## Abstract

Lens array arrangements are commonly used for the homogenization of highly coherent laser beams. These fly’s eye condenser configurations can be used to shape almost arbitrary input intensity distributions into a top hat. Due to the periodic structure of regular arrays the output intensity distribution is modulated by equidistant sharp intensity peaks which are disturbing the homogeneity. As a new approach we apply chirped microlens arrays to the beam shaping system. These are non-regular arrays consisting of individually shaped lenses defined by a parametric description which can be derived completely from analytical functions. The advantages of the new concept and design rules are presented.

© 2007 Optical Society of America

## 1. Introduction

In many applications the transformation of the intensity distribution of a coherent
laser beam into a top hat is required. This can be accomplished using single
diffractive or refractive micro-optical beam shaping elements which are designed to
direct certain fractions of the input distribution into specific angles in order to
generate the desired output intensity distribution[1, 2]. The first suffer from their dispersion behavior and are
therefore suited for a small spectral range only. In addition, the scattered light
caused by the non-continuous surface profile of the diffractive elements can be
perturbing. Refractive elements are suited for a wider spectral range and exhibit a
small amount of scattered light. Since they are designed for a given input intensity
distribution, any changes in this distribution will lead to a distorted output distribution[3]. Consequently, temporal fluctuations of the input
distribution cannot be tolerated and furthermore tight position tolerances during
alignment occur, which are obstacles in their practical application. Other widely
used solutions are fly’s eye condenser setups known from microscopy
illumination systems. The setup usually consists of a Fourier lens and two identical
regular microlens arrays (MLA) - often referred to as tandem lens array - where the
second one is placed in the focal plane of the microlenses of the first array. Due
to the use of refractive microlenses the dispersion and the generation of stray
light are much smaller compared to diffractive approaches. As a further advantage
the position of the MLA with respect to the radiant source can be chosen almost
arbitrarily. This allows for both, easy assembly and insensitivity to temporal
fluctuation of the input intensity distribution. Finally, the fabrication of
microlens arrays is much easier compared to refractive beam shaping elements. The
effect of homogenization is achieved by superposing the fractions of the input
radiation transmitted by each channel of the tandem microlens array in the focal
plane of the Fourier lens. From the point of view of geometrical optics the more
channels are used the better the homogenization. However, as a consequence of the
use of many optical channels in parallel multiple-beam interference will occur due
to the spatial coherence of the setup. Furthermore, in our considerations we focus
on temporally coherent radiation sources with infinite coherence length. From the
viewpoint of wave optics the far field distribution
*u*(*y*′′) in the focal plane of
the Fourier lens is given by the Fourier transformation of the transmission function
of the MLAs[4, 5]. In case of regular arrays the transmission function can be
written as the convolution of the transmission function of one channel and a comb
function which encodes the positions of the single lenses multiplied by the
transmission function of the aperture of the entire array

with p being the pitch of the lenses, N the number of lenses in the array and
*y*′ as the coordinate in the focal plane of the
microlenses of the first array. Since we limit our consideration to cylindrical
lenses, a one-dimensional examination is sufficient. The field amplitude in the
focal plane of the Fourier lens is therefore a series of equidistant peaks each with
a sinc-distribution and has the envelope of the Fourier transformation of the
transmission function of the optical channel of the MLA

The modulation due to the comb clearly derogates the homogenization. The transmission function of each channel when using a single MLA is given by the Fourier transform of the aperture of the lens where the microlens acts as Fourier lens. Consequently, the field has a sinc-like amplitude distribution and a spherical phase leading to a modulated far field pattern with soft shoulders (Fig. 1, left). In a tandem setup using two MLA the second lens of each channel flattens the phase in the focus of the first lens. Therefore, a field with an amplitude of a sinc-distribution but now with a plane phase is generated. This leads to the desired far field intensity distribution with the envelope of a top hat (Fig. 1, center). The periodic interference pattern in the far field is caused by the periodic structure of the MLA. Consequently, breaking the periodicity of the MLA will lead to a non-periodic far field pattern. For doing so, chirped microlens arrays (cMLA)[6, 7] can be employed which consist of individually shaped lenses defined by a parametric description of the cells optical function. The parameters for each cell can be derived completely from analytical functions. When using a tandem cMLA the undesired equidistant peaks are suppressed (Fig. 1, right) while the envelope of a top hat is maintained. The functions describing the differently shaped lenses of the cMLA depend on the respective geometrical arrangement. In Section 2 considerations for the design of the cMLA are given. The simulation results for different cMLA setups and design rules are presented in Section 5.

