Orbital angular momentum (OAM) of laser beams has potential application in free space optical communication, but it is sensitive against pointing instabilities of the beam, i.e. shift (lateral displacement) and tilt (deflection of the beam). This work proposes a method to correct the distorted OAM spectrum by using the mean square value of the orbital angular momentum as an indicator. Qualitative analysis is given, and the numerical simulation is carried out for demonstration. The results show that the mean square value can be used to determine the beam axis of the superimposed helical beams. The initial OAM spectrum can be recovered.
© 2008 Optical Society of America
The OAM is related to the rotation of the phase of the electric field distribution. It can be measured as lħ per photon for a beam with phase term of exp(ilφ), where l is the azimuthal number . In the past decade, the OAM of a beam was paid great attention in many applications, such as optical spanner , high density data transmission , quantum information processing , and etc. The OAM spectrum is an important parameter to characterize the beam. It was proposed firstly by Molina-Terriza  and discussed in detail by Torner et al . It is also called the spiral spectrum. The OAM spectrum is the power weighted average of the OAM eigen-states, i.e. the spiral harmonics. Since the OAM spectra of the input and output beam are changed by interaction, it finds use in object characteristics detection 6 and data storage .
In free space optical communication, misalignment, i.e. shift and tilt of the beam occur. The misalignment changes the referent frame, and the description of the beam structure will be influenced. When we expand the beam on the spiral harmonics, i.e. OAM modes, the changed referent frame will make the distribution of the power of these modes (OAM spectrum) different [3, 8]. This effect can be called the mode dispersion. In the free space optical communication, the mode dispersion is generally caused by the misalignment and the changed field distribution. Frank-Arnold et al derived an uncertainty relationship between the slice-aperture and the OAM spectrum . Their work shows the worse mode dispersion occurs with less azimuthal angle of the slice-aperture. Vasnetsov et al gave the analytical expression of the misalignment when the incident beam is the Lgauerre-Gaussian beam with radial number zero . The numerical results show the OAM spectrum disperses more seriously with worse misalignment. Gibson et al indicated that the uncertainty relationship and the misalignment increase the security of OAM in free space communication . To overcome the problem induced by the slice-aperture, a telescope with a sufficiently large aperture is needed. But the misalignment is more sophisticated, and to our knowledge no method is presented to correct this error till now.
Moreover a method is required to enhance the signal to noise ratio at the receiver. This method should deliver the beam axis of arbitrarily superimposed helical beams and align the receiver axis with the beam axis to eliminate OAM dispersion. The usual method is to measure the first intensity moments of the beam (center of gravity) . But for the beam in the OAM states, the intensity does not contain the complete information, the phase structure should be taken into account. As shown by Zambrini et al, the mean square value of the OAM can be used to determine the displacement of the beam axis in lateral direction .
In this work, we deduce in section 2 the OAM dispersion in the case of misalignment. Then we use the new variance proposed by Zambrini et al  as an indicator for the position and direction of beam axis, and give the qualitative analysis for shift and tilt in Section 3. To demonstrate this method the numerical simulation is given in Section 4. Finally in Section 5 a scheme to correct the OAM dispersion caused by misalignment is discussed.
2. OAM spectrum and misalignment
The OAM spectrum is the relative power of the spiral harmonics in the laser beam , and it is also called spiral spectrum . Following the work of Zambrini et al , we define a function of the azimuthal angle relative to the reference center as the ensemble average of the rotation operator
where |ψ〉 denotes the normalized complex field of the beam, and l̂z=x̂p̂y-ŷp̂x the angular momentum operator with x̂ and ŷ the position operators and p̂x and p̂y the momentum operators in the transverse plane. The OAM spectrum is the Fourier transform of M(θ) and reads
The Laguerre-Gaussian modes |p,l〉 with radial index p≥0 and azimuthal index l are complete and orthogonal
and the eigen-equation of the angular momentum operator is
Equation (1) can be rewritten as
where |Φ〉 is the wave function in the calibrated system, and T⃗ an operator representing the misalignment. Figure 1 shows two kinds of misalignment, shift and tilt. The arrow on the operator denotes the action on the function on the right hand. For example, the operator for shift reads
where r 0 is the distance between the two axes, η the azimuthal angle of the shifted axis and the superscript LD means shift (lateral displacement). The operator for tilt is
with γ the direction deflection and DD means tilt (direction deflection).
