7.19). Variation of reflectivity with change in angle of incidence 0 (deg) for tilt angle of a = 0, 30, 60, 75 degrees ( n1=1.5)

203

0.2 -

0.1

0.0

• a = 0 (Simulation Method) •

-

a = 30 (Simulation Method) -A- a = 60 (Simulation Method)

a = 75 (Simulation Method) - • a = 0 (Theoretical Method) -1-0. = 30 (Theoretical Method)

a = 60 (Theoretical Method) • 0- a = 75 (Theoretical Method)

• • •

6-

■

•

0.6 -

0.5 -

Table (7.9). Variation of reflectivity for incident angle 0 (deg) against tilt angles when n2=2.0

SI. No.

Incident angle 0 (deg)

Reflectivity (n2=2.0) Tilt angle a

(deg)(Theoretical) Tilt angle a (deg)

(Simulation) 0 30 60 75 0 30 60 75

1 0 0.11 0.11 1x10-3 1x10- ' 0.13 0.13 lx10-3 1x10-3 2 10 0.11 0.11 0.005 1 x10-3 0.13 0.13 0.007 1x10-3 3 20 0.11 0.10 0.008 1x10-3 0.13 0.12 0.01 1x10-3 4 30 0.11 0.09 0.05 1x10-3 0.13 0.11 0.07 1x10 5 40 0.11 0.10 0.08 1x10-3 0.13 0.12 0.09 1x10 6 50 0.13 0.10 0.10 0.005 0.15 0.12 0.11 0.008 7 60 0.16 0.11 0.10 0.01 0.16 0.13 0.11 0.03 8 70 0.24 0.12 0.10 0.05 0.26 0.14 0.11 0.07 9 80 0.50 0.12 0.10 0.10 0.52 0.13 0.11 0.11

0 10 20 30 40 50 60 70 80

Angle of incidence 0 (deg)

Figure (7.20). Variation of reflectivity with change in angle of incidence 0 (deg) for tilt angle of u = 0, 30, 60, 75 degrees ( n1 =2.0)

204

Table (7.10). Variation of reflectivity for incident angle 0 (deg) against tilt angles when n3=2.5

SI.No Incident a ngle 0 (deg)

Reflectivity (n3=2.5) Tilt angle a

(deg)(Theoretical) Tilt angle a (deg)

(Simulation) 0 30 60 75 0 30 60 75

1 0 0.18 I

0.18 _

1x10 3

1x10-3 0.19 0.19 1x10-3 1x10-3 -, 10 0.18 0.17 0.02 1x10-3 0.19 0.18 0.03 1x10-3 1 20 0.18 0.16 ' 0.03 1x10-3 0.19 0.17 0.04 1x10-3 4 30 0.18 0.15 0.04 1x10-3 0.20 0.16 0.04 1x10-3 5 40 0.19 0.16 0.06 1x10 3 0.20 0.17 0.07 1x10-3 6 50 0.20 0.17 0.13 0.005 0.20 0.18 0.14 0.005 7 60 0.22 0.18 0.18 0.02 0.23 0.19 0.19 0.04 8 70 0.28 0.20 0.18 0.12 0.29 0.21 0.19 0.14 9 80 0.45 0.20 0.18 0.18 0.48 0.21 0.19 0.20

0.50 -

0.45 -

0.40 -

0.35 -

• a = 0 (deg) Simulation method - o- a = 30 (deg) Simulation method

a = 60 (deg) Simulation method a = 75 (deg) Simulation method

- • a = 0 (deg) Theoretical method -4- a = 30 (deg) Theoretical method

a = 60 (deg) Theoretical method • a = 75 (deg) Theoretical method

• 1• • •

•

0.10 -

0.05 -

0.00 - 1 1 • 1 1 • I

• 0 10 20 30 40 50 60 70 80

Angle of incidence 0 (deg)

Figure (7.21). Variation of reflectivity with change in angle of incidence 0 (deg) for tilt angle of a = 0, 30, 60, 75 degrees ( n3=2.5)