## 2. cMLA design

#### 2.1. Geometrical considerations

Since the width of the top hat in the focal plane of the Fourier lens is given by
the numerical aperture (NA) *μ* of the lenses and the
focal length F of the Fourier lens, all channels must have the same numerical
aperture and thus f-number (f/♯) *η* for
achieving a top hat intensity distribution with sharp edges

with *f _{i}* and 2

*a*being the focal length in air and the width of the i-th lens, respectively. For small angles the NA can be approximated to

_{i}Obviously, the fill factor of the arrays being the ratio of the area covered with
lenses and the total area of the array has to be as close as possible to unity
for achieving maximum transmission efficiency of the system. Therefore, when
chirping the positions of the lenses the apertures have to be chirped as well
and consequently the focal lengths of the MLA have to be chirped for achieving
constant NA for all lenses. In theory almost arbitrary chirp functions are
possible meeting the constraints of maximum fill factor and constant NA for all
cells. However, the manufacturability of the components in praxis limits the
reasonable solutions. The most suitable setup results when placing both arrays
on two planar substrates. This allows for the use of reflow of photoresist as
fabrication method leading to very smooth and well defined surfaces[8, 9]. Consequently, a wedge configuration results where the
cell widths 2*a _{i}* and focal lengths

*f*of all lenses of the cMLA can be calculated using analytical equations having the wedge angle

_{i}*α*, the NA of the lenses and the minimum focal length

*f*

_{0}as parameters (Fig. 2). From simple geometrical considerations it is obvious that the focal length of the lens in air next to smallest lens is given by

with n being the index of refraction of the wedge material,
*a*
_{0} and *a*
_{1} the
semi-widths of the smallest and the adjacent lens, respectively. The semi-width
of the lens with index i=1 is therefore

The focal length of the lens with index i can be calculated to be

and the semi-width of the ith lens is consequently

The focal lengths of all cells can be calculated using Eq. 3 or 4. Using the above functions all parameters of the cMLA can
be determined which depend on geometrical conditions only. The wedge structure
acts as a prism and consequently the beams leaving the tandem array are
deflected by the angle *δ* which depends on the wedge
angle *α* [10]. In consequence, the intensity distribution in the
focal plane of the Fourier lens will not be centered at the optical axis of the
Fourier lens but being shifted laterally by Λ = *F*
∙ tan *δ*.

#### 2.2. Wave optical considerations

According to Eq. 2 the envelope of the far field distribution
*u*(*y*′′) in the focal
plane of the Fourier lens is given by the Fourier transformation of the
transmission function *T*(*y*′) of the
optical channels of the MLA. This can be calculated by a step-wise propagation
through the array. The field distribution in the focal plane of the first lens
is given by the Fourier transformation of the aperture of the first lens with
width 2a and focal length f

which yields

Here, *v* is the spatial frequency which holds

The amplitude of the field in the focal plane of the first lenses is therefore

Inserting Eq. 3 it’s obvious that the field depends on the f/# and the wavelength only

The field in focal plane of the first lens has a spherical phase distribution. Due to the second microlens which is located in the focal plane of the first lens a flat phase is generated and the field is clipped

The field *u*(*y*′′) in the
focal plane of the Fourier lens equals the Fourier transformation of Eq. 14 which yields

