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Independent light fields generated using a phase-only spatial light modulator Paul Leonard Hilario,* Mark Jayson Villangca, and Giovanni Tapang National Institute of Physics, University of the Philippines, Diliman, Quezon City, Philippines *Corresponding author: [email protected] Received January 15, 2014; revised March 4, 2014; accepted March 4, 2014; posted March 4, 2014 (Doc. ID 204774); published March 26, 2014 We present a method of reshaping light in three dimensions via phase modulation. The method calculates the target computer-generated hologram individually and adds to it an appropriate transfer function to translate the reconstruction to any location in space. We are able to simultaneously generate independently controlled arbitrary patterns at different positions using a single laser beam. © 2014 Optical Society of America OCIS codes: (090.1760) Computer holography; (140.3300) Laser beam shaping; (230.6120) Spatial light modulators. http://dx.doi.org/10.1364/OL.39.002036

The ability to modulate both the amplitude and phase of incoming light has a myriad of uses such as in optical communication [1,2], optical manipulation [3–5], and microfabrication [6]. Easier control over the phase and amplitude of the light field was made possible with the advent of spatial light modulators (SLM). Bringing this ability to shape and control an incident light field in three dimensions opens the possibility of optical trapping and microfabrication in multiple locations along the optical axis. One method that has been reported is the use of the angular spectrum method by Xia and Yin [7]. This is essentially a multiplane implementation of the Gerchberg–Saxton (GS) algorithm [8] in which they impose constraints at different planes beyond the Fourier plane and require the phase of the whole source aperture to satisfy these constraints. A major drawback of this method is that if, for example, we want to generate N target intensity patterns at different locations in 3D, we would require the hologram to satisfy N constraints for the same aperture. Since there are multiple constraints, rapid convergence to a solution is not always assured. Furthermore, we lose control of the reconstructed light fields. If we want to change one target, we need to redo the computation for the whole aperture. Another method is based on the linearity of the Fourier transform, where one can replace the Fourier transform of a sum with the sum of Fourier transforms of the individual components. To generate N target intensity patterns at the focus of a lens, we may calculate the computer-generated holograms (CGH) for each constraint using the GS algorithm and superpose the resulting fields [9–11]. This gives us independent control over each target. This also solves the problem of convergence since only one constraint at a time is required to be satisfied. However, the total power will be divided among the individual targets. In this Letter, we present a variation of the multiplane implementation of the GS algorithm based on the linearity of the Fourier transform. Using appropriate transfer functions, we can position the targets at arbitrary locations in 3D. The GS algorithm iterates only between the hologram and the reconstruction plane. To be able to satisfy the constraints at positions other than the Fourier plane, 0146-9592/14/072036-04$15.00/0

we combine the GS algorithm and the angular spectrum (AS) method. The AS method computes the complex field distribution at some distance z from an initial field distribution U
x; y; 0 and is given by Eq. (1)
U
x; y; z  F

−1

 q A
f x ; f y  exp ikz 1 − λ2 f 2x − λ2 f 2y ; (1)

where A
f x ; f y  is the Fourier transform of the field U
x; y; 0 [12] or its angular spectrum, and k is the wave number. For the far-field case, we multiply a circ function with the angular spectrum to exclude evanescent fields. Due to the implementation of the GS algorithm, the angular spectrum is given by A
f x ; f y   a
f x ; f y  exp
iϕGS
f x ; f y ;

(2)

where ϕGS
f x ; f y  is the hologram calculated using the GS algorithm, and a
f x ; f y  is the aperture bounding the hologram. Substituting Eq. (2) into Eq. (1) results in a target reconstruction located at a distance z from the Fourier plane given by
U
x; y; z  F −1 a
f x ; f y  exp
iϕGS
f x ; f y   q × exp ikz 1 − λ2 f 2x − λ2 f 2y :

(3)

The CGH, therefore, can be expressed as the sum of the hologram ϕGS
f x ; f y  and the phase of the angular q spectrum transfer function ϕtransfer  kz 1 − λ2 f 2x − λ2 f 2y . Last, to translate the reconstruction along the transverse direction, we use the Fourier shift theorem and multiply the transfer function H translate  exp
i2π
f x Δx  f y Δy:

(4)

The calculation for the input hologram for a single target is summarized in Fig. 1. The effective input field, u, therefore, is © 2014 Optical Society of America

April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS

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Fig. 2. Optical setup for holographic reconstruction of the target light fields. Fig. 1. Schematic diagram of the algorithm to generate an arbitrary light field situated at an arbitrary 3D location. The algorithm is repeated for other patterns. For multiple targets, the hologram is the phase of the superposed inputs.

u
f x ; f y   a
f x ; f y  exp
i
ϕGS  ϕtransfer  ϕtranslate : (5) To generate multiple independent patterns, we compute for the input field of each target then superpose all inputs, that is uinput 

N X i1

ui
f x ; f y ;

