In earlier work, we introduced new ways of generating a series of interference patterns formed from Laguerre–Gaussian (LG) beams, which are being used as advanced optical tweezers in creating and manipulating three-dimensional structures. In this work, we have succeeded in demonstrating, for the first time to our knowledge, double LG and LG beams with a Gaussian-beam interference using a Michelson interferometer. We have been able to observe LG interference of unequal azimuthal charge by using just two holograms. This is a new type of optical tweezers because the tweezers have the ability to transfer orbital angular momentum, spin angular momentum, and optical gradient force simultaneously to microparticles. This provides a great opportunity for investigating the force interaction within a single laser beam.
© 2002 Optical Society of America
In general, optical trapping and manipulation is based on three types of force or momentum. First, particles are trapped in the highest-intensity region of the beam, owing to optical gradient force for high-refractive-index particles with respect to its surrounding medium . Second, the orbital angular momentum of a beam is transferred to a particle because of the phase singularity within a beam . Third, spin angular momentum is transferred to the particles because of the circular polarization of the beam [3,4].
Currently, two types of optical tweezers are being used. The first type uses a highly focused Gaussian beam  to create optical gradient force on high-index particles. This method was first demonstrated by Ashkin in 1986  and has been widely used in the life sciences and in the study of colloid interactions. The method has frequently been employed in the three-dimensional micromanipulation of micrometer-sized colloid particles, DNA, chromosomes, and protein structures. In biomechanics, optical tweezers have been extremely useful in the force estimate for kinesin motors, the interaction of myosin and actin, and other molecular motors .
By coupling the first type of optical tweezers with a spatial light modulator (SLM) , Curtis et al. were able to generate dynamic multiple optical tweezers for multiple manipulation of microparticles. This has provided a nonmechanical approach to the steering of the beams. However, SLM is restricted by its present pixel size (resolution) and high cost.
The second type of optical tweezers uses the Laguerre–Gaussian (LG) beam, TEM*0l, also known as the optical vortex, to transfer the orbital angular momentum of the beam to the particles trapped in the region of zero intensity, which was demonstrated by He et al.  in 1995. This provides an additional dimension of rotation to optical trapping. Furthermore, the trapping of the particles of low refractive index is possible. Recently, Gahagan and Swartzlander  also demonstrated that both high- and low-refractive-index particles can be simultaneously trapped by use of a single optical vortex.
More recently, new optical tweezers have been introduced by St. Andrews University [8,9], in which two optical tweezers are combined. These new optical tweezers have the ability to manipulate and rotate multiple particles simultaneously in three dimensions.
The new tweezers are created when two LG beams of opposite helicity are interfered by use of a Mach–Zehnder interferometer. The outcome is a series of propagation-invariant intensity spots circling the circumference of the vortex. These intensity spots have been demonstrated to be able to stack microparticles and create three-dimensional structures . And with the addition of a wave plate at the output, each of the spots can transfer spin orbital momentum onto the particles . By use of these advanced optical-trapping geometries, enhanced micromachines can be easily created. The authors of Ref.  have embarked on creating an appropriate “optical toolkit.”
In this paper, we present new methods to achieve double LG interference and a single LG beam with Gaussian-beam interference. Instead of using the Mach–Zehnder interferometer used by most research groups, we attempted to use the other mirrored interferometer, the Michelson interferometer. The Michelson interferometer is a much more concise setup, since it uses less optical equipment while achieving the same results as the Mach–Zehnder interferometer. Furthermore, we present, for the first time to our knowledge, advanced optical tweezers that possess all three optical forces mentioned above. The setup, which essentially uses two holograms–one inverted with respect to another–is extremely simple, stable, and compact to use. The theory behind our interference setup is discussed in Section 2, and discussion of the setup and simulation and of the experimental results is presented in Sections 3 and 4, respectively.
2. Theory: Interference of Laguerre–Gaussian Beams
Collinear interference of LG beams with either its mirror image or the nondiffracted zeroth order generated from the hologram has resulted in a series of rotating intensity patterns [8,12]. These intensity spots have proven to be useful in manipulating chromosomes and silica spheres.
A Michelson interferometer can be utilized to obtain such a rotating intensity pattern. The approach, using an off-axis hologram, is pretty straightforward. When the zeroth order–a Gaussian beam–from one arm of the interferometer interferes with a +1 or -1 diffraction order–a LG beam from the other arm–a fan-shaped intensity pattern is formed. This fanshaped intensity pattern is the result of the mismatch of Guoy phase and the wave-front curvature.