Inserting Eqs. 4 and 11 finally yields

The first term of the convolution describes the top hat which has a width of
2*Fμ*. The second term results from the clipping
of the angular spectrum at the aperture of the second lens. In case of an
infinite aperture no clipping would occur and the sinc-function would converge
to a *δ*-function leading to a perfect top hat.
However, any finite extent of the aperture will clip the angular spectrum and,
due to the convolution of the rect-function with a sinc-function with finite
extent, some modulation will always be present in the far field. In analogy to
the Airy disk diameter as a measure for the spot size for circular apertures we
define the spot size *d _{spot}* for the cylindrical lens
with rectangular aperture as the width between the first zeros around the
maximum in the field amplitude. Figure 3 illustrates the influence of the clipping on the
appearance of the far field distribution. Here, different apertures with widths

*w*∙

_{norm}*d*which act as spatial frequency filter are placed in the focal plane of the first microlenses. Strong modulation can be noticed for widths less than 10∙

_{spot}*d*. For bigger values of

_{spot}*w*the far field is almost stable. However, due to the clipping no perfect top hat without any modulation can be achieved. In consequence, the width of the lenses should be at least 10 times larger than the spot diameter to avoid large deviations from the top hat distribution.

_{norm}## 3. Numerical simulation

For regular MLAs the far field distribution can be calculated very easily in a completely analytical manner using Eq. 2 . This is due to the fact that the Fourier transformation of a comb results in another comb-distribution but of different spacing. In case of a cMLA in general the transmission function of the array can be written as

$$\otimes \sum _{i=0}^{N-1}\{\delta \left(y\prime -\sum _{n}^{i-1}{a}_{0}+{2a}_{n}+{a}_{i}\right)\bullet \mathrm{rect}\left(\frac{y\prime}{{2a}_{i}}\right)\bullet \mathrm{exp}\left[i\frac{2\pi}{\lambda}\left(n-1\right)n\left({f}_{i}-{f}_{0}\right)\right]\}.$$

The first term describes the field of a single channel of the tandem MLA according to
Eq. 13 having a flat phase due to the second microlens. The field
directly behind the second lens is the same for all channels since it only depends
on the NA or f/♯ of the lens which is identically for all lenses of the
arrays. The second term describes the effect of diffraction at the aperture of the
entire array. The *δ*-function of the last term encodes
the center positions of the channels. The rect-function accounts for the width of
the aperture of the second lens. The channel-wise constant piston phase factor
results from the different propagation lengths of the channels in media with
different index of refraction (air and wedge material, respectively). The detailed
values of the widths 2*a _{i}* and focal lengths

*f*are determined by the parametric chirp functions and therefore the geometry of the array (see Section 2). For calculating the far field distribution Eq. 17 has to be Fourier transformed which is possible numerically only. Equation 17 describes a geometry as shown in Fig. 4. Here, the wedge between the two arrays is approximated as a staircase since such a geometry without tilted surfaces can easily be described by scalar diffraction theory. However, the only difference to the setup with two plane surfaces accommodating the arrays is a missing prism term which will lead to a deflection by an angle of

_{i}*δ*(Fig. 2).

In the first step for calculating the far field distribution the geometrical
parameters for all channels of the array were obtained from a given NA, minimum
focal length *f*
_{0} and wedge angle
*α* using Eqs. 7 and 8. According to Eq. 17 an array of sinc-functions clipped by the associated
aperture sizes and centered at the related lens vertex position was generated. Then
the channel-wise constant piston phase was multiplied to the amplitude distribution
of the array and finally the Fourier transformation was executed. In the simulation
the tandem cMLA is illuminated by a single plane wave meaning that we assume a
perfectly coherent beam without any divergence.