205

7.3.5 Numerical Aperture and Intensity Distribution

The intensity distribution can be calculated as the ratio between the

numbers of light ray leaving the microlens to those entering the microlens

as depicted in associated configuration. To obtain the N.A by finding

maximum angle between rays and optical axis, the given procedures have

been followed (i) the focal point by measuring the position of maximum

spot intensity, (ii) measuring the distance from focal point to the V-grooves

and diameter of the far field pattern as shown in the ray trajectory Figure

(7.22). The intention is to extrapolate data for these regions, then this has

been done via the theoretical modeling that is presented on these figures are

explicitly constructed by means of simulation

To evaluate the numerical aperture, the generalized Luneberg lens

model were considered as:

n(r) = n2expl2o ri9 ... (7.37)

r = ap exp {2co [p is the distance from the centre, n2 is the index of

the substrate and a is the radius of the lens, co [ a n p = —1 f dt t2-p2

nr is a function and p = n2 a is a parameter, which presents the dependency

of the diffusion coefficient on ion concentration[56]. The normalized

maximum refractive index n(0) is obtained by p = 0 in Equation (7.37)

206

microlens

(.4 0

(.1

f:4

4-

:.0 4.0 1. 0 3.0

70,6°

inicro:cructured 7ilicon

Axial distance Z (mm)

1.0

1.0

Figure (7.22). Intensity distributions of the ray trajectories of rear incidence and N.A by the intensity of far field pattern.

207

A = [n(0) — n21 c=-1 exp[2a — 1]

n2 ... (7.39)

n(r) = n2exp—n. 2 sin -1

fl

t2 p2

[ ta

dt ... (7.38)

n2exp [La] f

The relative index difference A is obtained by

Using Equations(7.38) and (7.39), the focal length f and N.A are obtained

as

2a 2a ... (7.40) 'z1 f L ,_ n(0)]

= rc In (1 + A)

I.' n2 1

n2a = E N.A = n2 In[n(0)] = 5- n2 ln[1 + A] 2

...(7.41) i 2

If A <-<-1, then N.A P-- 'I n2 A 2

...(7.42)

It is determined that the index distributions for low aberration

n2T 2 a 2 microlens with parameter of normalized lens radius

Xand —f

= 1" 2 3" 4 5 . It must be Gt. a

noted that the profile is dependent on the focal length f. From this model,

two ways of increasing N.A., (i) Increasing of maximum index difference

An and (ii) Reduction of aberration by controlling the index profile to fully

utilize the index difference An. The index of substrate n2 has assumed

n2 (0)—n22

208

theoretically to the 1.537. The maximum N.A of the microlens substrate n2

has assumed theoretically to the 1.537. The calculated and simulated results

of the relation between lens properties in terms of focusing efficiency and

Radial distance of microlenses are shown in Figure (7.23) and tabulated in

Table (7.11). Figure (7.24) shows the Measured and calculated light

trapping scheme as the energy distribution of light intensity (normalized)

versus V-grooves axial distance for focusing with and without microlenses

and numerical values were reported in Table (7.12). As long as the light

rays are bounded in a unit circle of phase space, those rays are fully guided

within the microlens. Those that end up reaching outside the unit circle

represent the light ray radiated from the microlens and they are associated

with loss in transmitted light caused by lateral offset between the

microlenses.

Table (7.11). Variation of focusing efficiency for change in radial distance

SI.No. Radial distance I1M

Focusing efficiency Simulated Theoretical

1 110 10 6 2 120 27 24 3 130 43 40 4 140 66 64 5 150 84 80 6 160 92 89

209

150 I • I

160

100

80 -i

Fo

cusi

ng e

ffic

ien

cy (

/o)

60 -

40 - 7• • Theoretical

—ID— Simulation

20 -

I • I • I • I •

110 120 130 140 0

Radial distance (gm)

Figure (7.23). Variation of Focusing efficiency with radial distance

210

1.0 -

•

I

Table (7.12). Normalized intensity without focusing and with Focusing by

microlens along the radial distance

SI. No.

