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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 9081–9090
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Despeckling fly’s eye homogenizer for single mode laser diodes

Yosuke Mizuyama, Nathan Harrison, and Riccardo Leto  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 9081-9090 (2013)
http://dx.doi.org/10.1364/OE.21.009081


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Abstract

A novel fly’s eye homogenizer for single mode laser diodes is presented. This technology overcomes the speckle problem that has been unavoidable for fly’s eye homogenizers used with coherent light sources such as single mode laser diodes. Temporal and spatial coherence are reduced simultaneously by introducing short pulse driving of the injection current and a staircase element. Speckle has been dramatically reduced to 5% from 87% compared to a conventional system and a uniform laser line illumination was obtained by the proposed fly’s eye homogenizer with a single mode UV-blue laser diode for the first time. A new spatial coherence function was mathematically formulated to model the proposed system and was applied to a partially coherent intensity formula that was newly developed in this study from Wolf’s theory to account for the results.

© 2013 OSA

1. Introduction

In the last decade, single mode UV-blue laser diodes have emerged into the consumer market such as in Blu-ray optical drives and are coming in handy. The cost-per-watt for laser diodes can be very low compared with other lasers if they are mass-produced. Also, they are becoming attractive owing to their deep absorption for many useful and functional materials and their compact size. One of the feasible ways to build a high power laser annealing machine is to assemble a massive number of laser diodes, each of which is shaped into a flat-top intensity. The only potential drawback of laser diodes is their wide variation in divergence angles among a production batch. This makes it difficult to combine with the refractive beam shaping method to get a flat-top intensity with a massive number of laser diodes because the refractive beam shaper is very sensitive to the input beam intensity profile, which requires individual design for each laser diode.

In order to alleviate this problem, the fly’s eye homogenizer could be best suited as the fly’s eye homogenizer is much less sensitive to the input beam variation. Yet, it is well known that the conventional fly’s eye homogenizer does not work well with coherent light sources such as single mode laser diodes since they are spatially and temporally coherent and the principle of the fly’s eye homogenizer, i.e., superimposing multi-beamlets, gives rise to speckle with a high intensity contrast. Therefore, the only light sources that have been successfully used in the conventional fly’s eye system so far are poor-coherence lasers such as excimer lasers [1

1. L. Erdmann, M. Burkhardt, and R. Brunner, “Coherence management for microlens laser beam homogenizers,” Proc. SPIE 4775, 145–154 (2002) [CrossRef] .

] and multimode laser diodes with very large M2.

This study aims to address this challenging problem to open a new path to expand the application of the fly’s eye homogenizer. Another purpose of this study is to model a new spatial coherence function and to extend Wolf’s partial coherence theory to explain the experimental results since neither conventional spatial coherence functions nor conventional partial coherence theories can be applied to the proposed system. To the best of our knowledge, we are the first to be successful in achieving these goals.

2. Standard fly’s eye homogenizer

3. Despeckling fly’s eye homogenizer

3.1. Optical configuration

Our novel despeckling fly’s eye homogenizer includes a method of controlling both the temporal and spatial coherence of a laser diode by simultaneously applying electronic and optical disturbances. In our proposed despeckling fly’s eye system (Fig. 1), the laser beam first passes through a collimator lens. The first microlens array is a one-dimensional element having a curvature only in the beam shaping (y) axis. The collimated beam is split into the same number of beamlets as microlenses by the microlens array. Each beamlet is focused near the first surface of a staircase element. The staircase element has the same number of steps as the number of microlenses and the same aperture size as the microlens aperture size in the y direction. Each step of the staircase element has a predetermined height in the z direction. Finally all the beam-lets are superimposed in the image plane after passing through the second microlens array and a field lens.

Fig. 1 Optical layout of proposed despeckling fly’s eye homogenizer. The system has a single mode (TEM00) laser diode, two fly’s eye lenses with N (=4) microlenses with aperture size d and focal length fML, a staircase element with four steps of height h and a field lens with focal length fFL. The laser beam is homogenized in the fast axis (top) and is focused in the slow axis (bottom).

3.2. Temporal coherence

The laser diode is pulsed at as short as one nano-second so the temporal coherence of the laser diode becomes substantially low. The optical path differences that the staircase elements generate are then long enough to make any pair of beamlets uncorrelated if the step heights are different for all the pairwise combinations. The resultant beam intensity at the image plane is a smoothed diffraction pattern determined by the microlens aperture depending on the degree of coherence reduction.

