Focal-plane irradiance tailoring using the concept of Woofer-Tweeter deformable mirrors

Zexin Feng; Lei Huang; Mali Gong

doi:10.1364/OE.22.008871

1. Introduction

One important beam shaping problem is how to shape a desired irradiance profile especially a uniform one in the focal-plane domains. Applications requiring such transformations include laser fusion, materials processing and medical procedures [1]. One effective approach to implement this problem is using reflective or refractive freeform optical elements. The desired freeform optical elements can be constructed by a variety of methods (see e.g [2–12].). We also attack the more general beam shaping problem of controlling both the irradiance and phase by firstly computing a ray mapping and then followed by a simultaneous point-by-point construction of two freeform optical surfaces [13, 14]. However, the resulting freeform optical elements designed according to all the above methods are “passive” and limited to static situations.

Smilie et al. proposed a refractive two-freeform-optical-element system which can convert a Gaussian laser beam into a uniform focal-spot with the capability to vary the spot diameter by lateral translation movement between the two elements [15]. Deformable mirror (DM) can release more degrees of controllable freedoms since its surface profile can be deformed by pressures from a number of actuators. The influence of one actuator on its surrounding surface can be described by a special type influence function. The DM surface can be represented by summing all the influence functions. For the most common DMs with piezoelectric (PZT) actuators, the influence function can be simply modeled with a quasi-Gaussian form shown in Eq. (1) [16]:

\exp [\ln ω \cdot {(‖ x - c ‖ / d)}^{α}],

where c represents the location of the actuator, ω is the coupling coefficient between neighboring actuator influence, α is the Gaussian exponent and d is the actuator spacing. It is a radial basis function (RBF), specifically a Gaussian RBF when α = 2, since its value only depends on the distance ||x-c||. The DM surface can then be considered as a combination of RBFs. Cakmakci et al. has been successfully employed Gaussian-RBF combinations to represent freeform optical surfaces in imaging applications [17].

Although stochastic optimization algorithms (see e.g [18–20].) can be conveniently used to implement beam shaping with DMs, they usually suffer from a low convergence speed and a large number of iterations. Some deterministic methods may implement beam shaping efficiently by firstly solving a required surface profile and then approximating it using a single DM [21, 22]. However, these methods suffer a limitation that both the incident and focal-plane beam irradiances must be factorized in two orthogonal transversal coordinates (separable). In addition, a single DM with a limited number of actuators may result in relative large irradiance deviations from the target. We will demonstrate later that the performance can’t be improved by only increasing the actuator number as they suggested.

We attack the above problem using dual DMs, where one DM has low-bandwidth and usually operates in high actuator stroke (Woofer) and the other one has high-bandwidth and usually operates in low actuator stroke (Tweeter). Such a concept has been used for effectively correcting wavefront aberrations (see e.g [23].). Our procedure for controlling focal-plane irradiance using this concept is presented in section 2. Based on Energy conservation and the stationary phase method, we firstly determine a required freeform reflective surface by numerically solving a standard Monge-Ampère type equation. Then, we approximate the required surface using the Woofer-Tweeter DM system based on least square solutions. In section 3, some examples and comparisons are presented to demonstrate the effectiveness of the Woofer-Tweeter system. A brief summary is given in section 4.

2. The concept of Woofer-Tweeter DM system for focal-plane irradiance tailoring

2.1 Generating the required freeform reflective surface

We suppose an incident beam at z = 0 has an irradiance distribution I_in(x) in a bounded domain Ω and a wavefront distribution z = w_in(x), where x = (x,y). The desired focal-plane irradiance distribution at z = f is prescribed as I_d(ξ) in a bounded domain T, where ξ = (u,v). The design problem is to determine the freeform reflective surface z = s(x) which is required to transform I_in(x) into I_d(ξ). Since the strokes of the DMs are usually in microns, we can describe the freeform reflective surface as a pure optical phase transformer, which is equivalent of the thin lens approximation. Considering this approximation, the most efficient way to implement this beam shaping problem may be firstly computing a ray mapping based on Energy conservation and then employing the stationary phase method [24, 25] to obtain s(x) (see e.g [2, 3, 21, 22].). A limitation of this method is that both the incident and focal-plane beam irradiances must be separable. For non-separable beams, the ray mapping is defined implicitly and may be difficult to obtain [2]. We attack this problem as follows.

We firstly derive the synthesis equation of w(x), the desired wavefront immediately behind the freeform reflective surface, by synthesizing the Energy conservation and the stationary phase method according to [24].

