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КОРОТКИЕ СООБЩЕНИЯ
Orbital angular momentum of the spiral beams
V.G. Volostnikov1
1Samara Branch of P.N. Lebedev Physical Institute of RAN, Samara, Russia
Abstract
At first sight, any rotation generates some angular momentum (it is true for a solid body). But these characteristics (rotation and orbital angular momentum) are rather different for optics and mechanics. In optics there are the situation when the rotation is important. On the other hand, there are the cases where the nonzero orbital angular momentum is necessary. The main goal of this article is to investigate a relationship between a rotation under propagation of spiral beam and its angular momentum.
It can be done the following conclusion: there is no any relation between rotation under propagation of spiral beam and its OAM.
Keywords: spiral beams, singular optics, orbital angular momentum.
Citation: Volostnikov VG. Orbital angular momentum of the spiral beams. Computer Optics 2019; 43(3): 504-506. DOI: 10.18287/2412-6179-2019-43-3-504-506.
Acknowlegdements: This research is financially supported by the Russian Foundation for Basic Research (project No. 17-42-630934).
Introduction
Spiral beams are a family of the lightfields whose intensity distribution remains invariant, up to scale and rotation, during propagation [1, 2]. It is seen that history of the spiral beams is sufficiently extensive. At first sight any rotation generates some angular momentum (it is true for a solid body) but it is not so for the spiral beams. These characteristics (rotation and orbital angular momentum) are rather different. There are the situation when the rotation is important [3, 4]. On the other hand, there are the cases where the nonzero orbital angular momentum is necessary [5]. The relationship between the rotation and orbital angular momentum (OAM) in some cases have considered partly. Therefore, in [1] the examples of the light fields with opposite rotation and equal OAM takes place. In [5] the example of light field with nonzero rotation but zero OAM has been presented. But there is no the general consideration of this question. In this article, there is the investigation of relationship between a rotation under propagation of spiral beam and its angular momentum.
1. Orbital angular momentum
Spiral beams are a family of the lightfields whose intensity distribution remains invariant, up to scale and rotation, during propagation, namely:
I(x,y,l) = D(l)I0 X f x cos 8 (l) - y sin 8 (l) x sin 8 (l) + y cos 8 (l) ^ (1)
d (l)
d (l)
where 8(l) - rotation parameter (see below), d (l) - scale parameter.
In accordance with [1] the density of OAM, M of light field is the following:
M = Im(F xFa),
(2)
where F is complex amplitude and a - polar angle.
The mode index condition for the spiral beams with Gaussian parameter p is [1, 2]:
2n + |m| + 80 m = const, (3)
where 80 - is so called constant of rotation, 2J
kp
8 (l) = 80arctg| —
(4)
l - variable of propagation, k - wavenumber; n, m is number of Laguerre-Gauss modes.
Imaginary part of real quantity is equal zero:
Im\j-FF | = Im(FFa+ FaF)
(5)
hence two complex conjugate light fields have opposite OAM:
Im\ Aff| = Im(FFa+ FaF) = 0.
,oa
(6)
Here a is polar angle.
Analytic formulae for the orbital angular momentum is by definition:
L = jjMdxdy / E, (7)
R2
where
E= jj FFdxdy -
R2
the energy. Any light field F with finite energy can be represented as decomposition of normalized Laguerre-Gauss modes LGnm, because it is hole system:
F = X c„mLG„m dxdy (8)
n,m
and OAM is the following:
L = z|c»,»f m / sic»,m f. (9)
n,m n, m
2. OAM and rotation of the spiral beams
To investigate the question let us consider the following compositions:
L F = cLG0 0 + c2LG0,n ;
2. F2 = ciLGn,0 + c2LG0,n ;
3. F3 = ci LG2n+1, - n + C2LGn,n , n * 0;
4. F4 = ciLGn,0 + c2LG0,2n .
(10)
In Fig. 1-4 the distributions of intensities and phase of these compositions are shown. It is convenient because any sum of two Laguerre-Gauss modes is some spiral beam with constant of rotation:
2n, -2n* + \m\-ImJ
2 1 111 21. (ii)
00 =
m* - m
Two first compositions have opposite rotation parameters but equal OAM:
00 =-02 =-1? L1 = L2 = ni\ = ln2
(12)
The third composition is rotated without OAM:
d,n * 0. (13)
û n +1 T n I
00 ,L = 0, n
In [6] the example of lightfield with nonzero rotation but zero OAM has been presented also.
And fourth composition is not rotated of all but has some OAM:
00 = 0, L = n, nJ = n2
(14)
It is seen from these examples the following: in the first place, there is no any relation between rotation of spiral beam and its OAM, secondly, it is difficult to reveal OAM directly from these distributions, because OAM is integral characteristic of lightfields. There is no any optical method of detecting OAM at present.
From formal point of view, it is clear: rotation parameter is defined only indexes of mode whereas OAM depends on weight coefficients too.
To do this more physical clearly let us consider two modes: real Laguerre-Gauss mode and complex one with timedependent components:
F = Re LG0i x exp(-/wt),
F2 = Re(LGoi x exp(-/wt)).
F = Re(r cos(a)exp(-r2) x
x exp(-/wt)) = (15)
= r cos a cos(-wt) exp(-r2),
F2 = Re(r exp ia x exp(-/wt)exp(-r2)) = = r cos(a -rot )exp(-r2).
It is seen that second distribution is rotated. Namely, this rotation defines OAM.
Of course, this rotation is not the rotation under propagation.
4
-2 0 2 4 (b) -4 -2 0 2 4
Fig. 1. The distributions oof intensity (a) and phase (b) for the first composition (at n=1)
-4 -41
(a) -4 -2 0 2 4 (b) -4 -2 0 2 4
Fig.2. The distributions oof intensity (a) and phase (b)
for the second composition (at n=1) 4 4
-4 mmmmmmmm _4\
(a) -4 -2 0 2 4 (b) -4 -2 0 2 4
Fig.3. The distributions oof intensity (a) and phase (b) for the third composition (at n=2)
-2 0 2 4 (b) -4 -2 0 2 4
Fig.4. The distributions oof intensity (a) and phase (b) for the fourth composition (at n=2)
Conclusion
The main purpose of the article is to consider the question: is it some relation between the rotation under propagation of spiral beam and its OAM. We consider four type of spiral beams. The two types of beams have opposite rotation parameters but equal OAM. The third typerotated without OAM. The fourth type of spiral beam is not rotated of all but has some OAM. From these examples it can be done the following conclusion. There is no any relation between rotation under propagation of spiral beam and its OAM. It should be noted [7], where the possibility of OAM measurement has been considered but only for the special case of light-
fields in the form of superposition of Laguerre-Gauss
modes with zero lower index.
References
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[7] Volyar AV, Bretsko MV, Akimova YaE, Egorov YuA. Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams. Computer Optics 2018; 42(5): 736-743. DOI: 10.18287/2412-6179-2017-42-5-736-743.
Author's information
Vladimir Gennadievich Volostnikov (b. 1951), Phys.-math. Doctor, professor, chief researcher in Samara Branch of Lebedev Physical Institute of RAS. The area of scientific interest: phase retrieval in optics, Gaussian beam optics, singular optics, optical vortices.
Code of State Categories Scientific and Technical Information (in Russian - GRNTI)): 29.31.15 Received - November 27, 2018. The final version - June 13, 2019.