(6)

where ui is the input field for the ith target and is calculated using Eq. (5), and N is the number of desired targets. The effective input CGH into the SLM is given by the phase of uinput . Note that for the case of paraxial approximation, ϕtransfer results in a quadratic phase similar to those presented by Haist et al. [9]. In addition to this, we may define a to be a subregion of the aperture. This results to a piecewise implementation similar to that presented by Belloni and Monneret for optical trapping [11]. Defining a to be a subregion will lead to a loss of resolution due to limitations in the allowed frequencies. The implementation of the Fourier transforms means either the input and output planes are separated by a distance approaching infinity, or that the input and output planes are positioned at the front and back focal planes of the lens or vice versa such as the case in our experiment. For cases other than these, we may use other transformation methods such as the Fresnel transform [13,14]. We built the setup shown in Fig. 2 to implement the method described above. We used a He–Ne laser operating with wavelength 632.8 nm and a beam expander to cover the aperture of the SLM (Hamamatsu Photonics PPM-X8267, 80% fill factor). The SLM consists of parallel-aligned nematic liquid crystal cells which modulates the phase of the input beam and which can be controlled by the video output of a computer. A half-wave plate was used to ensure linear polarization of the laser, which is a requirement for the operation of the SLM. After modulating the phase, the beam is passed through a lens (f  300 mm) which performs the Fourier transform. A charge-coupled device (CCD) camera which scans

along the z axis was used to capture the images at different axial positions. Our targets for the three dimensional light modulation consists of the letters S, Y, N, and C. The target positions for the hologram are shown in Table 1. Figure 3 shows the numerical simulation of the technique. The field for each target was calculated according to Eq. (5), and the whole hologram was calculated using Eq. (6). The fields at different axial planes were calculated using the AS method described in Eq. (1), with z representing the axial distance from the focus of the lens. Figure 4 shows our experimental results. Both Figs. 3 and 4 demonstrate that we are able to simultaneously create arbitrary patterns at different axial positions using our method. The bright spot in the middle of each image is the zeroth-order diffraction or the DC term. This can be removed with the methods described by Palima and Daria [15]. Another option is to add a blazed grating phase to move the reconstructions away from the DC. Note that the total intensity on one target accounts for 15% of the total intensity captured by the CCD camera. This is comparable to the highest possible efficiency of 20% per target assuming that only 80% of the incident light is modulated. We look into the sensitivity of the reconstructions to the axial position. We calculate for the normalized mean square error (NMSE) of the reconstructions for varying z. The plot is shown in Fig. 5. We see from the figure that the NMSE increases slightly as we try to reconstruct the targets at larger distances from the focus. However, this increase is very small and no significant changes in the overall shape of the reconstructions are visible. This shows that the reconstructions are robust to z.

Table 1. Summary of Positions for the Experimental Targetsa Target S Y N C

Δx (mm)

Positions Δy (mm)

z (mm)

−1.0 1.0 −1.0 1.0

−1.0 −1.0 1.0 1.0

0 10 20 30

Negative Δx denotes leftward position and negative Δy denotes upward position. a

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OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014

Fig. 3. Numerical simulation of generation or arbitrary light fields using our proposed method. The hologram window size is 20 mm, while the reconstruction window size is around 7.3 mm.

Fig. 6. Independent control of the generated light fields along the transverse direction. The letters Y, N, and C were moved to different locations, while the letter S was held in place.

Fig. 4. Experimental generation or arbitrary light fields using our proposed method. The size is around 4.8 mm.

Fig. 5. Plot of the NMSE versus z. The NMSE changes slightly with axial position from the focus.

Another limitation of the method is the cross talk between axial planes as shown in both Figs. 3 and 4. Since the input hologram is a superposition of independent fields, the reconstructions at different planes are also

superpositions of light fields from the other planes. This proves that, indeed, the fields are independent of each other, and one can control one target without affecting the others. We can increase the separation of the target planes to improve the signal-to-noise ratio of the images. Additional constraints can also be added to reduce the noise introduced by the light coming from other planes at the expense of the fast convergence of the single-GS implementation for each target. To further demonstrate independent control of the light fields, instead of using static phase images for the input, we vary the input holograms to the SLM corresponding to a particular configuration in the Fourier plane according to Eq. (5). In this experiment, the letter S was held in place while the letters Y, N, and C were moved to arbitrary locations along the transverse direction. The reconstructions are shown in Fig. 6. Since the phase for each region is computed independently of the others, moving one target does not affect the motion of the other targets. Figure 6 shows that we can individually control the generated light structures. Such capability is important in applications requiring precise control of the incident light such as optical manipulation and optical microfabrication. We have presented a method to modulate light in three dimensions using the Gerchberg–Saxton algorithm extended with the use of the linearity of the Fourier transform and the angular spectrum method. We have shown that the method allows us to project arbitrary, independently controlled, and dynamic patterns simultaneously at different axial positions. This would be useful for applications such as multiplane optical trapping and three-dimensional lithography. Experiments were performed at the Instrumentation Physics Laboratory of the National Institute of Physics.

April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS

The spatial light modulator was purchased through the Institution Development Program of the Philippine Council for Advanced Science and Technology Research and Development of the Department of Science and Technology (PCASTRD-DOST). Additional support was received from the University of the Philippines Office of the Vice President for Academic Affairs through the OVPAA-EIDR-VISSER::SM project. M. Villangca is currently affiliated with DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, DK-2800, Kgs. Lyngby, Denmark. We thank C. A. Saloma and L. I. Tapang for useful discussions in the preparation of the manuscript. References 1. F. Gonte, A. Courteville, and R. Dandliker, Opt. Eng. 41, 1073 (2002). 2. E. Alon, V. Stojanovic, J. M. Kahn, S. Boyd, and M. Horowitz, in IEEE Global Telecommunications Conference, 2004 (IEEE, 2004), Vol. 2, pp. 1023–1029.

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