For a lossless system, the Energy conservation must be satisfied as Eq. (2):

\int_{Ω} I_{i n} (x) d x = \int_{T} I_{d} (ξ) d ξ .

Introduce the ray mapping ξ = m(x) into the right side of Eq. (2), we can obtain Eq. (3):

\begin{matrix} I_{i n} (x) = I_{d} (m (x)) \det (\nabla m (x)), & x \in Ω \end{matrix} .

The relationship between the complex amplitude E_d(ξ) at the focal plane and the complex amplitude E_in(x) of the incident beam can be computed by the Fraunhofer approximation of the diffraction integral, as shown in Eq. (4):

E_{d} (ξ) = \frac{e^{i k f}}{i λ f} e^{i k (ξ \cdot ξ) / 2 f} \int_{Ω} E_{i n} (x) e^{- i k w (x)} e^{- i k (x \cdot ξ) / f} d x,

wherein k = 2π/λ. The above integral is of the form shown in Eq. (5):

I (k) = \int_{Ω} E_{i n} (x) e^{i k F (x)} d x,

wherein F(x) = –w(x) – (x∙ξ)/f. According to the stationary phase method [24, 25], the major contribution to the integral will come from portions right around the stationary points of F(x), which can be found by Eq. (6):

\nabla F (x) = 0.

Thus, the relationship between the ray mapping ξ = m(x) and the desired wavefront w(x) can be approximated as Eq. (7):

\begin{matrix} m (x) = - f \nabla w (x), & x \in Ω \end{matrix} .

Substitute Eq. (7) into Eq. (3), we can obtain Eq. (8), a standard Monge-Ampère Equation of w(x):

\begin{matrix} I_{d} (- f \nabla w (x)) \cdot \det (D^{2} w (x)) = I_{i n} (x) / f^{2}, & x \in Ω \end{matrix},

where D²w is the Hessian of w. The equation comes with the Neumann boundary condition as shown in Eq. (9):

\begin{matrix} \nabla w (x) \cdot n (x) = - \frac{1}{f} m (x) \cdot n (x), & x \in \partial Ω \end{matrix},

where n(x) is the unit outward normal to the boundary ∂Ω.

The fully non-linearity of the Monge-Ampère Equations make them a difficult task to solve analytically. A few numerical methods have been proposed to solve these equations (see e.g. [26–28].). For uniform focal-plane irradiance distributions, we focused on using a simple explicit iterative finite difference method based on [27].

Equation (10) shows the natural finite difference discretization of Eq. (8) on a uniform Cartesian grid of x with the side length h:

\frac{w_{i, j + 1} + w_{i, j - 1} - 2 w_{i, j}}{h^{2}} \frac{w_{i + 1, j} + w_{i - 1, j} - 2 w_{i, j}}{h^{2}} - {(\frac{w_{i + 1, j + 1} + w_{i - 1, j - 1} - w_{i - 1, j + 1} - w_{i + 1, j - 1}}{4 h^{2}})}^{2} = \frac{I_{i n, i, j}}{f^{2} I_{d}} .

Solve the above quadratic equation for w_i,j and select the larger root, we can obtain Eq. (11):

w_{i, j} = \frac{a_{1} + a_{2}}{2} + \sqrt{{(\frac{a_{1} - a_{2}}{2})}^{2} + {(\frac{a_{3} - a_{4}}{4})}^{2} + \frac{h^{4}}{4 f^{2}} \frac{I_{i n, i, j}}{I_{d}}},

wherein the notations a₁, a₂, a₃ and a₄ are given as follows:

\begin{matrix} a_{1} = \frac{w_{i + 1, j} + w_{i - 1, j}}{2}, & a_{2} = \frac{w_{i, j + 1} + w_{i, j - 1}}{2}, \\ a_{3} = \frac{w_{i + 1, j + 1} + w_{i - 1, j - 1}}{2}, & a_{4} = \frac{w_{i - 1, j + 1} + w_{i + 1, j - 1}}{2} . \end{matrix}

Equation (11) is rewritten as Eq. (12) which is more suitable for iteration:

w_{i, j} = w_{i, j} + σ [\frac{a_{1} + a_{2}}{2} + \sqrt{{(\frac{a_{1} - a_{2}}{2})}^{2} + {(\frac{a_{3} - a_{4}}{4})}^{2} + \frac{h^{4}}{4 f^{2}} \frac{I_{i n, i, j}}{I_{d}}} - w_{i, j}],

wherein the parameter σ lies in (0, 2). The iterations of the boundary grid points must be must be enforced with Neumann boundary conditions. It is worth noting that -w(x) can also be a solution to Eq. (8) for rectangular focal spot symmetric about both u and v axes. After obtaining w(x), the required freeform reflective surface can be simply calculated as s(x) = (w_in(x)-w(x))/2.

