Controlling total spot power from holographic laser by superimposing a binary phase grating

Xiang Liu; Jian Zhang; Yu Gan; Liying Wu

doi:10.1364/OE.19.008498

1. Introduction

Laser-based optical traps use radiation pressure proportional to the incident laser power to manipulate particles on microscopic scale [1]. Since phase-only spatial light modulators (SLMs) allow for dynamic modulation of wavefronts with high accuracy and resolution, it is a good choice to use them for holographic beam steering [2,3]. Some optimized characteristics of manipulation can be obtained if the power of multiple 3D spots focused from the holographic beam can be made controllable, for example, dissociated neurons can be moved as fast as possible and kept alive by just adjusting the power to an optimized value [4].

It can be seen from experiment results that if the total power of spots can be directly controlled by tuning the driver of a laser source [4], the stability of laser is poor, sometimes even its intensity profile can be changed [5]. So, such a tunable optical component as tunable attenuator [6], acousto-optical modulator [7], polarization attenuator and so on, is usually added in front of the laser source to produce a stable power-tunable laser, even a high-accuracy power control. For example, as a kind of polarization attenuator, a half-wave plate placed between two polarizers can be used to achieve an power accuracy, which is as high as 10 times the accuracy of a three-polarizer attenuator (0.001) [8]. However, the additional optical component incorporated inevitably causes an insertion loss, and makes the optical manipulation system more complex in structure. A SLM-based system would have a less power loss, simple structure, even better performance while the powers of spots can be individually controlled by the SLM via a computer generated hologram (CGH) algorithm. A modified spherical wave approach [9] can be used to directly obtain multiple 3D spots, and the power of these spots can be separately controlled using the coefficients used to describe the intensity of spots. However, an obvious intensity error can be seen from the simulated results. The relative intensity error can be kept below 5% through a large number of acceptable single-pixel changes in the refresh interval of a typical liquid crystal SLM (LCSLM) using Direct Search algorithm (DS) [10]. A higher accuracy needs a longer computing time.

Weighted Gerchberg-Saxton (GSW) [11] is proposed in recent years to generate arbitrary 3D spots, because of its efficiency higher than DS after only 10 iterations, which can be completed in about 7 ms [12] with the help of a consumer GPU. From the viewpoint of energy conservation, a dummy area [13] is needed for containing the unwanted power. So, we used a GSW [3] with a fixed dummy area to adjust the power of a single spot with a resolution of 0.002, as shown in Fig. 1 . However, too many iterations are needed to support the real-time applications in this case. It would take a longer time to complete the power adjustment of multiple spots with the same resolution because more than one spot must be controlled. Boulder Nonlinear Systems (BNS), Inc. has now fabricated high-speed SLM at a switching frequency up to 500 Hz. So what we need most is to reduce the computing time needed for a high-accuracy algorithm. It is therefore proposed in this paper to superimpose a tunable binary phase grating [14] with a conventional phase-only hologram to control the total spot power with high accuracy to a random value for real-time optical manipulation without extra power loss by changing the phase depth of grating.

Fig. 1 Simulation results of linearly controlling the power of a single spot with a resolution of 0.002 by GSW. (a) Use the same weighting as Eq. (1) in Ref. [3]. and set the convergence parameter to 0.25 to linearly adjust the power at a single spot (Media 1) located in signal area 362 × 362 centering in hologram 512 × 512; during the iterations per aiming power, the generated complex amplitudes in dummy area are kept until the desired power is obtained, there are two spots containing most power of dummy area. (b) 500 equidistant powers of spot (128, 128, 0) are obtained in more than 9 hours by Matlab in PC with 2.8GHz Intel Pentium D 820 CPU and 512MB Memory, and (c) the number of iterations per aiming power is less than 500, and by average, 105 iterations.

