## Abstract

We put forward a technique to manipulate the size of orbital angular momentum (OAM) beams based on space diffraction compensation. Paraxial Fresnel diffraction which carries a negative spatial quadratic phase distribution can be regarded as a negative diffractive effect. To compensate the negative diffraction, we employ a *4f* Fourier lens system containing a phase mask to generate an inverse quadratic phase. The size of OAM beams can be easily controlled by designing the phase mask profile without changing the OAM. The applications of space diffraction compensation in OAM demultiplexing, ring fiber coupling for OAM beams and optical manipulation of micro particles are also discussed.

© 2014 Optical Society of America

## 1. Introduction

There has been increasing interest in orbital angular momentum (OAM) modes, in which the beams comprise a transverse angular phase profile equal to $\text{exp}\left(il\theta \right)$, where $\theta $ is the angular coordinate and $l$ is the azimuthal index, defining the topologic charge (TC) of the OAM modes [1]. These beams have an OAM of $l\hslash $ per photon ($\hslash $ is Planck’s constant $h$ divided by 2$\pi $) and consist of a ring of intensity with a null at the center. OAM beams have been widely used in a variety of interesting applications, such as in optical microscopy [2], micromanipulation [3, 4], quantum information [5, 6] and optical communication [7, 8].

For most OAM beam generators [3, 4], the optical vortex’s radius, which can be tuned via varying transmission distance in free space manually or automatically, increases with the TC. The manual control of transmission distance is not precise, while automatic control is very complex and costly. In addition, it is very hard to freely control several OAM beams with different TCs simultaneously. This means that the OAM beam radius cannot be freely tuned for a fixed transmission system without changing the TC. Nevertheless, the TC of OAM beams characterizes the corresponding mode [7, 9] and the magnitude of optical torque [10, 11]. Especially in some cases, it is necessary to precisely control the beams size without changing the inherent nature of OAM beams (distinguished by TC), such as OAM demultiplexing [12], fiber coupling [7, 9] and optical particle manipulations [10, 11]. For OAM demultiplexing or fiber coupling, the OAM beams radius need be matched with the coherent receiver or the ring fiber to obtain maximum coupling efficiency. And in terms of optical particle manipulation, by freely tuning the OAM beam radius, we can manipulate the particle in radial direction without changing the optical torque.

In this paper, we present a technique to manipulate the beam size of certain OAM based on space diffraction compensation. Paraxial Fresnel diffraction which carries a negative spatial quadratic phase distribution can be regarded as a negative diffractive effect. To compensate the negative diffraction, we employ a 4*f* Fourier lens system containing a phase mask to generate an inverse quadratic phase. The size of OAM beams can be easily and precisely controlled by designing the phase mask profile without changing the OAM. This technique of space diffraction compensation can be applied to OAM demultiplexing, ring fiber coupling for OAM beams and optical manipulation of micro particles.

## 2. Principle of operation

In order to explain space diffraction compensation of OAM beams, Fig. 1 illustrates the time-space duality of dispersion compensation. The dispersion of single-mode fiber (SMF) in frequency domain can be approximately treated as a quadratic phase distribution, expressed by $\text{exp}\left(j\kappa {\omega}^{2}\right)$, where $\omega $ is the angular frequency and $j=\sqrt{-1.}\kappa $ is a constant relative to dispersion parameter. The quadratic phase term will cause pulse broadening during propagation [13]. To compensate the accumulation of harmful dispersion, the same amount of reverse dispersion supplied by a dispersion compensating fiber (DCF) is used [13]. And the output temporal pulse becomes as narrow as its initial state. The basic principle for use of DCF is shown in Fig. 1(a).