2.2 Approximating the required surface using the Woofer-Tweeter DM system

Now that the required freeform reflective surface z = s(x) has been derived, we will approximate it using the Woofer-Tweeter DM system shown schematically in Fig. 1. In this system, an 4f optical system was employed to arrange the two DM surfaces z = s₁(x) and z = s₂(x) in conjugate positions. Assume that the incident angle is sufficiently small and thus we can ignore its influence.

Fig. 1 Sketch of the Woofer-Tweeter DM system.

Download Full Size | PDF

The approximation problem can be formulated as Eq. (13):

s (x) \approx s_{1} (x) + s_{2} (x) = \sum_{p = 1}^{n} η_{p} \cdot l_{p} (x) + \sum_{q = 1}^{m} τ_{q} \cdot h_{q} (x),

where η_p denotes the weight of l_p (x), the p-th influence function of the Woofer DM with n actuators, and τ_q represents the weight of h_q(x), the q-th influence function of the Tweeter DM with m actuators.

After determining the influence functions, the approximation problem can then be transformed to find the weight vectors η = [η₁, η₂,…, η_n]^T and τ = [τ₁,τ₂,…,τ_m]^T. We firstly obtain the weight vectors η by matching s(x_t) with s₁(x_t) at the t-th discretization point x_t, where t = 1,2,…N. We can then determine τ by compensating the residual error s(x_t)-s₁(x_t) with s₂(x_t), where t = 1,2,…N. The above processes can be accomplished by solving Eqs. (14) and (15) in sequence:

L η = S,

H τ = S - L η .

where L, H and S are given as follows:

\begin{matrix} L = [\begin{matrix} l_{1} (x_{1}) & l_{2} (x_{1}) & \dots & l_{n} (x_{1}) \\ l_{1} (x_{2}) & l_{2} (x_{2}) & \dots & l_{n} (x_{2}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ l_{1} (x_{N}) & l_{2} (x_{N}) & \dots & l_{n} (x_{N}) \end{matrix}], H = [\begin{matrix} h_{1} (x_{1}) & h_{2} (x_{1}) & \dots & h_{m} (x_{1}) \\ h_{1} (x_{2}) & h_{2} (x_{2}) & \dots & h_{m} (x_{2}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ h_{1} (x_{N}) & h_{2} (x_{N}) & \dots & h_{m} (x_{N}) \end{matrix}], \\ S = {[s (x_{1}), s (x_{2}), \dots, s (x_{N})]}^{T} . \end{matrix}

The above Equations are over determined since the discretized-point number N is usually greatly larger than both n and m. However, η and τ can be conveniently acquired with least squares solutions, as shown in Eqs. (16) and (17):

η = {(L^{T} L)}^{- 1} L^{T} S,

τ = {(H^{T} H)}^{- 1} H^{T} (S - L η) .

To make the approximation more accurate, we suggest using measured influence functions and eliminating the original DM surface profiles in the calculation. The effectiveness of above method may also depend on the DM performances, the measurement accuracy of the incident beam and the calibration of the system, etc.

3. Simulation experiment

The above concept was numerically tested by shaping a rectangular uniform focal spot from a high structured Gaussian laser beam with a distorted wavefront aberration (see Fig. 2). The system parameters are shown in Table 1. The discretized-point number N is set as 128 × 128. The focal-spots are obtained based on the Fast Fourier Transform (FFT) simulation with an FFT size of 2048 × 2048. The performances are assessed in terms of the relative root-mean-square-deviation (RRMSD) from the desired focal-plane irradiance distribution and the light energy efficiency (LEE) defined as the ratio of the energy concentrated in the desired region.

Fig. 2 The (a) wavefront and (b) irradiance of the input laser beam.