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2. Tunable binary phase grating

The controlling of total spot power is accomplished via binary phase grating BG shown in Fig. 2 above. In order to yield a maximal angle of view, the period of grating is made two pixel widths 2d _pixel, during which one pixel is modulated to phase zero and the other to phase φ.

Fig. 2 Tunable binary phase grating. (a) Phase profile. (b) 2D gray image.

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The complex far-field amplitude resulting from a monochromatic plane wave which is normal incident on the binary phase grating can, according to Fraunhofer diffraction integral [15], be written as

\begin{matrix} U (ξ) = C F [e^{i φ_{BG} (x)}] \\ = \frac{C}{2} \sum_{n = - \infty}^{\infty} δ (ξ - \frac{n}{2 d_{pixel}}) sinc (\frac{n}{2}) (1 + e^{i φ} e^{i n π}) e^{\frac{i n π}{2}} \end{matrix}

where

F [•]

represents Fourier Transform, C is a complex constant, ξ is spatial frequency, and

sinc (x) = \sin (π x) / π x

.

As reported in [16], the intensity of diffractive zero order (n = 0) of binary phase grating BG is a simple cosine function of phase depth φ between two adjacent pixels,

\begin{matrix} I_{0} (φ) = {| U (ξ) |}^{2} \\ = I_{0} (1 + \cos φ) / 2 \end{matrix}

where I ₀ is the maximal intensity at the zero order when the phase profile of the grating becomes a plane wave φ _BG(x) = 0 with an efficiency of 100% at the zero order.

The power of diffractive order is proportional to its intensity [17]. So, the power of the zero order follows the same cosine function, and it can therefore be continuously adjusted in a range from 0 to maximal power without an extra power loss by just modulating phase depth φ.

3. Controlling the total power of 3D multiple spots

In a holographic laser system [2,18], generated by a SLM placed in a plane conjugate to the input pupil of a lens such as a microscope, phase-only hologram inducing a phase modulation of φ(x,y) can be used to transform an incident collimated laser beam with wavelength λ into multiple 3D spots. These 3D spots are generated by the beams with an individual quadratic phase factor, i.e. they act as point sources of converging/diverging spherical waves. For example, as shown in Fig. 3 above, when a converging beam with quadratic phase factor $- π (x^{2} + y^{2}) / (λ f^{'})$ is just away from the SLM, it would focus at spot $1^{'}$ with distance $f^{'}$ from the SLM plane. The quadratic phase factor can be considered as a lens against the SLM [15,19], through Fourier Transformation of the lens with focal length $f^{'}$ , the complex field of spot $1^{'}$ can be expressed as

U_{1^{'}} (x^{'}, y^{'}) = \frac{1}{λ f^{'}} F [e^{i φ (x, y)} e^{\frac{i π}{λ f^{'}} (x^{2} + y^{2})}]

and then, after a lens with focal length f images spot

1^{'}

to spot 1 at position

z = - f^{2} / f^{'}

from the front focal plane of the lens f with a magnification of

M = f / f^{'}

, as reported in [18], the complex field of spot 1 can be expressed by

U_{1} (ξ, η, z) = \frac{1}{λ f} F [e^{i φ (x, y)} e^{\frac{- i π z}{λ f^{2}} (x^{2} + y^{2})}]

where (ξ,η) is the spatial-frequency coordinates with a unit λf/D for convenience, and D is the aperture of SLM.

Fig. 3 Schematic diagram of Fourier optics propagation from SLM plane (back focal plane) to imaging plane.

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The field of multiple needed 3D spots U _H(ξ,η,z) generated by phase-only hologram H inducing a phase modulation of φ _H(x,y) can be calculated using a 3D hologram generation algorithm, for example, Fresnel-modified iterative Fourier transform algorithm (IFTA) [19]. In order to adjust the total power of these spots, a new phase profile displayed on a SLM is formed by directly superimposing binary phase grating BG with original phase-only hologram H as,

φ (x, y) = φ_{H} (x, y) + φ_{BG} (x)

After Fourier transform [Eq. (4)], each complex amplitude of these 3D spots can be described by

U (ξ, η, z) = U_{H} (ξ, η, z) * \frac{1}{2} \sum_{n = - \infty}^{\infty} δ (ξ - \frac{n}{2 d_{pixel}}) sinc (\frac{n}{2}) (1 + e^{i φ} e^{i n π}) e^{\frac{i n π}{2}}

where ∗ presents the convolution operation.