Radial distance

Normalized Intensity (without focusing)

pm

Theoretical Simulated 1 40 0.00 0.00 2 30 0.20 0.25 3 20 0.55 0.58 4 10 0.92 0.96 5 0 .096 0.94 6 -10 0.92 0.96 7 -20 0.55 0.58 8 -30 0.20 0.25 9 -40 0.00 0.00

Radial distance

p.m

Normalized Intensity (with focusing)

Theoretical Simulated 20 0.002 0.004 15 0.130 0.150 10 0.300 0.320 05 0.860 0.840 0 0.990 0.997 -5 0.860 0.840 -10 0.300 0.320 -15 0.130 0.136 -20 0.002 0.004

-30 CI

d is t3nc (WTI )

20 .30 • ',Mill ou t 'roc u inj(Thil ori, Mc al) • outtocu ling iiImuIafloti ) A -0:1111 Fowling (no ro tk; a I )

lylthFou 3 in g ula Mon )

Figure (7.24). Variation of normalized intensity with axial distance of solar cell with and without microlens

2 1 1

7.3.6 High throughput of cylindrical microlenses

There has been great progress to concentrate the beam onto the

steering reflectors performance using microlens. Although microlenses are

often used in imaging applications, in many cases the goal of the microlens

is the projection of light from one point to another. Knowing where the

light will go is only the first step in designing a light-projecting system; it is

just as important to know how much light is transmitted. Typically the

light-gathering capability of a microlens is quantified as follows:

f f #- D

... (7.43)

where, f is the focal length of the microlens and D is the diameter of the

microlens

NA (Numerical Aperture) o.s

...(7.44)

When using a microlens as a tool to convey light from an emitter to a

detector, it is important to consider what is known as throughput (TP), a

quantitative measurement of transmitted light energy. V-grooves reflectors

(Emitters and detectors) are areas regardless of the effect of the

microlenses, because microlens diameter affects throughput even when the

[ remains constant [57,58]. Referring to Figure (7.25), D1 is the

diameter of the focusing microlens, and D2 is a measurement of the V-

groove i.e., the size measured perpendicular to the beam axis with respect to

212

the V-groove and the expressions to determine throughput of a microlens

are:

D2 2F X =

f Y =— and Z = 1+(1 + r). Y2

2

4 X2Y2)1/2 G =

2

Throughput (7'P) = 2.4649 xG x D12

... (7.45)

... (7.46)

.(7.47)

Calculating TP for cylindrical microlens computed yields the best

performance between diameter and throughput. 165 pm optimum radial

distance is being used in this work which has the maximum N.A of the

microlens corresponding to throughput (TP) = 12.3 is 0.25 as shown in the

Figure (7.26) and represented in Table (7.13).

213

half fieil of view

Microlens

focal length f

X =D2 /21 Y=21/D, v /

i \

Ib000

li / , 1

,

V - groove reflector 1), V

Figure (7.25). Theoretical measuring setup of high throughput microlens

214

20 -

18 -

16 -

• Throughput (simulated) • Throughput (Theoretical)

4

I I I

0.1 0.2 0.3

Numerical Aperture NA

00 0.4

8 -

6

Table (7.13). Variation of throughput for different numerical aperture

Sl.No Numerical

aperture (NA) Throughput

Theoretical Simulated 1 0.00 5.00 5.400 2 0.05 7.20 7.500 3 0.10 8.10 8.500 4 0.15 9.20 9.600 5 0.20 10.00 10.40 6 0.25 12.30 12.70 7 0.30 14.00 14.50 8 0.35 16.20 16.80 9 0.40 19.60 19.90

Figure (7.26). Variation of throughput with numerical aperture

215

7.3.7 Quantum efficiency

The internal quantum efficiency (QE) is calculated approximated by

A QE=

. (7.48) A + Aft

I (1— R) 12(W) (PL)

Where (PL) is the path-length enhancement factor, R is rear surface

reflectivity, W is the average substrate thickness and at wavelength A the

absorption coefficient of undoped silicon is small [59,60]. A f, is the free-

carrier absorption coefficient. Across the visible region, silicon is not

transparent; it might have been compared to the thickness of the cell to

make a significant absorption coefficient. It has components from both the

bulk of the cell and from the diffusion regions. Its value can be estimated

from the results of Schmid [61] as 3x10'8 2.2 a-CM-1, where X, is wavelength

in irn and 3 is the average doping level in the cell in cm-3. For the silicon

cells, 3 has estimated as 1.7x1016 cm-3. PL takes into account the oblique

passage of light across the wafer. A value of 1.35 has assigned in the

present calculations [1/cos (45.5 )]. The angle of 45.50 corresponds to the

most important initial double passage across the wafer, although different

angles would apply for subsequent passes.