Download Full Size | PDF

Table 1. System parameters

View Table | View all tables in this article

Here, we adopt -w(x) as the desired wavefront and the resulting freeform reflective surface is shown in Fig. 3(a). It is fully continuous and will not cause spiral phase dislocations. Figure 3(b) shows the resulting focal-plane irradiance distribution. The RRMSD from the desired focal-plane irradiance distribution is ~18.4%, and the LEE is ~93.1%. To make a comparison, we approximate the required reflective surface shown in Fig. 3(a) using an 9-actuator DM, an 64-actuator DM and the conjunction of them i.e. the Woofer-Tweeter DM system, respectively. The fitting capability is quantified by the residual peak-to-valley (PV) value and residual root-mean-square (RMS) error. We can see from Figs. 3(c)–3(h) and Table 2 that the Woofer-Tweeter DM system can best approximate the required surface and yields best results in terms of the RRMSD and LEE of the focal spot. It seems that the 64-actuator DM doesn’t perform better than the 9-actuator one although it brings improvements in performances of the PV value and RMS error. The probably major reason is that the 64-actuator DM can produce more “discrete” surface fluctuations (see Fig. 3(e)) which may result in more “hot-spots” on the focal plane.

Fig. 3 (a) The required freeform reflective surface and its (b) resulting focal-spot; approximation using the 9-actuator DM: its (c) residual error and (d) focal-spot; Approximation using the 64-actuator DM: its (e) residual error and (f) focal-spot; Approximation using the Woofer-Tweeter DM system: its (g) residual error and (h) resulting focal-spot.

Download Full Size | PDF

Table 2. Performance parameters

View Table | View all tables in this article

Comparisons are also performed on influences of the focal spot size and coupling coefficient of the actuator influence function. The focal-spot is enlarged (reduced) just by scaling up (scaling down) the required freeform reflective surface. We can see from Figs. 4(a) and 4(b) that the Woofer-Tweeter DM system performs the nearest curves with the required reflective surfaces as the focal-spot size changes form 0.04mm to 0.36mm. For the Woofer-Tweeter DM system, the smaller the focal spot size, the larger the RRMSD value and the lower the LEE value. This phenomenon is mainly caused by the diffraction effect. The 9-actuator DM has the lowest LEE values, which indicates that 9 actuators are insufficient for confining the energy to the desired region. For the 64-actuator DM, the RRMSD value increases severely when the focal spot size is larger than 0.15mm. It may be caused by the fact that the 64-actuator DM surface is becoming more and more “rough”. Figures 4(c) and 4(d) show the influences of the coupling coefficient ω on the RRMSD and LEE values. As ω changes from 0.05 to 0.25, the Woofer-Tweeter DM system almost produces the lowest RRMSD values and the highest LEE values. However, the difference gaps between the Woofer-Tweeter DM system and single DMs become narrowing as ω increases.

Fig. 4 Variations of the (a) RRMSD and (b) LEE with respect to the focal spot size; Variations of the (c) RRMSD and (d) LEE with respect to the coupling coefficient.

Download Full Size | PDF

4. Conclusion

To our knowledge, we introduce the concept of Woofer-Tweeter DM system to control the focal-plane irradiance profiles for the first time. We firstly construct a freeform reflective surface required for the desired transformations by numerically solving a standard Monge-Ampère equation. This equation can be conveniently synthesized based on Energy conservation and the stationary phase method. Then, we can obtain the Woofer-Tweeter DM surface profiles by determining the control weights of all the actuators based on least square solutions for approximating the required surface. Simulation results show that, compared with single DMs, the Woofer-Tweeter DM system offers the best fitting capability in terms of PV and RMS of the residual errors and brings best focal-spot performances in RRMSD and LEE values.

Although we focused on producing a desired irradiance profile on the focal plane, this concept can also be applied for beam shaping systems without focusing lenses. In addition, this concept may provide a more accurate way for representing a given freeform optical surface.

Future work may include developing faster and more stable methods for determining the required freeform surface, finding better approximation way to obtain the actuator weights within the achievable limits of the two DMs and exploring adaptive control algorithms without measuring the incident beam.

Acknowledgment

The study was sponsored by the National Natural Science Foundation of China (Grant No.61178055).

