## Beam homogenizers based on chirped microlens arrays

Optics Express, Vol. 15, Issue 10, pp. 6218-6231 (2007)

http://dx.doi.org/10.1364/OE.15.006218

Acrobat PDF (871 KB)

### 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

2. C. Kopp, L. Ravel, and P. Meyrueis, “Efficient beam shaper homogenizer
design combining diffractive optical elements, microlens array, and random
phase plate,” J. Opt. Soc. Am. A: Pure
Appl. Opt. **1**, 398–403
(1999). [CrossRef]

3. H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, T. Beth, and F. Wyrowski, “Analytical beam shaping with
application to laser-diode arrays,” J.
Opt. Soc. Am. A **14**, 1549–1553
(1997). [CrossRef]

*u*(

*y*′′) in the focal plane of the Fourier lens is given by the Fourier transformation of the transmission function of the MLAs[4

4. A. Büttner and U. D. Zeitner, “Wave optical analysis of
light-emitting diode beam shaping using microlens
arrays,” Opt. Eng. **41**, 2393–2401
(2002). [CrossRef]

5. N. Streibel, U. Nölscher, J. Jahns, and S. J. Walker, “Array generation with lenslet
arrays,” Appl. Opt. **30**, 2739–2742
(1991). [CrossRef]

*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

6. J. Duparré, F. Wippermann, P. Dannberg, and A. Reimann, “Chirped arrays of refractive
ellipsoidal microlenses for aberration correction under oblique
incidence,” Opt. Express **13**, 10539–10551
(2005). [CrossRef] [PubMed]

## 2. cMLA design

### 2.1. Geometrical considerations

*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

*f*and 2

_{i}*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}8. D. Daly, R. F. Stevens, M.C. Hutley, and N. Davies, “The manufacture of microlenses by
melting photo resist,” Meas. Sci.
Technol. **1**, 4729–4735
(1990). [CrossRef]

9. P. Dannberg, G. Mann, L. Wagner, and A. Braüer, “Polymer UV-molding for micro-optical
systems and O/E-integration,” Proc. SPIE **4179**, 137–145
(2000). [CrossRef]

*a*and focal lengths

_{i}*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

*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

*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

*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

*v*is the spatial frequency which holds

*u*(

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

*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*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

_{spot}*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

*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*and focal lengths

_{i}*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).

## 4. Evaluation of homogenization

*d*=

*p*/

*b*. The quality factor

*q*simplifies to

_{a}*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.

## 5. Simulation results

*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}*α*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

_{c}*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.

## 6. Conclusions and outlook

## References and links

1. | F. M. Dickey and S. C. Holswade, “Laser beam shaping: Theory and Techniques,” Marcel Deller, New York, (2000). |

2. | C. Kopp, L. Ravel, and P. Meyrueis, “Efficient beam shaper homogenizer
design combining diffractive optical elements, microlens array, and random
phase plate,” J. Opt. Soc. Am. A: Pure
Appl. Opt. |

3. | H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, T. Beth, and F. Wyrowski, “Analytical beam shaping with
application to laser-diode arrays,” J.
Opt. Soc. Am. A |

4. | A. Büttner and U. D. Zeitner, “Wave optical analysis of
light-emitting diode beam shaping using microlens
arrays,” Opt. Eng. |

5. | N. Streibel, U. Nölscher, J. Jahns, and S. J. Walker, “Array generation with lenslet
arrays,” Appl. Opt. |

6. | J. Duparré, F. Wippermann, P. Dannberg, and A. Reimann, “Chirped arrays of refractive
ellipsoidal microlenses for aberration correction under oblique
incidence,” Opt. Express |

7. | F. Wippermann, J. Duparré, P. Schreiber, and P. Dannberg “Design and fabrication of a chirped
array of refractive ellipsoidal micro-lenses for an apposition eye camera
objective,” Proc. of SPIE |

8. | D. Daly, R. F. Stevens, M.C. Hutley, and N. Davies, “The manufacture of microlenses by
melting photo resist,” Meas. Sci.
Technol. |

9. | P. Dannberg, G. Mann, L. Wagner, and A. Braüer, “Polymer UV-molding for micro-optical
systems and O/E-integration,” Proc. SPIE |

10. | E. Hecht, Optics, 2nd Edition, (Addison-Wesley Publishing Co., Reading, Mass, USA, 1987) |

11. | U.-D. Zeitner and E.-B. Kley, “Advanced lithography for
micro-optics,” Proc. SPIE |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(030.6140) Coherence and statistical optics : Speckle

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(140.3300) Lasers and laser optics : Laser beam shaping

(220.4830) Optical design and fabrication : Systems design

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: March 21, 2007

Revised Manuscript: May 3, 2007

Manuscript Accepted: May 3, 2007

Published: May 4, 2007

**Citation**

Frank Wippermann, Uwe-D. Zeitner, Peter Dannberg, Andreas Bräuer, and Stefan Sinzinger, "Beam homogenizers based on chirped microlens arrays," Opt. Express **15**, 6218-6231 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6218

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### References

- F. M. Dickey and S. C. Holswade, "Laser beam shaping: Theory and Techniques," (Marcel Deller, New York, 2000).
- C. Kopp, L. Ravel, and P. Meyrueis, "Efficient beam shaper homogenizer design combining diffractive optical elements, microlens array, and random phase plate," J. Opt. Soc. Am. A: Pure Appl. Opt. 1, 398-403 (1999). [CrossRef]
- H. Aagedal, M. Schmid, S. Egner, J. Muller-Quade, T. Beth, and F. Wyrowski, "Analytical beam shaping with application to laser-diode arrays," J. Opt. Soc. Am. A 14, 1549-1553 (1997). [CrossRef]
- A. Buttner and U. D. Zeitner, "Wave optical analysis of light-emitting diode beam shaping using microlens arrays," Opt. Eng. 41, 2393-2401 (2002). [CrossRef]
- N. Streibel, U. Nolscher, J. Jahns, and S. J. Walker, "Array generation with lenslet arrays," Appl. Opt. 30, 2739- 2742 (1991). [CrossRef]
- J. Duparre, F. Wippermann, P. Dannberg, and A. Reimann, "Chirped arrays of refractive ellipsoidal microlenses for aberration correction under oblique incidence," Opt. Express 13, 10539-10551 (2005). [CrossRef] [PubMed]
- F. Wippermann, J. Duparre, P. Schreiber, and P. Dannberg "Design and fabrication of a chirped array of refractive ellipsoidal micro-lenses for an apposition eye camera objective,"Proc. SPIE 5962, 723-733 (2005).
- D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, "The manufacture of microlenses by melting photo resist," Meas. Sci. Technol. 1, 4729-4735 (1990). [CrossRef]
- P. Dannberg, G. Mann, L. Wagner, and A. Brauer, "Polymer UV-molding for micro-optical systems and O/Eintegration," Proc. SPIE 4179, 137-145 (2000). [CrossRef]
- E. Hecht, Optics, 2nd Edition, (Addison-Wesley Publishing Co., Reading, Mass, USA, 1987)
- U.-D. Zeitner and E.-B. Kley, "Advanced lithography for micro-optics," Proc. SPIE 6290, 629009-1 - 629009-8 (2006).

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