This analysis suggests that the main mechanism for improving the

internal quantum efficiency of the cell is to improve the rear surface

reflectivity. R, which presently appears to be about 97%. Ultimately, this

216

efficiency will be limited by free-carrier absorption, at least at cell operating

voltage[62]. The external quantum efficiency also depends on the reflection

from the cell, which approaches a value of 67% at long wavelengths. Ray

tracing shows that most of the light reflected at these wavelengths arises

from light escaping after one "double pass" across the cell.

Improving the cell's light-trapping scheme by incorporating

pyramids on both top and rear surfaces[63] or by tilting the top surface

pyramids[64] would decrease this reflection. The latter is the preferred

option since the former would involve multiple bounces of light on some

rear reflections, reducing the absorption.

Figure (7.27) shows that the theoretical and simulated internal

quantum efficiency of the cell together with the microlens reflectance using

(quantity of light transmitted) throughput technique. The calculated internal

quantum efficiency is nearly to peak at about 98% and the simulated value

attained to peak around 95% for measurement from 1.0 to 1.6nm

wavelength ranges and does not remain in the region 100-62% as stated by

the reference [65]. The maximum harvesting sun radiation under standard

air mass index (AM 1.5D) the calculated and simulated results quite agree

with reported values were tabulated in Table (7.15) and a comparative

graph is drawn and reported in Figure (7.28).

217

Table (7.14). Normalized quantum efficiency and reflectivity for incident wavelength A ( um)

Sl.No Incident

wavelength A, (ttm)

Quantum efficiency normalized

Reflectivity normalized

Theoretical Simulated Theoretical Simulated 1 1.0 0.65 0.57 0.45 0.43 2 1.1 0.69 0.64 0.31 0.36 3 1.2 0.75 0.70 0.25 0.30 4 1.3 0.85 0.80 0.15 0.20 5 1.4 0.86 0.81 0.14 0.19 6 1.5 0.88 0.82 0.12 0.18 7 1.6 0.89 0.84 0.11 0.16

1.0

0.9 -

0.8 -

0.7 -

0.6 -

0.5 -

0.4 -

0.3 -

0.2 -

0.1 -

• A

A Quantum efficiency (Theoretical) Reflectivity (Theoretical)

A- Quantum efficienpy (Simulation) Reflectivity (Simihtion)

A

e _ -

•

•

0.0 10

•

11.1 11.2 11.3 11.4

Wavelength 2, (pin)

1.5

16

Figure (7.27). Normalized quantum efficiency and normalized reflectivity with change in incident angle.

218

85

Theoretical ---0-- Simulated

A. Reporterd

---- 80

65- f)

55

1.2 1.3 1.4 1.5 50

10 11.6

Table (7.15). Harvesting of solar radiation in ')/0 with change in incident wavelength (um)

SI.No Incident

wavelength )■-• (gm)

Harvesting of solar radiation (/o)

Theoretical Simulated Reported1381

1 1.0 54.7 53.5 53.0 2 1.1 62.4 61.9 61.0 3 1.2 68.2 67.5 67.4 4 1.3 71.3 70.9 70.2 5 1.4 73.4 73.0 72.7 6 1.5 76.8 75.8 75.0 7 1.6 79.9 78.8 78.0

Incident Wavelength 2, (gm)

Figure (7.28). Variation of harvesting of solar radiation with incident angle.

219

7.3.8 Results and Discussion

V-groove thin silicon solar cell can give raise to a significant degree

of light trapping within the solar cells. This effect is particularly relevant to

maximizing the efficiency of silicon photovoltaic devices, where texturing

has commonly used for reflection control. A theoretical and simulation

model was proposed to get efficient light trapping in V-grooved thin silicon

solar which is schematically demonstrated in Figure (7.14). A solar cell

with a negligible reflectivity requires a departure from conventional design.

An anisotropically etched (V-grooved) surface modifies the geometry of the

cell to permit multiple reflection and enhanced absorption.

A ray tracing model is developed to analyze solar cells with

V-grooved surfaces. For a structure with V-grooves formed by intersecting

(111) plane, the maximum conversion efficiency is increased about 38%

when compared to planar structure. V-grooved texturing of the top surface

of a solar cell combined with a planar reflective rear surface gives

significantly poorer light trapping than the corresponding where the top

surface is Lambertian[66]. This is due to the high percentage of rays which,

on their first pass across the cell, are reflected from rear reflector onto a face

of a V-groove which couples them straight-out. A random layout of

pyramids gives slightly better performance. This percentage is very high, if

the V-grooves are all the same size and are positioned on a regular interval.