In Eq. (10) 〈p 0 l 0|Φ〉 is the expansion coefficient of the normalized laser beam on the Laguerre-Gaussian beams and 〈pl|T⃗|p 0 l 0〉 can be interpreted as the probability amplitude for the transition |p 0 l 0〉to|pl〉 caused by the misalignment operation T⃗. The matrix element is similar to the transition matrix element in atomic physics. In this work we call this matrix the transition matrix.
The operator for lateral displacement is given by Eq. (8). The calculation of the transition matrix element is difficult in the spatial domain, but much easier in the frequency domain. The transition matrix element in the integral representation reads
F(*) and F -1(*)are the Fourier transform and its inverse. The symbol ⊗ denotes the convolution.
The Laguerre-Gauss functions are the eigenfunctions of the Fourier transform, therefore holds
where the subscripts w 0 and 2/w 0 are the waist radii of the Laguerre-Gaussian beams in the spatial domain and frequency domain, respectively. The spatial shift in the space domainwill cause a phase term in the frequency domain
with x=rcosφ and y=rsinφ. In the Dirac notation Eq.(15) reads
With the relation
and the δ-function
the transition matrix element becomes
The mode dispersion caused by lateral displacement is governed by the Bessel function of the first kind. The dispersion will increase with larger lateral displacement r 0.
The operator for shift is given by Eq. (9). The calculation is similar to that for tilt and delivers
Again the mode dispersion depends on the Bessel function. The OAM spectrum defined by Eqs. (10), (19) and (20) is the general case of the relation derived in Ref. , which holds only for a single initial Laguerre-Gaussian beam with zero radial number. Using Eqs. (10), (18) and (19) we can - obtain the orbital momentum spectrum for any superposition of initial Laguerre-Gaussian beams and with any misalignment.
In general the resulting misalignment operator reads
Finally we obtain for the transition element
Then the calculation of the OAM is simpler and more suitable for computer processing.
From Eqs. (10), (19) and (20), we can find that misalignment will cause serious mode dispersion, the signal to noise ratio is lower, and larger errors will occur. In the following Sections we will develop a method to measure and to correct the misalignment.
3. Indicator for beam axis determination and interpretation
To correct the misalignment, we should firstly determine the beam axis position and the beam axis direction. For the case of shift, the beam center has to be defined. The usual method is based on the first intensity moments, which defines the center of gravity. But this is inapplicable here, because we are interested in the angular momentum, which is a phase structure. To demonstrate the difference, an example is given in Fig. 2. The intensity distribution of the beam is a coherent superposition of two Laguerre-Gaussian beams with radial number and azimuthal number (p, l)=(0,0) and (0,1), respectively The dark cross and the white cross denote the beam axis defined by the intensity momentum and the real axis, respectively. It’s clear that the two axes are not collinear. Here the real axis denotes the axis of the pre-set helical beams which are superposed collinearly.
The OAM is both intrinsic and extrinsic , also called quasi-intrinsic , which means that the OAM of a wave front with helical structure is definite and independent on the position of reference axis. Zambrini et al propose a quasi-intrinsic-variance 
This variance is the mean-square value of the OAM and describes the degree of mode dispersion. Although the OAM of a helical beam is unchanged when the axis is shifted, the V value varies with the position of axis. We will use the variance to find the beam center. For the single Laguerre-Gaussian beam well aligned, V is zero, but with misalignment V reads 
where k is the wave number, zR the Rayleigh range, p 0 the radial number of the Laguerre-Gaussian beam, and l0 the azimuthal number. Eq. (23) demonstrates that the variance V is an indicator of the relevance for the choice of the rotation axis position within the spatial beam distribution, even when the average OAM is unchanged.
For the superimposed Laguerre-Gaussian beams, the variance V is not zero because of the non-helical wave front. The OAM spectrum at the receiver is dispersed from the incident superimposed Laguerre-Gaussian beam through Eq. (10). When the axis of the receiver is not collinear with the incident beam, the dispersion of the OAM spectrum becomes more serious with increasing lateral displacement, and the V value larger. Only when the two axes are collinear the variance V will reach its minimum. Then we can define the position with the minimum variance V value as the center of the beam. This method also can be interpreted using the information entropy. It is defined unit bits as
When the incident beam is decided, the OAM spectrum is also decided. If we take the spectrum as the probability of the emergence of the OAM states, the information entropy of the incident beam is ascertained. Misalignment adds additional information to the original information and increases the information entropy. If we change the alignment to the correct state, the information entropy will reach the minimum only if not any additional information is included, i.e. the system should be well aligned.