Each intensity of these spots can be synchronously controlled by the diffractive zero order of binary phase grating (n = 0) as

I (ξ, η, z) = {| U_{H} (ξ, η, z) |}^{2} (1 + \cos φ) / 2

and it has the same modulating factor of

(1 + \cos φ) / 2

. Logically, the total power of all concerned spots can be easily controlled by the common factor

(1 + \cos φ) / 2

in the range from 0 to the original maximal total spot power governed by U _H(ξ,η,z) without extra power loss. It can be seen through analyses above that a binary phase grating generates a power-controllable zero order, which can be immediately used to control the power of multiple 3D spots generated by original phase-only hologram H. However, because of the two-pixel period structure of the grating, the maximal angle of view in the direction ξ is reduced by 1/2 to (-N _p/4, N _p/4) in order to avoid the overlap of the concerned spots controlled by other diffractive orders of the grating (n≠0), where N _p is the matrix size of SLM in pixels.

Since being normalized by its maximal value, the total spot power can be adjusted without extra power loss by just modulating phase depth φ in the following common factor

P_{Total} (ξ, η, z) = (1 + \cos φ) / 2

so we just concentrate on the use of Eq. (8). The adjustment using Eq. (8) is symmetrical to π radian in [0,2π] and changed monotonically from 1 to 0 in [0, π], as shown by the solid line in Fig. 4(a) . Furthermore, a phase-only SLM only allows M equidistant phase levels between 0 and 2π, which could be limited by the resolution of driving voltage and the stability of modulated phase [20], it thus makes the binary phase grating to adjust the power with steps in reality, as shown by the dashed line in Fig. 4(a). Meanwhile, the maximal power step △P _max, decreasing with number M described in Fig. 4(b), is inverse proportion to a power-adjustment resolution, and in the constraint of power error ε smaller than △P _max/2, as shown in Table 1 , the power resolution of 0.002 is expected with a phase-only SLM of M = 1600, and it took only 240 seconds ( = 9 hours/135, equivalent to 0.78 iteration of the GSW used in Fig. 1 per aiming power) to generate all phase-only holograms under the same conditions with Fig. 1, during which 500 desired phase depth calculated by solving Eq. (8) in [0, π] can be obtained in 142 μs.

Fig. 4 Normalized simulated power adjusted by a tunable binary phase grating.

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Table 1. Simulation Results of Adjusting Normalized Power by Eq. (8)*

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4. Experiments

A P512-0635-DVI phase-only LCSLM (N _P = 512 and M = 1600 supplied by BNS Inc.) is used to generate the composite phase profile Eq. (5) moduled by 2π because of the limited phase modulation depth (phase stroke) of LCSLM. The polarizer shown in Fig. 5 is oriented such that phase-only modulation is obtained from the LCSLM. A 5 mm-diameter expanded beam originating from a HeNe laser (λ = 632.8 nm, 2 mW) is incident on the LCSLM, subsequently, the beam away from the LCSLM is focused into multiple 3D spots by a lens with f = 150 mm. One of these spots is sampled by a pinhole of 0.6 mm in diameter and measured by a dual-channel power meter (Newport 2832-C), where one channel measures the power of laser spot P ₂ and the other measures the branch power of laser source P ₁ in order to eliminate the effect on measured results from the instability of laser source with result P ₂/P ₁. Therefore, only one individual laser spot can be measured each time, and multiple spots can be measured one-by-one.

Fig. 5 Measurement setup, LCSLM is placed in the back focal plane of the lens.