## 4. Evaluation of homogenization

In order to compare the quality of homogenization of different optical setups a quality measure describing the degree of homogenization has to be found. A possible measure could be the ratio of standard deviation to the mean value of the intensity distribution which is sampled at M supporting points

with

and

For a perfect top hat q equals 0. Otherwise q is larger than 0 and consequently the larger q, the poorer the degree of homogenization. Assuming a periodic intensity distribution according to Fig. 5(a) the mean value and standard deviation yield

and

with *d* = *p*/*b*. The quality factor
*q _{a}* simplifies to

In a fly’s eye condenser based on a regular tandem MLA the far field intensity distribution will look like in Fig. 5(b). The distance p between adjacent peaks can be calculated using the paraxial grating equation

with F being the focal length of the Fourier lens and 2a the pitch of the array. On the other hand the width b of a peak results from diffraction at the aperture of the entire array and is given according to Eq. 2 by

Therefore, the factor d equals N being the number of lenses in the array. The difference in the shape of the peaks of the periodic intensity distributions have to be taken into account using a factor k leading to the simple equation for the quality factor q for regular arrays

The defined quality factor is independent of the pitch and the NA of the microlenses
as well as of the focal length F of the Fourier lens. It is only dependent on the
number of lenses of the array. In Fig. 6 the simulation results using the wave optical
propagation software *Virtual Lab*
^{TM}
*3* and
the graph according to Eq. 26 are plotted. Both graphs match perfectly for k=0.66. Further
on, this is a confirmation of the accuracy of the simulation and gives confidence
that the results for the cMLA are correct, too.

On the one hand many lenses of the array have to be used in order to achieve a good homogeneity independent of the input field distribution. On the other hand, due to the interference effects the peaks in the far field intensity distribution will narrow down when using more lenses which in consequence leads to a decreased homogeneity.

## 5. Simulation results

Based on the proposed numerical algorithm we calculated the far field distribution as
a function of the NA of the microlenses, the minimum focal length
*f*
_{0}, the angle of the wedge
*α*, and the number of illuminated lenses N. In Fig. 7 examples of the calculated far field distribution are
given for a wedge angle of 0° (regular array) and 7°. In case
of using a cMLA a speckle pattern with smaller and non-regular peak distances
compared to regular arrays results which is connected with an improved
homogenization. Figure 8 shows a plot of the quality factor q as a function
of the wedge angle *α* and the number of lenses N when
using microlenses with NA=0.03 and a minimum focal length
*f*
_{0} of 2.63mm. For a wedge angle of 0° a
regular MLA results leading to a graph according to Eq. 26. With increased wedge angle the quality factor decreases
indicating a better degree of homogenization. The larger the number of illuminated
lenses, the smaller the wedge angle can be allowing for a constant degree of
homogenization. However, for increasing wedge angle and number of lenses the minimum
quality factor never significantly drops beyond unity meaning that the standard
deviation and the mean value of the intensity distributions are about the same. The
proposed quality factor q is a global measure of the entire intensity distribution
which is not significantly influenced by local hot spots in the distribution. It is
therefore necessary to have a detailed look at the resulting far field intensity
distribution. In Fig. 9 different plots of the far field intensity
distribution as a function of the wedge angle are given for different numbers of
illuminated lenses. Each line in the graph is normalized to the maximum intensity
value for that specific angle. For a wedge angle of zero the regular peak pattern
results. Firstly, the appearance of the patterns becomes more stable when increasing
the number of illuminated lenses [Fig. 9 from 9(a) to 9(h)]. Since the distributions result from multiple-beam
interference effects the change in the number of involved beams has a stronger
impact when dealing with a small number of beams or lenses, respectively. Secondly,
a region of higher intensity peaks is shifted towards the left side of the
distribution when increasing the wedge angle [Fig. 10(b)]. This is caused by the shift of the point where
the zeroth orders of the lenses interfere constructively. This shift is due to the
channel-wise different piston phase caused by the staircase geometry of the tandem
cMLA which results in a deflection of the zeroth orders. If this deflection angle
exceeds the divergence angle of the microlenses which is determined by their NA, no
peak will appear in the intensity distribution. The staircase geometry acts like a
prism for the zeroth orders. Therefore, no hot spots will be present in the
intensity distribution for wedge angles larger than the critical angle
*α _{c}*