As a result of Section 2, we know that the mode dispersion by tilt is similar to that od shift. Since the variance V can be used to determine the beam axis position, we also can expect it can be used to determine the beam axis direction.
4. Numerical simulation and discussions
In the numerical simulation, we take only some Laguerre-Gaussian beams into account. These Laguerre-Gaussian beams have radial number p ranging from 0 to 8, and azimuthal number l from -10 to +10 for sake of simplicity To reduce the error resulting from truncating the number of mode, the initial beam is assumed as a the superposition of Laguerre-Gaussian beams with radial number 0, 1 and 2, and radial number from -2 to +2. The coefficients have random normalized amplitude and random phase. The random coefficients are introduced for the calculation of the general case.
Table 1 shows the expansion coefficients of the incident normalized beam. These coefficients are complex values, chosen randomly. Fig. 3 shows the OAM spectrum of the incident beam. The OAM spectrum in Fig. 3 has a meaning in two aspects, one for the probability of the photon in a spiral mode state and other for the weights of the energy in all the spiral modes simultaneously. For such a beam, the field distribution is not smooth but irregular. For example, the amplitude distribution is not a circular spot (for l=0) or bright circle (for l≠0), as shown in Fig. 4. It is obvious that the amplitude distribution is not symmetric and the phase structure not helical. There are more than one singularity and they are not centered. Then the beam axis will be determined. As discussed in Section 3, the beam center will not have the minimum V value, but the OAM spectrum is the same as the initial one shown in Fig. 3. The following calculation will point out that the beam axis defined by the minimum V value is indeed the real axis.
To demonstrate the relation between the V value and the shift, Fig. 5 shows a contour plot (a), a mesh plot (b) and plots along the x, y-directions through the center (c). Here x and y are scaled by w 0, the beam radius of the fundamental Laguerre-Gaussian beam. From Fig. 4(a) and (b) we can find out that the V value changes with the shift in the transverse plane. All contour lines enclose the beam center, which indicates no other extremes exist, except the central value. The mesh plot shows a parabolic -like surface without ripples along the radial direction, which indicates that the V value increases with the distance between the beam center and the axis of the receiver system. These two point are consistent with the qualitatively analysis. The grads of V value in special cases along the x, y-directionsare plotted in Fig. 5(c). The lines are conic and reach the minimum at the position (x, y)=(0, 0), just the beam axis used of the incident beam. This result means that we can find out the position with the minimum V value and this position is just the axis position of the incident beam. The beam center defined by the intensity moment is about (x, y)=(-0.02, 0.07), which is not the real incident beam center in free space optical communication using OAM. And at this center the received OAM spectrum is dispersed.
Figure 6 shows the information entropy I varying with the lateral displacement, (a) contour plot, (b) mesh plot, and (c) x, y-directions. The variation od the information entropy I is similar to that of the mean square value of the OAM, as we expect in Section 3. The information entropy will find its minimum at the position (x, y)=(0, 0), i.e. at the axis of the incident beam. But the information entropy is not so excellent for the indicator of the beam axis than the mean square value. Comparing Fig. 5(c) and Fig. 6(c) we find the mean square value changes more sharply with the x and y coordinates than the information entropy. This is because the information entropy does not care for the azimuthal number, which will make the mean square value even larger with increasing dispersion. In case of more azimuthal numbers used for optical communication, the effect of the mean square value is more obvious.
For the tilt, because of the similarity of Eq. (8) and (9), we can also use the V value to indicate the direction of incident beam’s axis. And it is easy to understand that in this case the V value will reach the minimum, when the axis of the receiver system is collinear with the incident beam. For the complicated case that both misalignments occur together, we can make the analysis in two steps, first shift and then tilt or vice versa. In any case the V value can reach the minimum only when the axis of the receiver system is collinear with the incident beam. This conclusion can be used to correct the OAM spectrum in free space optical communication. In the next Section we will give an algorithm and design a schematic setup for OAM spectrum correction.