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As shown in Fig. 6(a) and 6(b), a phase-only hologram calculated by a Fresnel-modified version of GSW [11] generated four equal-bright 3D spots, and their projections formed a 2×2 spot array and centered at the optical axial in the front focal plane of lens. After tunable binary phase grating was added to the original hologram in the form of Eq. (5), relative power P ₂/P ₁ at each spot can be adjusted almost synchronously, while there is a little misalignment between them as shown in Fig. 6(c), which might be caused by the phase error of LCSLM [21–23]. The little misalignment makes the minimal total spot power to be shifted above 0, which will shorten the adjustment range of the total spot power. For example, the minimal total spot power pointed by an arrow in Fig. 6(d) is shifted up to 6.53% of the maximal total power, as shown in the column of “Min. total spot power up-shifted” in Table 2 , so the power-adjusted range will be decreased by 6.53%. In the effective adjustment range, the normalized total spot power is in good agreement with the simulation results, as shown in Fig. 6(e). However, as shown in Fig. 6(f), only 208 equidistant normalized powers can be obtained in the range of phase depth φ from 0 to π, and is about 2/5 of the simulated result 500. Expanding the phase depth to [0,2π], a power resolution of nearly 0.002 is obtained as shown in Fig. 6(g), and similar results can be also achieved at other 4 3D spots, as shown in Table 2.In order to control the total spot power to a random value, 100 random phase depths were chosen from 0 to 2π, as shown in detail in Fig. 7(a) . Their corresponding powers shown in Fig. 7(b) can be controlled. This 100-step measurement was repeated 10 times, 95% standard deviations of normalized total spot power are less than 0.001, and an arithmetic mean of these standard deviations is less than 0.0006, as shown in Fig. 7(c). It can be known from ISO 5725-2:1994(E) that the arithmetic mean can be considered as the repeatability standard deviation of test results obtained under repeatability conditions. A little repeatability standard deviation of 0.0006 can guarantee the total spot power to be controlled to a random value under the same conditions.

Fig. 6 Control the total power of four 3D spots with a projection spacing 128 by superimposing binary phase grating with a conventional phase-only hologram by sweeping the phase depth of grating from 0 to 2π with a step of 2π/1600. A 100-iteration phase-only hologram displayed by (a) a 8-bit gray image generated (b) 4 3D spots marked in turn by 1, 2, 3 and 4 (Media 2 captured nearby the front focal plane in Fig. 5 one-by-one by a CCD of PointGrey GRAS-20S4M-C). (c) Measured relative powers of these 4 spots. (d) Total spot power, a sum of the relative powers of these spots, has up-shifted the minimal total power above zero. (e) Total spot power, normalized in range of 0-1, can be controlled linearly in range of (f) phase depth [0,π] and (g) phase depth [0,2π].

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Table 2. Measurement Results of Four Equal-Bright 3D Spots

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Fig. 7 Control the total spot power to random values by superimposing binary phase grating with the same hologram as shown in Fig. 6(a) by changing the phase depth of grating to the sequence of random values going from 0 to 2π. (a) 100 phase depths of grating chosen for the following step-by-step measurement. In order to make sure the largest power adjustment range to be measured for the normalization of total spot power, the phase depth of the first step was set to 0, the phase depth of the 50th step was set to π, and the phase depth of the last step was set to 2π; the other 97 phase depths were chosen using an uniformly distributed random function. (b) Normalized total spot power can be controlled randomly using phase depth shown in (a). (c) The standard deviation of normalized total spot powers per step can be obtained upon completion of 10 times 100-step measurement, and the arithmetic mean of all standard deviations is less than 0.0006.