References and links

1. F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications (CRC, 2005).

2. 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(7), 1549–1553 (1997). [CrossRef]

3. Y. Arieli, N. Eisenberg, A. Lewis, and I. Glaser, “Geometrical-transformation approach to optical two-dimensional beam shaping,” Appl. Opt. 36(35), 9129–9131 (1997). [CrossRef] [PubMed]

4. W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3482, 389–396 (1998). [CrossRef]

5. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef] [PubMed]

6. V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE 5942, 594207 (2005). [CrossRef]

7. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]

8. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef] [PubMed]

9. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation usingsource-target maps,” Opt. Express 18(5), 5295–5304 (2010). [CrossRef] [PubMed]

10. D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011). [CrossRef] [PubMed]

11. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012). [CrossRef] [PubMed]

12. R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21(18), 20974–20989 (2013). [CrossRef] [PubMed]

13. Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013). [CrossRef] [PubMed]

14. Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013). [CrossRef] [PubMed]

15. P. J. Smilie and T. J. Suleski, “Variable-diameter refractive beam shaping with freeform optical surfaces,” Opt. Lett. 36(21), 4170–4172 (2011). [CrossRef] [PubMed]

16. W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991). [CrossRef]

17. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008). [CrossRef] [PubMed]

18. R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, “Adaptive beam profile control using a simulated annealing algorithm,” Opt. Express 13(16), 6085–6091 (2005). [CrossRef] [PubMed]

19. P. Yang, Y. Liu, W. Yang, S. Hu, M. Ao, B. Xu, and W. Jiang, “An adaptive laser beam shaping technique based on a genetic algorithm,” Chin. Opt. Lett. 5(9), 497–500 (2007).

20. H. Yang, Y. Liu, X. Li, and X. Bi, “Technique of adaptive laser beam shaping based on stochastic parallel gradient descent,” Proc. SPIE 7282, 72821W (2009). [CrossRef]

21. K. Nemoto, T. Fujii, N. Goto, T. Nayuki, and Y. K. Kanai, “Transformation of a laser beam intensity profile by a deformable mirror,” Opt. Lett. 21(3), 168–170 (1996). [CrossRef] [PubMed]

22. Z. Zeng, N. Ling, and W. Jiang, “The investigation of controlling laser focal profile by deformable mirror and wave-front sensor,” J. Mod. Opt. 46(2), 341–348 (1999). [CrossRef]

23. S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt. 45(12), 2638–2642 (2006). [CrossRef] [PubMed]

24. F. M. Dickey and H. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).

25. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64(8), 1092–1099 (1974). [CrossRef]

26. G. Loeper and F. Rapetti, “Numerical solution of the Monge-Ampére equation by a Newton’s algorithm,” C. R. Math. Acad. Sci. Paris 340(4), 319–324 (2005). [CrossRef]

27. J. Benamou, B. D. Froese, and A. M. Oberman, “Two numerical methods for the elliptic Monge-Ampére equation,” ESAIM: Math. Model. Numer. Anal. 44(4), 737–758 (2010). [CrossRef]

28. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge- Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011). [CrossRef]

Parameters	Value
Beam wavelength	1064nm
Input beam power	100W
Focal spot size	0.2mm × 0.2mm
Focal length	2000mm
DM aperture	200mm × 200mm
Actuator array of the first DM	3 × 3
Actuator spacing of the first DM	100mm
Actuator number of the second DM	8 × 8
Actuator spacing of the second DM	(200/7)mm
Gaussian exponent α of the IF	2
Coupling coefficient ω of the IF	0.15

Systems	PV(λ)	RMS(λ)	RRMSD	LEE
Required reflective surface	/	/	18.4%	93.1%
9-actuator DM only	0.87	0.114	34.3%	81.9%
64-actuator DM only	0.45	0.033	34.9%	91.2%
Woofer-Tweeter DM system	0.27	0.022	20.6%	93.4%

Parameters	Value
Beam wavelength	1064nm
Input beam power	100W
Focal spot size	0.2mm × 0.2mm
Focal length	2000mm
DM aperture	200mm × 200mm
Actuator array of the first DM	3 × 3
Actuator spacing of the first DM	100mm
Actuator number of the second DM	8 × 8
Actuator spacing of the second DM	(200/7)mm
Gaussian exponent α of the IF	2
Coupling coefficient ω of the IF	0.15

Systems	PV(λ)	RMS(λ)	RRMSD	LEE
Required reflective surface	/	/	18.4%	93.1%
9-actuator DM only	0.87	0.114	34.3%	81.9%
64-actuator DM only	0.45	0.033	34.9%	91.2%
Woofer-Tweeter DM system	0.27	0.022	20.6%	93.4%

Focal-plane irradiance tailoring using the concept of Woofer-Tweeter deformable mirrors

Abstract

1. Introduction

2. The concept of Woofer-Tweeter DM system for focal-plane irradiance tailoring

2.1 Generating the required freeform reflective surface

2.2 Approximating the required surface using the Woofer-Tweeter DM system

3. Simulation experiment

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (4)

Tables (2)

Equations (19)

Optics Express