This fan-shaped intensity pattern is a direct representation of the azimuthal variation of the LG beams. It indicates the presence of orbital angular momentum and its rotation direction. As mentioned by Basistiy et al. , the helical revolving directions of the +1 and -1 diffraction orders from an off-axis hologram are opposite in direction. As seen in Fig. 4, an off-axis hologram illuminated with a plane wave produces a single zeroth-order Gaussian beam and two LG beams of the same numerical charge but opposite helical/topological sign.
Therefore when these two opposite diffraction orders interfere, the resulting intensity pattern is the same when the LG beam interferes with its mirror image. This resulting beam will allow effective stacking of microparticles and the creation of three-dimension structures, as demonstrated by MacDonald .
However, we believe that such invariant spots may essentially allow for two types of optical force to be transferred to particles, namely, spin angular momentum and gradient forces . Because of the equal but opposite orbital angular momentum within the beam, no orbital angular momentum will be transferred to the particles. Therefore, for the orbital angular momentum to be transferred onto the particles, interference of unequal LG modes has to occur.
Modifying the conventional interferometer to obtain the unequal interference will be difficult. Furthermore, the interferometer is sensitive to the surrounding environment, which will restrict the type of environment in which it can be deployed. Therefore, we embarked on a novel interference setup that we call “double-hologram” interference.
In the double-hologram interference, we managed to obtain two sets of rotation spots, at the two diffraction orders, instead of one single set of rotating intensity spots by use of a conventional interferometer. In this setup, instead of using the beam splitter as the form of interfering mechanism, we use far-field diffraction to allow each of the orders to interfere with one another. All the results of single and double LG interference are shown in Section 3.
The interference of LG modes can be easily understood when we employ the theory of interference as expressed below in Eq. (1):
The LG full mode in cylindrical coordinates is expressed as follows :
where is a generalized Laguerre polynomial, w(z) is the radius of the beam at the position z, and zR is the Raleigh range. In this context, since only the single annular LG beam will hence be considered to be p=0,
where (z)=1, .
In general, the LG interference can be expressed as
where , and (±l 1∓l 2)ϕ is the phase variation and the resulting orbital angular momentum. Hence the orbital angular momentum per photon in the interference beam will be (±l 1∓l 2)h̄. By use of Eq. (4) we obtain the resulting software simulation results with varying l 1 and l 2; the results are shown in Fig. 1.
Phase variation within the beam is illustrated in Fig. 1A. In Figs. 1B and 1C, the interference between LG charge 3 and a Gaussian beam is shown at the beam waist and far field, respectively. Figure 1D shows the opposite helical phase front of two opposite-charge LG beams. The resulting interference intensity between the LG beam with charge -2 and LG beam with charge 3 is shown in Fig. 1E. Figure 1F shows the intensity pattern of charge 3 and charge -1.
3. Interferometer Setup
In Fig. 2 a Michelson interferometer has a hologram acting as the input source into the beam splitter instead having it placed in either one of the interferometer arms. Mirrors E and D form the two arms. By means of manipulating the reflected intensity beam from these two arms, the interference of the LG beam with its mirrored image and the Gaussian beam can be easily achieved.
In Fig. 3 the double-hologram setup is illustrated. The critical part in the setup is the two holograms, namely, D1 and B1. For interference of oppositely charged LG beams, one hologram must be inverted with respect to the other. However, for interference of same-charged LG beams, both holograms are in the same direction. When two holograms are inverted, it means that the direction of the fork pattern on the hologram is pointing in different directions. Furthermore, for single LG interference with a Gaussian beam, a normal diffraction grating may be used. The diffraction grating on all computer-generated holograms (CGHs) must be the same. The main reason for the interference is that at far field, both the +1 and the -1 from the CGH converge because they have the same diffraction angle. A piece of glass plate, C1, is used to control the rotation of the intensity patterns by changing the optical path from B1 to C1. The resulting intensity pattern is shown in Fig. 6.
The interference results of +1 and -1 diffraction orders with zeroth order, using the setup shown in Fig. 2, are shown in Fig. 4. The helical wave front of the -1 diffraction orders (red arrow), is opposite to the +1 diffraction (blue arrow). This shows that the helicities of the two diffraction orders are opposite. The zeroth order is misaligned at a small angle with the respective diffraction orders (green arrow and yellow arrow). This shows that the charge numbers of the diffraction orders are equal.