The principle of space diffraction compensation is quite similar to that of dispersion compensation, as shown in Fig. 1(b). With the Fresnel approximation of the scalar diffraction theory [14], the transfer function for Fresnel diffraction can be expressed as

where*k*is the wavenumber,$\lambda $ is the wavelength in free space,

*z*is the propagation distance and is the $(\mu ,\nu )$ coordinate of spatial frequencies. Equation (1) indicates Fresnel diffraction induces a negative quadratic phase to the angular spectral function of input beams, corresponding to a negative diffractive effect. We can add a positive quadratic phase by using a 4

*f*Fourier lens system containing a phase mask before transmission. Define that

*U*,

_{in1}*U*are the optical field at the front (left) focal plane of Lens 1, Lens 2 respectively,

_{in2}*U*,

_{out1}*U*are the optical field in the back (right) focal plane of Lens 1, Lens 2 respectively and

_{out2}*U*is the output optical field after propagation a distance of

_{out}*z.*Assume that the focal length of two Fourier Lenses, whose function is used to implement the spatial Fourier transform, are $f$, the transfer function of phase mask is $M(x,y)$, where $(x,y)$ are two-dimensional rectangular coordinate. We define that$F[\cdot ]$ denotes the spatial Fourier transform operation only on the transverse coordinates $(x,y)$ and subscript

*id*=

*in1, in2, out1, out2, out.*With the Fourier optics method [14], it is derived that

We input an OAM beam at the front focal plane of Lens 1 and then tune the quadratic phase via the phase mask to control the beam size. For simplicity, we assume the input beam is an ordinary Gaussian vortex beam, expressed as [15]

We set the wavelength of input beams as 1550 nm. The waist size of input beam is
*w* equal to 0.2 mm and focal lengths of two Fourier Lenses are 200 mm. The
optical vortex's radius R is defined as the distance from the maximum intensity position to
the center, as shown in the inset of Fig. 2.Figure 2 presents the dependence of optical
vortex's radius R on topological charge (TC = *l*) and RCQ (|.|denotes the
absolute value operation). It indicates that the optical vortex's radius is not relative to
the sign of TC and the sign of RCQ. We also see that the optical vortex’s radius
increases with the TC, consistent with previous Refs [3,
4]. Nevertheless, the optical vortex’s radius also
increases with the absolute value of RCQ. So we can keep different OAM beams at a same optical
vortex’s radius by tuning the RCQ of space diffraction. For example, if one expects to
keep the vortex’s radius at 2 mm, the RCQ should be 1.16 m, 0.58 m, 0.41 m for OAM beams
whose TC is 1, 4, and 8 respectively. Figure 3 simulates
the normalized intensity and the phase of OAM beams on different focal planes. The optical vortex radii of output fields are all controlled at 2 mm via using space
diffraction compensation. The pattern sizes of *U _{in1}* and

*U*are 2 mm$\times $2 mm and 8 mm$\times $8 mm respectively. The pattern sizes of phase mask and

_{out}*U*are both 3 mm$\times $3 mm.

_{out1}One can see the input beams (*U _{in1}*) have different size, and the phase patterns indicate the corresponding OAM modes. We add distinct quadratic phases via a phase mask in the angular spectrum (

*U*) to make the output beams (

_{out1}*U*) have a same radius of 2 mm. It is notable that the phases of

_{out}*U*are composed of a helical phase and a quadratic phase caused by the space diffraction compensation, which does not affect the OAM modes. In addition, the size of phase masks in Fig. 3 are 3 mm × 3 mm. The pixel sizes of commercially available spatial light modulators (SLMs) are typically as large as 1.6–32 μm [16, 17] and the phase plates [18] are low-cost with a pixel sizes of 5.2 μm or smaller and can be easily integrated in different optical devices. They are both precise enough to generate these phase patterns. Furthermore, if the RCQ has to be designed as a large value in some cases, we can add partly the propagation distance with an opposite sign so as to obtain a smaller ACQ (ACQ = RCQ +

_{out}*z*).