The turn-on characteristics of laser diodes are described by the multi-longitudinal mode rate equation [6

6. K. Petermann, Laser Diode Modulation and Noise (Kluwer, 1988) [CrossRef] .

]. Typically, the side modes quickly diminish after a couple of nano-seconds and only the main modes remain excited. By pulsing the injection current quickly enough, more longitudinal modes can exist, which results in a wider power spectrum and further means a narrower fringe visibility according to the Wiener-Khintchine theorem. Thus, short pulsing realizes poorer temporal coherence. Figure 2 shows the power spectrum of a laser diode lasing at a center wavelength of 0.4025 μm and the corresponding fringe visibility for CW driving and driving conditions of different pulse widths. The fringe visibility was computed from the Fourier transform of the power spectrum owing to the Wiener-Khintchine theorem. The fringe visibility undergoes significant reduction as the pulse width becomes short whereas the CW condition exhibits a lingering visibility over a wide range.

Fig. 2 (a) Measured power spectra and (b) computed fringe visibilities for various pulse widths (blue: 1 ns, green: 10 ns,yellow: 100 ns) compared to CW operation (red). Fringe visibility revives at the multiple of 2nLDL = 5.5 mm optical path difference corresponding to the measured free spectral range of c/(2nLDL)= 55 GHz. Here nLD and L are the refractive index of the laser diode material and the laser cavity length, respectively.

In more detail, the power spectrum 𝒫 in Lorentzian shape and its complex fringe visibility 𝒱 for a Fabry-Pérot laser with the center frequency ν0, free spectral range and bandwidth δν can be written as a function of time delay τ as [7

7. J. W. Goodman, Statistical Optics (John Wiley & Sons, 2000).

]
𝒫(ν)=m=0N01Imπδν/2(νν0+mdν)2+(δν/2)2
(2)
𝒱(τ)=Aexp[πδν|τ|]m=0N01Imexp[2πi(ν0mdν)τ]
(3)
where A is an appropriate constant, N0 is the number of longitudinal modes and Im is the mode intensity, which is determined by the laser gain function. The fringe visibility has a revival period given by 2nLDL. This periodicity is attributed to the free spectral range of the laser cavity = c/(2nLDL) where c is the speed of light. Thus the interference ability of a laser diode decays with a decay constant δν but it reappears at frequency .

3.3. Spatial coherence

Now that the temporal coherence is appreciably degraded, insertion of a staircase element spatially introduces lower coherence over the entire beam size than the inherent spatial coherence of the laser diode. The inherent spatial coherence is then superseded and modified by the temporal coherence. If the step positions of the staircase element completely match with the microlenses aperture positions and if all N!/{2!(N − 2)!} combinations for choosing two beamlets out of the entire N beamlets have at least one step height difference, the correlation drops in accordance with the degraded temporal coherence to a value given by 𝒱 ((nST − 1)mh) from unity, where nST is the refractive index of the staircase material and m(> 0) is the number of step difference that the two beamlets undergo as in Fig. 3.

Fig. 3 Logarithmic close-up plot of Fig. 2(b). The abscissa is normalized by the unit optical path difference corresponding to a single step height. Circles show the degree of coherence corresponding to each step difference.

It is essential that there be no misalignment between the staircase element and the microlens arrays. Otherwise even a small leak of light that misses a step from an adjacent lenslet causes interference and amazingly destroys the despeckling effect. Owing to short pulsing, the optical path differences that the staircase element needs to create in order to make spatial coherence disturbances between the beamlets are realistic in size. Otherwise, one must insert an extremely large staircase having long step heights, which would introduce various nontrivial problems such as optical aberration and geometrical restriction.

The theoretical background described in Eq. (3) allows us to devise the staircase element and facilitates the design of the staircase element, e.g., one can determine the number of steps of a staircase element that avoids not only impractical element size as previously mentioned but also high fringe visibility revival. Note that the visibility revival can not be significantly reduced even with short pulsing. To know the actual spatial coherence, we need actual visibility data that are directly obtained by experiment or calculated from the power spectrum.

The modified spatial coherence as a whole can be described by a completely new type of function that we see for the first time and will be derived in detail in the later part of this paper.