For the optimum value of angle BDG of 54.7° and width of the V-groove of

220

80 pm, minimum pitch of the V-groove is calculated as 160 pm for our

proposed model of cylindrical microlens coupled V-groove thin silicon

solar cells. Using this design geometry optical path length is calculated to

determine thickness of solar cell to improve efficiency of solar cells.

Reduction in reflection loss using V-grooved surface is mainly due

to enhancement of multiple reflections, reduction of the angle of incidence

and utilization of total internal reflections. Three parameters such as tilt

angle of V-groove surface, incident angle and refractive index were taken

and analyzed to reduce reflectivity. The critical angles for V-groove

reflections were reported in Table (7.5). For some particular critical angles

the total number of rays getting reflections is doubled. The same attempt

was made for three different refractive indices of 1.5, 2.0 and 2.5 and results

were summarized in Figures (7.19 - 7.21) and tabulated in Table (7.8 -7.10),

respectively.

Reflectivity takes a step like behavior for various tilt angles and it

found to be near zero for higher tilt angles represented in Figure (6.17). For

higher index values the reflectivity loss increases and it attains minimum

reflectivity only at higher tilt angle of 60 degrees. Lesser reflectivity loss

can be achieved by reducing the refractive index such as 1.5 yields at

around 42 degrees. Even for lower tilt angles reflectivity reaches a

maximum of 0.04 for n = 1.5. Lower refractive index is suggested to get

221

minimum reflectivity loss even at optimum tilt angle a of 40 degrees. The

daily average reflectivity monotonically decreases with increasing tilt angle

from 30° to its upper limit (59.1° for ni=1.5) in this analysis. The three tilt

angle regions were observed 0-300, 30-600, 60-750, above 750 in the above

study. Based on this, an attempt was made to study the reflectivity by

varying the incident angle for tilt angles 0°, 30°, 60° and 75° . The similar

graph was plotted for index value of 1.5, 2.0 and 2.5 represented in

Figure (7.19-7.21).

Tilt angle a = 0° takes appreciable reflectivity for an incident angle

00 degree and increases exponentially after 0 = 600 degree. V-groove tilt

angle of 30° maintain a steady reflectivity of 0.04, 0.11 and 0.19 for

refractive index 1.5, 2.0 and 2.5 respectively. The near zero reflectivity is

maintained for higher tilt angle of 750 for vast incident angle variation from

0 to 650 for the refractive index value of 1.5. In general, the V-grooved

surface of large tilt angle reduces the reflectivity at a small angle of

incidence due to multiple reflections between opposite faces of groove.

Concentrator has been considered to have two important role ie., to

increase efficiency and to reduce the cell area. The first one can be

accomplishes by light confinement that already discussed in chapter VI and

in section 7.4. Cavities with a small entry aperture and a bigger cell area

makes the light from a concentrator goes into the cavity through the entry

222

aperture impinges on the cells and diffusely reflecting walls and it totally

randomized in direction. The decreased reflectivity of the V-groove is

considered to increased light confinement beyond the one that can be

achieved by simple randomization of light by normal textured solar cells.

The radial distance between the concentrator and the solar cells is

varied from 110 pm to 160 pm and its focal performance is analyzed

theoretically and simulation method results were reported in Table (7.11)

and Figure (7.23). From the reported figure it is clear that the focusing

efficiency increases linearly as radial distance increases and it attains a

maximum efficiency of 92% at a radial distance of 160 tim. When this

theoretical approach was extended to microlens arrays for practical

applications, a worst alignment with respect to solar cells having axial

distance of 80 pm is considered as + 20 itm. Since the axial distance is

80p.m the interference effect due to microlens arrays focusing to solar cells

greatly reduces to give better focusing efficiency. By fixing the radial

distance the normalized intensity for solar cell by using with and without

microlenses were compared and results were plotted as Figure (7.24) and

given in Table (7.12). The intensity is broadly distributed in solar cells if no

microlens had been used. But as we expected the light coming through

concentrators focused at the center and increases the intensity light. This

has been confirmed with increase in throughput values shown in

Figure (7.26). The excess heat exists in the solar cell may be removed and

223

utilized by up-conversion process which has applications as infrared

indicator, bio-labels or three dimensional displays[67,68] etc.