5. Schematic design of an OAM spectrum correction
Figure 7 shows the schematic routine for an OAM spectrum correction. The receiver system is separated into two parts, the self adaptive adjustment unit and the information processing unit. The self adaptive adjustment unit is the key element of this setup. The optical system is mounted on a mechanical system, which can provide four dimensional adjustments, two for the shift (x, y) and two for the tilt (u, v). At first we adjust the system to a place with rough calibration, set the two bits binary number C, which is used for controlling the dimension to optimize, set S which is used for deciding the direction to optimize, and set V 1 and V 2 as the initial V value to make the system work automatically. Then the system performs the mechanical adjustment to a new state, measures the OAM spectrum and calculates the V value. There are only two states for V, V≠V 1 and V=V 1. The first case denotes the optimization in one dimension and is a first approach. The second one means that the V value reaches its minimum in this dimension but it does not indicate the minimum in all the four dimensions, as shown in Fig. 7(a). When V≠V 1, the loop will go on for the same parameter till V=V 1. Then the second decision will be made whether V≠V 1. If not, it means that the optimization of the second dimension is not complete, and the loop will go on for till V=V 1=V 2. Then the first misalignment is probably corrected and the procedure jumps to the other misalignment. In our consideration, the optimization is carried out for each parameters of the same misalignment repetitively, and then for the two misalignments in the same manner. Only when the minimum V value is known, we can perform the correction of the misalignment till the minimum V value are obtained. The geometry of these dimensions is shown in Fig. 7(b).
To control the direction of optimization, we use a parameter S to control the adding or subtracting of an increment. Take as an examplethe optimizing of x. When S=‘+’ and the V value at x+Δx is larger than that at x, it mean the direction with increasing x is wrong, and we should change S to ‘-’. Generally S should be changed to S×sign (V 1-V) where sign(*) is the sign function. This mechanism proposes negative feedback to minimize the V value minimum.
To control the jump between the different dimensions of the same misalignment or between the different misalignments, the two binary number C is useful because of inversing the first bit for different misalignment and the second one for different dimension of the same misalignment. C is also used for selecting the dimension to optimize, i.e. 00 for u, 01 for v, 10 for x and 11 for y.
In the perfect calibration, the information processing unit starts working. When processing the information, the self adaptive adjustment unit is always working on to hold the state of perfect calibration. . The most difficult unit is the OAM spectrum measurement, which requires high precision. Only the cascaded Mach-Zehnder interferometers can achieve this goal  today. Although the rotational Doppler Effect [14, 15] can also be used the extremely short wavelength of the laser beam make it very difficult. Up to now the rotational Doppler Effect was used in the mm wave range .
This work proposes the mean square value of the OAM spectrum as an indicator to realize the misalignment correction. This method gives a different insight on the beam center and axis direction which differs from the well known ISO definition, using first intensity moments. A mathematical analysis and numerical calculation show the validity of this method. Finally a schematic setup and an algorithm are proposed for OAM spectrum correction. But we should admit that this work is insufficient because the monotone dependence of the V value on the shift r 0 and tilt angle γ is merely qualitatively and numerically validated without a rigid mathematical deduction.
This work is supported by the National Science Foundation of China (Grant No. 60778002), the Doctoral Fund of Ministry of Education of China (Grant No. 20050007027), and the Program for New Century Excellent Talents in University (NCET).
References and links
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992). [CrossRef] [PubMed]
2. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular Momentum to Absorptive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]
3. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]
5. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. 88, 013601/1–4 (2002). [CrossRef] [PubMed]
7. Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, “Realizing high density optical data storage by using orbital angular momentum of light beam,” Acta Phys. Sin. 56, 854–858 (2007) (in Chinese).
8. M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, “Analysis of orbital angular momentum of a misaligned optical beam,” New J. Phys. 7, 46/1–17 (2005). [CrossRef]
9. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103/1–7 (2004). [CrossRef]
10. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992). [CrossRef]
12. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. 88, 053601/1–4 (2002). [CrossRef] [PubMed]
14. G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996). [CrossRef]
15. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational Frequency Shift of a Light Beam,” Phys. Rev. Lett. 81, 4828–4830 (1998). [CrossRef]