3.4. Experimental results

Under driving conditions of 1, 10, 100 ns pulsing and CW and an optical setup with fML = 40 mm, fFL = 40 mm, d = 1 mm, h = 1 mm, nST = 1.51 and N = 4, the beam intensity profile in the image plane was measured by a CCD camera. The input beam was collimated and of laterally single mode for both axes as in Fig. 4. The laser diode was oriented in such a way that the fast axis was homogenized along the y axis and the slow axis was focused along the x axis. The image data were summed up in each column to represent the integral average data along the row direction, i.e., x direction. The local speckle contrast is defined by (ImaxImin)/(Imax + Imin), where Imax and Imin are the maximum and minimum intensities in a speckle period. The average speckle contrast was calculated for all speckle periods of 16.1 μm for the entire flat-top width of 1 mm. The results were shown in the insets of Fig. 5. Drastic improvements can be seen for the simultaneous combination of the shortest pulsing and the staircase element (Fig. 5(f)) compared with either the case with the shortest pulsing alone (Fig. 5(a)) or with the staircase element alone (Fig. 5(b)), which represents the conventional fly’s eye homogenization technology. It is worth noting that fly’s eye lenses with only four microlenses achieved a flat-top profile with sufficient uniformity although one typically needs much more microlenses to have good homogenization. This is another virtue of our system using single mode laser diodes.

Fig. 4 Far field intensity of the input beam measured by a CCD camera and cross section plots showing the Gaussian mode for both axes. The fast axis was used for homogenization while the slow axis was focused by the field lens.
Fig. 5 CCD camera images and the integral row data. (a) 1 ns driving without staircase element, (b) CW driving without staircase, (c) – (f) CW, 100 ns, 10 ns and 1 ns with staircase. The insets are magnified images showing the detail structure of the speckle. The numbers in the figure denote the average local speckle contrasts.

3.5. Mathematical modeling of the spatial coherence function

In order to calculate the partially coherent intensity, one needs to know the spatial coherence function and to apply it in the framework of Wolf’s partial coherence theory, which is already established and has proven to be correct for some practical cases [8

8. R. A. Shore, B. J. Thompson, and R. E. Whitney, “Diffraction by apertures illuminated with partially coherent light,” J. Opt. Soc. Am. 56(6), 733–738 (1966) [CrossRef] .

]. In most cases, the spatial coherence functions are not known a priori and therefore they were conventionally ”assumed” [8

8. R. A. Shore, B. J. Thompson, and R. E. Whitney, “Diffraction by apertures illuminated with partially coherent light,” J. Opt. Soc. Am. 56(6), 733–738 (1966) [CrossRef] .

]. In our case, however, the spatial coherence function can be mathematically formulated and can be quantitatively evaluated with the aid of experimental measurement of the temporal coherence as mentioned in the preceding subsection since the original inherent spatial coherence is replaced by our artificial spatial coherence. Unlike the conventionally assumed spatial coherence functions such as Gaussian, exponential and sinc functions, our spatial coherence function is no longer such a function. Also the conventional assumption that the spatial coherence function depends only on the coordinate difference of the two integration points for which the mutual coherence is calculated does not hold for our case. This fact makes one unable to use the convenient Schell’s theory [7

7. J. W. Goodman, Statistical Optics (John Wiley & Sons, 2000).

, 9

9. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15(1), 187–188 (1967) [CrossRef] .

]. Yet, we are able to derive a convenient formulation for the partial coherent intensity for our case once the spatial coherence function has been determined, which will be shown below and in the following subsection.

Before deriving the spatial coherence function, let us define the mutual coherence modulus gm, (m = 0, ⋯ ,N − 1), which is a normalized positive number that determines the degree of coherence between two arbitrarily chosen points on the diffraction aperture, which is in the plane after the second microlens array. When the projections of the two points to the staircase element stay on the same step surface, there is no path difference, namely the original spatial coherence predominates, i.e., g0 is unity. Here we have ignored very small optical path differences that the microlenses yield. If they cross a step discontinuity, an optical path difference of (nST − 1)h appears and therefore the temporal coherence replaces the original spatial coherence. Here we have assumed that the inherent spatial coherence distance, which is described by the van-Cittert-Zernike theorem, is much larger than the step width d. The spatial coherence for the two points in this case is equal to the temporal coherence at the time delay of (nST − 1)h/c, i.e., g1 = 𝒱 ((nST − 1)h) and it remains constant as long as the two point projections cross only one discontinuity. This applies to the case of multiple steps, i.e., gm = 𝒱 ((nST − 1)mh), m = 0,⋯,N − 1.

