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
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. 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 fi and 2ai being the focal length in air and the width of the i-th lens, respectively. For small angles the NA can be approximated to
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 2ai and focal lengths fi of all lenses of the cMLA can be calculated using analytical equations having the wedge angle 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 . 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
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
The first term of the convolution describes the top hat which has a width of 2Fμ. 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 dspot 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 wnorm ∙ dspot 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∙dspot. For bigger values of wnorm 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.
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
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 2ai and focal lengths fi 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 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
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
with d = p/b. The quality factor qa 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 αc
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 αc 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 55μm while the aperture width of the second lens is 263μm. 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. 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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