V-groove structures treated on the surface of silicon solar cells

proposed in section 7.2 and 7.3 are effective in lowering the reflection loss

as discussed in section section 6.4. When using a concentrator and

optimizing its radial distance, numerical aperture of solar cell etc., high

density diffusion occurs at a sharpened position in a textured surface. This

non-uniform diffusion reduces the conversion efficiency. To solve this

problem, the proposed theoretical model has been used and its quantum

efficiency with respect to wavelength has been reported in Table (7.14) and

in Figure (7.27). Both quantum efficiency and reflectivity had been

normalized and a appreciable numerical values had been obtained. The

maximum quantum efficiency of 0.89 is calculated for an incident

wavelength of 1.6 prn and minimum values of 0.65 for 1.0 pm and the

corresponding reflectivities are 0.45 and 0.11 respectively. Harvesting the

solar radiation at air mass (AM1.5G) responsible for increasing the

quantum efficiency also reported as 54% for A = 1.0 pm and the

maximum harvesting was achieved as 79.9% for sun radiation of

wavelength A = 1.6 urn. The harvesting efficiency of the solar radiation by

the solar cells using cylindrical microlens array as concentrator for the

incident wavelength range from 1.0 pm to 1.6 pm insteps of 0.1 pm

224

reported as 54% to 79.9% well agrees with reported values[69] and

represented in Table (7.15) and in Figure (7.28).

7.4 Conclusion

The optical design and analysis software ZEMAX° had been used to

design our cylindrical microlens system and simulation was done to study

the aberration using spot diagram, point spread function and modulation

transfer function. Theoretical calculations were attempted to calculate the

RMS radii of spot size, normalized intensity and encircled energy, and

compared with the simulated results. All these simulation and theoretical

results primarily gives the aberration of our lens system. When wavelength

increases from 1.0 to 1.6 j.tm for a field angle near to optical axis less than

two degrees aberration gets reduced. This lens system combination well

suited for collimating incident wavelength to the near field angle and to

focus the light with high intensity to the image plane. Hence, this designed

simulated cylindrical microlens system can be used as a microlens array

with minimum interference effect as an efficient light trapper for solar cell

applications. A ray tracing model is developed to analyze solar cells with

V-grooved surfaces. For a structure with V-grooves formed by intersecting

(111) plane, the maximum conversion efficiency is increased about 38%

when compared to planar structure. The minimum pitch of the V-groove is

225

calculated as 160 gm for our proposed model of cylindrical microlens

coupled V-groove thin silicon solar cells.

Lesser reflectivity loss can be achieved by reducing the refractive

index such as 1.5 yields at around 42 degrees. Lower refractive index is

suggested to get minimum reflectivity loss even at optimum tilt angle a of

40 degrees. The near zero reflectivity is maintained for higher tilt angle of

750 for vast incident angle variation from 0 to 650 for the refractive index

value of 1.5. In general, the V-grooved surface of large tilt angle reduces

the reflectivity at a small angle of incidence due to multiple reflections

between opposite faces of groove.

The focusing efficiency increases linearly as radial distance

increases and it attains a maximum efficiency of 92% at a radial distance of

160 gm. The intensity is broadly distributed in solar cells if no microlens

had been used. When using a concentrator and optimizing its radial

distance, numerical aperture of solar cell etc., high density diffusion occurs

at a sharpened position in a textured surface. This non-uniform diffusion

reduces the conversion efficiency. Both quantum efficiency and reflectivity

had been normalized and a appreciable numerical values had been obtained.

The maximum quantum efficiency of 0.89 is calculated for an incident

wavelength of 1.6 p.m and minimum values of 0.65 for 1.0 p.m and the

corresponding reflectivities are 0.45 and 0.11 respectively. Harvesting the

226

solar radiation at air mass (AM1.5G) responsible for increasing the

quantum efficiency also reported as 54% for A = 1.0 gm and the

maximum harvesting was achieved as 79.9% for sun radiation of

wavelength 2. = 1.6 pm. The harvesting efficiency of the solar radiation

by the solar cells using cylindrical microlens array as concentrator for the

incident wavelength range from 1.0 [im to 1.6 p.m insteps of 0.1 gm

reported as 54% to 79.9% well agrees with reported values[69] and

represented in Table (7.15).

227

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