Next, to derive our spatial coherence function, let y′ and y″ denote the two points and let us note that our spatial coherence function depends on the absolute coordinate of one of the two points, say y′, as well. Due to the discontinuities, our spatial coherence function γ(y′,y″) can be expressed by a shifting piecewise constant function of y″ for N piecewise domains of y′ as illustrated in Fig. 6 for given N (=4) and d for a specifically chosen y axis that is aligned along the bottom edge of the staircase element. Note that we avoided the singular points between the domains, which correspond to the valley of the microlens arrays and the discontinuity of the staircase element. Now the spatial coherence function can be mathematically formulated as below. Notably, it is worth mentioning that it is found to be a separable function, which simplifies to a great extent not only the formulation of the partial coherent intensity but also the computational load as will be seen in the later subsection.
γ(y,y)={γ1(y),0<y/d<1γ2(y),1<y/d<2γ3(y),2<y/d<3γ4(y),3<y/d<4
(4)
where
(γ1,γ2,γ3,γ4)={(g0,g1,g2,g3),0<y/d<1(g1,g0,g1,g2),1<y/d<2(g2,g1,g0,g1),2<y/d<3(g3,g2,g1,g0),3<y/d<4
(5)
and
gm={1,m=0𝒱((nST1)mh),m=1,,3.
(6)

Fig. 6 Spatial coherence functions γ(y′, y″) at the second microlens array for proposed despeckling fly’s eye system versus normalized coordinate of y″.

3.6. Mathematical derivation of the partially coherent intensity

The diffraction theory for partially coherent light can be formulated by Wolf’s theory [10

10. M. Born and E. Wolf, Principles of Optics, 7th ed. (McGraw-Hill, 1968).

]. Let P(y;z) be a point in the image plane at distance z from a diffraction aperture 𝒜 in one dimension and P′(y′) and P″(y″) be two positions on 𝒜. For a given mean wavelength λ̄, the partial coherent intensity Ipc at the point P is calculated by the following double integral formula under the Fresnel or the Fraunhofer approximation.
Ipc(y;z)=1λz𝒜u(y)u*(y)γ(y,y)eik(yy)y/zdydy
(7)
where k = 2π/λ̄ and u is the complex transmission amplitude of the diffraction aperture and the asterisk stands for the complex conjugate. Note that u includes the quadratic phase for the Fresnel approximation. This formula is used to propagate the field from the second microlens array at z = z1, which is the diffraction aperture 𝒜, to the image plane at z = z2 in our system.

To determine u(y,z1) in Eq. (8), we simply need to calculate the complex amplitude after the second microlens array by the Fresnel approximation. This can be done as is done for the conventional double fly’s eye system. One may refer to Büttner et al. for derivation [12

12. A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41(10), 2393–2401 (2002) [CrossRef] .

]. The only difference here is that, to a good approximation for low NA microlenses, we can neglect the spherical aberration introduced by the thickness of the staircase element when N is small.

The complex transmission amplitude for the microlens array TML is defined as
TML(y)=m=1Nrect(y(m12)dd)eik{y(m12)d}2/(2fML).
(10)
Let u0 and u0+ be the complex amplitude before and after the first microlens array, and let u1 and u1+ be the complex amplitude before and after the second microlens array. Then, overlooking trivial constants, we get each amplitude as follows.
u0(y0)=Ae(y02d)2/σ2u0+(y0)=u0(y0)TMLy0u1(y1)=eiky12/(2fML)fML[u0+(y0)TML(y0)eiky02/(2fML)]u1+(y1)=u1(y1)TML(y1)
(11)
where σ and A are constants. Up to trivial constants, the complex amplitude u(y1, z1) is given by
u(y1,z1)=u1+(y1)eiky12/(2fFL).
(12)
Inserting Eq. (12) into Eq. (8), we can numerically compute the partially coherent intensity for the proposed despeckling fly’s eye homogenizer as in Fig. 7. Compared to the coherent intensity, smoothing effects are evident for the case of the staircase element alone and the case of the combination of staircase element and short pulse driving as well. In particular, the latter case exhibits dramatic improvement of the speckle contrast. The computational results with regard to the speckle contrast qualitatively follow the trend of experimental results very well.

Fig. 7 (a) Computational results of the intensity at the image plane z = z2 for the proposed despeckling fly’s eye homogenizer with various laser driving conditions (1, 10, 100 ns and CW) versus coherent intensity indicated as Dirichlet2. Horizontal axis is normalized by the flat-top width dfFL / fML, (b) Close-up plot versus relative position normalized by the speckle period λfFL / d. The numbers in the legend are the local speckle contrast in the center.

