## Despeckling fly’s eye homogenizer for single mode laser diodes |

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

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

^{2}.

## 2. Standard fly’s eye homogenizer

*(sometimes called the Dirichlet kernel) as where*

_{N}*N*is the number of microlenses and

*p*represents the periodicity of the main peaks of the coherent intensity. In a fly’s eye system, it holds that

*p*=

*f*

_{FL}

*λ*/

*d*,

*f*

_{FL},

*λ*and

*d*are the focal length of the field lens, wavelength and the aperture size of the microlenses, respectively [2]. Following previous literature [3

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] .

## 3. Despeckling fly’s eye homogenizer

### 3.1. Optical configuration

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

*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.

### 3.2. Temporal coherence

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

*μ*m 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.

*𝒱*for a Fabry-Pérot laser with the center frequency

*ν*

_{0}, free spectral range

*dν*and bandwidth

*δν*can be written as a function of time delay

*τ*as [7] where

*A*is an appropriate constant,

*N*

_{0}is the number of longitudinal modes and

*I*is the mode intensity, which is determined by the laser gain function. The fringe visibility has a revival period given by 2

_{m}*n*

_{LD}

*L*. This periodicity is attributed to the free spectral range of the laser cavity

*dν*=

*c*/(2

*n*

_{LD}

*L*) 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

*dν*.

### 3.3. Spatial coherence

*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*((

*n*

_{ST}− 1)

*mh*) from unity, where

*n*

_{ST}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.

### 3.4. Experimental results

*f*

_{ML}= 40 mm,

*f*

_{FL}= 40 mm,

*d*= 1 mm,

*h*= 1 mm,

*n*

_{ST}= 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 (

*I*

_{max}−

*I*

_{min})/(

*I*

_{max}+

*I*

_{min}), where

*I*

_{max}and

*I*

_{min}are the maximum and minimum intensities in a speckle period. The average speckle contrast was calculated for all speckle periods of 16.1

*μ*m 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.

### 3.5. Mathematical modeling of the spatial coherence function

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] .

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] .

*g*, (

_{m}*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.,

*g*

_{0}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 (

*n*

_{ST}− 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 (

*n*

_{ST}− 1)

*h/c*, i.e.,

*g*

_{1}=

*𝒱*((

*n*

_{ST}− 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.,

*g*=

_{m}*𝒱*((

*n*

_{ST}− 1)

*mh*),

*m*= 0,⋯,

*N*− 1.

*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. where and

### 3.6. Mathematical derivation of the partially coherent intensity

*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

*λ*̄, the partial coherent intensity

*I*

_{pc}at the point

*P*is calculated by the following double integral formula under the Fresnel or the Fraunhofer approximation. where

*k*= 2

*π*/

*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*=

*z*

_{1}, which is the diffraction aperture

*𝒜*, to the image plane at

*z*=

*z*

_{2}in our system.

*u*(

*y*,

*z*

_{1}) 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] .

*N*is small.

*T*

_{ML}is defined as Let

*σ*and A are constants. Up to trivial constants, the complex amplitude

*u*(

*y*

_{1},

*z*

_{1}) is given by 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.

## 4. Conclusions and discussion

## Acknowledgments

## References and links

1. | L. Erdmann, M. Burkhardt, and R. Brunner, “Coherence management for microlens laser beam homogenizers,” Proc. SPIE |

2. | M. Zimmermann, N. Lindlein, R. Voelkel, and K. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE |

3. | F. Wippermann, U. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express |

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 |

5. | F. M. Dickey, S. C. Holswade, and D. L. Shealy, |

6. | K. Petermann, |

7. | J. W. Goodman, |

8. | R. A. Shore, B. J. Thompson, and R. E. Whitney, “Diffraction by apertures illuminated with partially coherent light,” J. Opt. Soc. Am. |

9. | A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. |

10. | M. Born and E. Wolf, |

11. | Y. Mizuyama, R. Leto, and N. Harrison, “A new fly’s eye homogenizer for single mode laser,” Proc. SPIE |

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

**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

- L. Erdmann, M. Burkhardt, and R. Brunner, “Coherence management for microlens laser beam homogenizers,” Proc. SPIE4775, 145–154 (2002). [CrossRef]
- M. Zimmermann, N. Lindlein, R. Voelkel, and K. Weible, “Microlens laser beam homogenizer - from theory to application,” Proc. SPIE6663, 1–13 (2010).
- 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]
- 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]
- F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications (Taylor & Francis, 2006).
- K. Petermann, Laser Diode Modulation and Noise (Kluwer, 1988). [CrossRef]
- J. W. Goodman, Statistical Optics (John Wiley & Sons, 2000).
- 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]
- 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]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (McGraw-Hill, 1968).
- Y. Mizuyama, R. Leto, and N. Harrison, “A new fly’s eye homogenizer for single mode laser,” Proc. SPIE8429, 842915 (2012). [CrossRef]
- 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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