## 3. Applications

#### 3.1. OAM demultiplexing

The first potential application is OAM demultiplexing, where all OAM modes are expected to have
similar mode sizes for high efficient detection. Taking the digital coherent demultiplexing as
example, Ref [12]. has employed a circular array of
coherent receivers to map the entire optical field into the digital domain. The effective area
of receiver is a narrow annulus in which the coherent receivers are placed. For example, the
maximum receiving energy is about 33% in coherent detection, because there are different
optical vortex's radii shown as *U _{in1}* in Fig. 3, when the beams contain three modes of TC = 1, 4, 8 respectively. But
if we employ a space diffraction compensation to the input OAM beams, the maximum receiving
energy will be greatly improved. The scheme is shown in Fig.
4(a).The input beams contain several OAM modes (such as TC = 1, 4, 8) and every mode has a
distinct radius. We add different RCQs to the corresponding OAM modes to keep all the radii
equivalent. The inset shows the assigned intervals of RCQs in which the boundaries are the
intersections of their intensity and the intervals inside the boundaries can cover most of the
intensity. When the middle interval is very narrow, this assignment is still accurate because
the TCs are adjacent in this case and the RCQs are approximately equal. Figure 4(b) shows the simulated results where the parameters are the same
with the ones in Fig. 3 and the radii of all OAM modes
equal 2 mm after adding RCQs. One can see the original beams are distributed in a large area
and the beams after space diffraction compensation have 72% energy limited in a narrow annulus
(2 ± 0.5 mm). In addition, the latticed distributions of intensity are caused by the
interference of these OAM modes.

#### 3.2. Ring fiber coupling for OAM beams

Secondly, the ring fiber coupling for OAM beams is discussed. The structure of ring fiber and the
definitions of HE_{m, 1} and EH_{m, 1} are the same with ones in [9]. The inner radius is 4 μm and the outer radius is 5
μm shown as Fig. 5(a).The refractive index of ring region is 1.494 at 1550 nm and the refractive index of
cladding region is 1.444. Figure 5(b) shows the
intensity and phase distribution of the low-order OAM modes of the fiber, which are from the
proper combination of the even and odd HE_{m, 1} (m = 1~4) modes. In fact, there are
another group of OAM modes which are the combination of the even and odd EH_{m, 1}
modes. But for simplicity, we just discuss the coupling of the former group. Assume the initial
beams expressed by Eq. (4) is strong focused with
*w* = 2 μm, so the radius equals $w\sqrt{\left|TC\right|/2}$. Figure 5(c) shows the
dependence of optical vortex's radius R on TC and RCQ which is similar to the one in Fig. 2 except the magnitude of beam radius. We define the
coupling efficiency as

#### 3.3. Optical manipulation

For traditional optical manipulation, the automatic control of the optical vortex’s radius is very complex and costly. Furthermore, the optical vortex’s radius increases with the TC, but the TC characterizes the optical torque [10, 11]. Figure 5(c) illustrates the beam size can be tuned without changing the optical torque when the beam radius is about several micrometer. This kind of beams are still strongly quasi-focused to manipulate micro particles. So the particles can be manipulated in radial direction through engineering the RCQ into the phase mask (SLMs). If we directly generate the helical phase via the same phase mask, the azimuthal direction is also tunable. So this technique shows great potential to offer a flexible and programmable way to control the micro particles both in radial and azimuthal direction.

## 4. Conclusion

We put forward a technique to manipulate beam size of certain OAM based on space diffraction compensation. We employ a 4*f* Fourier lens system containing a phase mask to generate an inverse spatial quadratic phase to compensate the space diffraction during free-space propagation. The OAM beams size can be controlled by designing the phase mask and the phase patterns can be generated by commercial SLMs. This technique of space diffraction compensation can be used to improve the receiving efficiency of OAM demultiplexing and ring fiber coupling for OAM beams, this technique also has potential to offer a flexible and programmable way to control the micro particles both in radial and azimuthal direction.

## Acknowledgments

This work was partially supported by the Program for New Century Excellent Talents in Ministry of Education of China (Grant No. NCET-11-0168), a Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 201139), the National Natural Science Foundation of China (Grant No. 11174096)

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