4. Conclusions and discussion

We have developed a novel despeckling fly’s eye homogenizer and obtained a significant reduction of speckle that has never been achieved with any static method in the past. The results presented here will provide a very useful and cost effective method for making a laser beam homogenizer for laser diodes of spatially and temporally high coherence. We also formulated a new class of spatial coherence function and established a convenient formula for the partially coherent intensity for our proposed system.

Acknowledgments

Authors are grateful for fruitful discussions with Professor Frank Wyrowski at Friedrich Schiller University Jena, Germany and Professor Masahisa Tabata at Waseda University in Tokyo, Japan.

References and links

1.

L. Erdmann, M. Burkhardt, and R. Brunner, “Coherence management for microlens laser beam homogenizers,” Proc. SPIE 4775, 145–154 (2002) [CrossRef] .

2.

M. Zimmermann, N. Lindlein, R. Voelkel, and K. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE 6663, 1–13 (2010).

3.

F. Wippermann, U. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express 15(10), 6218–6231 (2007) [CrossRef] [PubMed] .

4.

P. Di Lazzaro, S. Bollanti, D. Murra, E. Tefouet Kana, and G. Felici, “Flat-top shaped laser beams: reliability of standard parameters,” Proc. SPIE 5777, 705–710 (2005) [CrossRef] .

5.

F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications (Taylor & Francis, 2006).

6.

K. Petermann, Laser Diode Modulation and Noise (Kluwer, 1988) [CrossRef] .

7.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 2000).

8.

R. A. Shore, B. J. Thompson, and R. E. Whitney, “Diffraction by apertures illuminated with partially coherent light,” J. Opt. Soc. Am. 56(6), 733–738 (1966) [CrossRef] .

9.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15(1), 187–188 (1967) [CrossRef] .

10.

M. Born and E. Wolf, Principles of Optics, 7th ed. (McGraw-Hill, 1968).

11.

Y. Mizuyama, R. Leto, and N. Harrison, “A new fly’s eye homogenizer for single mode laser,” Proc. SPIE 8429, 842915 (2012) [CrossRef] .

12.

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41(10), 2393–2401 (2002) [CrossRef] .

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.6140) Coherence and statistical optics : Speckle
(030.6600) Coherence and statistical optics : Statistical optics
(140.2020) Lasers and laser optics : Diode lasers
(140.3300) Lasers and laser optics : Laser beam shaping

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: January 10, 2013
Revised Manuscript: March 28, 2013
Manuscript Accepted: March 29, 2013
Published: April 4, 2013

Citation
Yosuke Mizuyama, Nathan Harrison, and Riccardo Leto, "Despeckling fly’s eye homogenizer for single mode laser diodes," Opt. Express 21, 9081-9090 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-9081


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References

  1. L. Erdmann, M. Burkhardt, and R. Brunner, “Coherence management for microlens laser beam homogenizers,” Proc. SPIE4775, 145–154 (2002). [CrossRef]
  2. M. Zimmermann, N. Lindlein, R. Voelkel, and K. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE6663, 1–13 (2010).
  3. F. Wippermann, U. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express15(10), 6218–6231 (2007). [CrossRef] [PubMed]
  4. P. Di Lazzaro, S. Bollanti, D. Murra, E. Tefouet Kana, and G. Felici, “Flat-top shaped laser beams: reliability of standard parameters,” Proc. SPIE5777, 705–710 (2005). [CrossRef]
  5. F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications (Taylor & Francis, 2006).
  6. K. Petermann, Laser Diode Modulation and Noise (Kluwer, 1988). [CrossRef]
  7. J. W. Goodman, Statistical Optics (John Wiley & Sons, 2000).
  8. R. A. Shore, B. J. Thompson, and R. E. Whitney, “Diffraction by apertures illuminated with partially coherent light,” J. Opt. Soc. Am.56(6), 733–738 (1966). [CrossRef]
  9. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.15(1), 187–188 (1967). [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics, 7th ed. (McGraw-Hill, 1968).
  11. Y. Mizuyama, R. Leto, and N. Harrison, “A new fly’s eye homogenizer for single mode laser,” Proc. SPIE8429, 842915 (2012). [CrossRef]
  12. A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng.41(10), 2393–2401 (2002). [CrossRef]

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