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5. Conclusion

The total spot power from a holographic laser can be controlled without any extra power loss by superimposing tunable binary phase grating with a conventional phase-only computer-generated hologram. Realized by a LCSLM, the total spot power has an up-shifted minimal value to shorten the effective power-adjusted range. Furthermore, after being normalized, the total spot power follows a simple cosine function of a phase depth of the grating, thus a simulated phase depth of the binary phase grating can be obtained in an order of μs. A simulated resolution of 0.002 and a random power control companied with a good repeatability precision has been demonstrated by experiments. Future research is scheduled for separate fine adjustment of spot power.

Acknowledgments

This work is funded by National Natural Science Foundation of China under grant 60878048, and China Postdoctoral Science Foundation under grant 20080440898.

References and links

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M	16	256	314	1024	1600
△P _max	0.1913	0.0123	0.0100	0.0031	0.0020
N	5	81	100	332	500
ε	0.0913	0.0061	0.0050	0.0015	0.0010
*N is number of equidistant normalized powers.

	Positions (ξ,η,z) of 4 spots marked in turn by				Phase Depth		Min. total spot power up-shifted (%)
	Positions (ξ,η,z) of 4 spots marked in turn by				[0,π]	[0,2π]
	1	2	3	4	N	N
1	(64,64,0)	(64,-64,0)	(−64,-64,-1)	(−64,64,1)	199	498	3.55
2*	(64,64,0)	(64,-64,0)	(−64,-64,-10)	(−64,64,10)	208	499	6.53
3	(64,64,0)	(64,-64,0)	(−64,-64,-20)	(−64,64,20)	222	499	6.52
4	(32,32,0)	(32,-32,0)	(−32,-32,-1)	(−32,32,1)	368	500	0.48
5	(32,32,0)	(32,-32,0)	(−32,-32,-10)	(−32,32,10)	339	500	1.79
6	(32,32,0)	(32,-32,0)	(−32,-32,-20)	(−32,32,20)	386	500	1.21
7	(96,96,0)	(96,-96,0)	(−96,-96,-1)	(−96,96,1)	340	497	0.51
8	(96,96,0)	(96,-96,0)	(−96,-96,-10)	(−96,96,10)	294	499	0.81
9	(96,96,0)	(96,-96,0)	(−96,-96,-20)	(−96,96,20)	305	491	0.95

M	16	256	314	1024	1600
△P _max	0.1913	0.0123	0.0100	0.0031	0.0020
N	5	81	100	332	500
ε	0.0913	0.0061	0.0050	0.0015	0.0010
*N is number of equidistant normalized powers.

	Positions (ξ,η,z) of 4 spots marked in turn by				Phase Depth		Min. total spot power up-shifted (%)
	Positions (ξ,η,z) of 4 spots marked in turn by				[0,π]	[0,2π]
	1	2	3	4	N	N
1	(64,64,0)	(64,-64,0)	(−64,-64,-1)	(−64,64,1)	199	498	3.55
2*	(64,64,0)	(64,-64,0)	(−64,-64,-10)	(−64,64,10)	208	499	6.53
3	(64,64,0)	(64,-64,0)	(−64,-64,-20)	(−64,64,20)	222	499	6.52
4	(32,32,0)	(32,-32,0)	(−32,-32,-1)	(−32,32,1)	368	500	0.48
5	(32,32,0)	(32,-32,0)	(−32,-32,-10)	(−32,32,10)	339	500	1.79
6	(32,32,0)	(32,-32,0)	(−32,-32,-20)	(−32,32,20)	386	500	1.21
7	(96,96,0)	(96,-96,0)	(−96,-96,-1)	(−96,96,1)	340	497	0.51
8	(96,96,0)	(96,-96,0)	(−96,-96,-10)	(−96,96,10)	294	499	0.81
9	(96,96,0)	(96,-96,0)	(−96,-96,-20)	(−96,96,20)	305	491	0.95

Controlling total spot power from holographic laser by superimposing a binary phase grating

Abstract

1. Introduction

2. Tunable binary phase grating

3. Controlling the total power of 3D multiple spots

4. Experiments

5. Conclusion

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (7)

Tables (2)

Equations (8)

Optics Express