In Fig. 5 the interference of the +1 and −1 diffraction orders with the setup shown in Fig. 2 is shown in detail. Figures 5A and 5B show the interference of two LG beams misaligned at a small angle, with charges 3 and 4, respectively. Collinear interference of two LG beams of opposite helicity with charges 2, 3, 4, and 5 are shown in Figs. 5C, 5D, 5E, and 5F, respectively. Hence this shows that the diffraction orders are mirror images of one another.
Next we present the interference pattern achieved with the double-hologram method shown in Fig. 3. In Figs. 6A and 6B the interferences of the misdirected LG beams of charge 1 and charge 2 with a LG beam of charge 3 at the two diffraction orders are shown. Figures 6C. 6D, 6E, and 6F show the collinear interference of the LG beam with charge 1, 2, 4, and 5, respectively, with a LG beam of charge 3. We can see that the interference intensity compared with the results when the LG beam interferes with its mirror image or diffractive orders is the same, and that different number level of helicity is being interfered.
Figure 7A shows an enlarged image of the misaligned LG beam of charge 2 with a LG beam of charge 3, which shows five distinct fork fringes. The interference is obtained from double-hologram methods. To show the existence of orbital angular momentum, it is interfered with the zeroth order by use of a Michelson interferometer. The interference pattern is shown in Figs. 7B and 7D. We can see that within the fork interference there is also a third interference pattern in the form of a spiral arm. In Figs. 7B and 7D, there are two and three spiral arms, respectively. These spiral arms are the same as those in Figs. 7C and 7E where the LG beam of charge 2 and charge 3 interferes collinearly with the zeroth order. These indicate the presence of oppositely directed orbital angular momentum. In Fig. 8 the movie clip will show that Figs. 7B and 7D are rotating in opposite directions.
By comparing the simulation results with the experimental results obtained, we show that our theory is consistent with experimental results. However, an additional test is performed to prove that these intensity patterns from the double-hologram output are due to opposite helical-mode interference of different charge. In view of this, the resulting intensity pattern is directed into a Michelson interferometer, and one of the diffraction orders is interfered with the zeroth order.
In Fig. 7 we can observe the phase wave front of the resulting interference beam from the double-hologram setup. Since there are two sets of opposite-directed spiral patterns, two spiral arms in Fig. 7A and three spiral arms in Fig. 7B are embedded with the fork intensity pattern. The spiral interference pattern is a direct representation of the phase azimuthal variation, and the phase variation is the reason for the existence of orbital angular momentum .
In this paper, consequently, we are able to show that there exists opposite-directed orbital angular momentum. Since the orbital angular momentums are opposite to one other and unequal, the remaining orbital angular momentum, lR , experienced will be equivalent to
where N= intensity spots around the vortex .
However, we note that the control mechanism, a glass plate that is used in the experiments, is coarse and not of suitable precision. Therefore the next step is to bring either a piezoelectric mirror or a LCD for automated changing of optical path length. This will allow for a more precise control mechanism, since optical tweezers are required for working in microscopic scales.
We have shown that a Michelson interferometer can be an alternative tool that researchers can employ to study LG-mode interaction as well as to construct advanced optical tweezers systems instead of using the usual Mach–Zehnder interferometer.
Furthermore, we have demonstrated, for the first time to our knowledge, observation of unequal LG charge interference while orbital angular momentum is conserved after interference, by use of the proposed Michelson interferometer. Most important, it is noted that unequal LG charge interference creates a new laser beam that will enable the transfer of spin, orbital angular momentum, and optical gradient force. Once the intensity pattern is directed onto a microscopic sample, we will be able to observe how the three forces/momentum interact with one another.
In physics, this provides a great opportunity for investigating the force interaction within a single laser beam. Whereas, in the field of life science, it will be a new optical tool that allows dual rotation within the beam, since we have a remaining orbital angular momentum while having external rotating intensity spots. The method can be used in the simultaneous angular deformation of cell structures.
The double hologram can be easily transformed into an attachable laser module system. This will be much more feasible than using an expensive SLM or a conventional interferometer.
We thank Tao Shaohua, Zhang Dianwen, and Cheong Wai Chye for their valuable input. We also extend our thanks to Eddie Tan and Yap Waylin for helping us with the image acquisition and production of the holograms.
References and Links
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