In case of air between the two cMLA of the tandem array, no channel-wise different
piston phases would occur. Consequently, the zeroth orders would always interfere
constructively in the center of the distribution. Fig. 10(a) shows the calculated far field pattern again as a
function of the angle between the two arrays but with air in between the lenses.
Beginning from a regular array on top resulting in a regular interference pattern
the homogenization of the distribution in general improves with increasing angle.
However, since the zeroth orders always coincide in the center of the distribution a
hot spot is always present which is of course undesired. In conclusion, it is not
sufficient to look at the quality factor for choosing the appropriate amount of
required chirp since hot spots can still be present caused by the zeroth orders of
the cMLAs channels. For avoiding these intensity peaks a wedge angle larger than the
critical angle *α _{c}* has to be applied. Using
configurations with even larger wedge angles will of course change the specific
speckle pattern but not improve the overall homogeneity (Fig. 11). In Fig. 12 plots of the far field intensity distribution as a
function of the wedge angle are given for different NAs of the microlenses but with
constant minimum focal length

*f*

_{0}=2.63mm and number of illuminated lenses N=50. In case of a small NA, the ratio of the spot size in the focal plane of the first microlens and the width of the lenses becomes too small and the aperture of the second lens clips large fractions of the angular spectrum (see Sec. 2). Consequently, the far field distribution deviates considerably from the desired top hat as can be seen in Fig. 12(a). Here, the spot diameter in the focal plane of the first lens is about 55

*μm*while the aperture width of the second lens is 263

*μm*. According to Eq. 27 the critical angle increases with increasing lens NA (see Fig. 12(b) to (e)). The critical angle becomes 10° for a lens NA of 0.09 when a wedge material with index of refraction of 1.52 is used. Consequently, in the diagrams the hot spot will not be shifted enough to be outside the distribution. Due to the different NAs of the configurations plotted in Fig. 12 the distributions have different extension which is given by the NA of microlenses and the focal length of the Fourier lens (Eq. 16) but are shown with different magnification in x-direction for better illustration. Furthermore, since the minimum focal length is kept constant the lens widths of the different configurations increase with increasing NA and consequently according to Eq. 24 the distance between adjacent intensity peaks becomes smaller. The extent of the modulated region around the hot spot is therefore decreasing with increasing NA.

Finally, in Fig. 13 plots of the far field intensity distribution as a function of the wedge angle are given each using 50 illuminated microlenses with NA of 0.05 but different minimum focal lengths. Since the NA is constant the lens widths scale with the minimum focal length according to Eq. 24. Consequently, the distance of adjacent intensity peaks is smaller for larger minimum focal lengths and the modulated region around the hot spot sharpens. Due to the constant NA for all configurations the hot spot leaves the distributions at the same critical angle of 5.5°.

## 6. Conclusions and outlook

The use of non-regular MLAs in tandem fly’s eye condenser setups under coherent illumination leads to non-periodic far field intensity distributions with improved homogeneity compared to regular MLA. Wedge configurations based on planar substrates are especially suited since established fabrication technologies for MLA can be employed.

Channel-wise varying phase differences results from unequal propagation lengths in media with different indices of refraction. In case of a wedge the phase differences lead to a shift of the point where the zeroth orders of the microlenses interfere constructively and form a hot spot in the distribution. In order to avoid these hot spots in a wedge configuration the wedge angle has to be chosen larger than a critical angle which is given by the NA of the microlenses and the index of refraction of the wedge material. Beside the accommodation of the arrays on plane substrates it is also possible to use e.g. curved ones[11]. Here, the phase differences the single channels accumulate during propagation are not linearly depending on the distance to the optical axis like in a wedge configuration. Consequently, there is no point in the distribution where all zeroth orders constructively interfere. The use of substrates with of non-planar surfaces for the accommodation of the microlenses in a tandem cMLA is under current investigation.

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