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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 2 — Feb. 1, 2014
  • pp: 274–282
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Hermite–Gaussian modal laser beams with orbital angular momentum

V. V. Kotlyar and A. A. Kovalev  »View Author Affiliations


JOSA A, Vol. 31, Issue 2, pp. 274-282 (2014)
http://dx.doi.org/10.1364/JOSAA.31.000274


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Abstract

A relationship for the complex amplitude of generalized paraxial Hermite–Gaussian (HG) beams is deduced. We show that under certain parameters, these beams transform into the familiar HG modes and elegant HG beams. The orbital angular momentum (OAM) of a linear combination of two generalized HG beams with a phase shift of π/2, with their double indices composed of adjacent integer numbers taken in direct and inverse order, is calculated. The modulus of the OAM is shown to be an integer number for the combination of two HG modes, always equal to unity for the superposition of two elegant HG beams, and a fractional number for two hybrid HG beams. Interestingly, a linear combination of two such HG modes also presents a mode that is characterized by a nonzero OAM and the lack of radial symmetry but does not rotate during propagation.

© 2014 Optical Society of America

1. INTRODUCTION

In 1992 it was shown by Allen et al. [1

1. L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992). [CrossRef]

] that the Laguerre–Gaussian (LG) modes have an orbital angular momentum (OAM). The OAM is also characteristic of all optical vortices or singular laser beams with a phase singularity or wavefront dislocation [2

2. M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979). [CrossRef]

]. The power flux (Poynting vector) of such beams rotates in a spiral about phase singularity points. For the first time, a laser beam with phase singularity was generated in 1979 [3

3. J. M. Vaughan and D. Willetts, “Interference properties of a light-beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979). [CrossRef]

]. The interference of two Hermite–Gaussian (HG) modes, denoted as HG01 and HG10, in a chromium ion laser’s cavity was shown to produce a mode LG01. In 1989, the term “optical vortex” was proposed [4

4. P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989). [CrossRef]

]. In 1990, an optical vortex was experimentally generated by Bazhenov et al. [5

5. V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).

] using an amplitude diffraction grating with a fork. In 1992, a singular laser beam was generated using a spiral phase plate [6

6. S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992). [CrossRef]

].

In 1991, the conversion of a higher-order HG mode into a LG mode with phase singularity using an astigmatic mode converter was reported by Abramochkin and Volostnikov [7

7. E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]

]. An interference laser mode π/2-converter was proposed in Ref. [8

8. M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

]. The laser beams with the OAM have found use in particle micromanipulation, quantum telecommunications, microscopy, interferometry, metrology, and so on. A most recent review of the OAM phenomenon can be found in Ref. [9

9. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). [CrossRef]

]. A variety of application areas for the vortex laser beams with OAM were also described in a monograph [10

10. V. V. Kotlyar and A. A. Kovalev, Vortex Laser Beams (Novaya Tekhnika, 2012) [in Russian].

]. The generation of the optical vortices by means of interferometers was dealt with in Refs. [11

11. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (2007). [CrossRef]

13

13. B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012). [CrossRef]

], in which an optical vortex array was generated through the interference of three plane waves. Another interferometric way to generate an optical vortex was considered in the very recent Ref. [14

14. P. Vaity, A. Aadhi, and R. Singh, “Formation of optical vortices through superposition of two Gaussian beams,” Appl. Opt. 52, 6652–6656 (2013). [CrossRef]

], in which a Mach–Zehnder interferometer has been used to obtain a superposition of two Gaussian beams with a tilt between these beams and with different beam axes. It is worth noting that only vortices with a unitary topological charge can be generated in this way.

In Ref. [7

7. E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]

], a relationship enabling a LG mode to be represented as a finite superposition of HG modes was derived. By way of illustration, to obtain a LG mode with topological charge 2, one needs to employ the superposition of at least three HG modes with definite complex coefficients. In this work, we demonstrate that it is possible to obtain a light field with an arbitrary integer OAM through the interference of just two HG modes with definite indices.

It is noteworthy that recently an optical laser vortex with the largest OAM and the largest topological charge of 5050 was generated with the aid of an aluminum reflection diffractive optical element [15

15. Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013). [CrossRef]

]. There are different methods for generating laser beams with fractional OAM [16

16. J. B. Gotte, K. O’Holleran, D. Precce, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008). [CrossRef]

18

18. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. C. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010). [CrossRef]

]. In [16

16. J. B. Gotte, K. O’Holleran, D. Precce, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008). [CrossRef]

], laser beams with half-integer OAM were formed as a linear combination of the LG modes. In [17

17. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004). [CrossRef]

], Hermite–Laguerre–Gauss beams with a fractional OAM were generated using an astigmatic mode converter. In [18

18. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. C. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010). [CrossRef]

], a Bessel beam with a smaller-than-unity OAM was generated based on the conical diffraction of a circularly polarized Gaussian beam.

In this work, we calculate the OAM for a linear combination of two HG beams whose double indices are composed of adjacent integer numbers taken in direct and inverse order with a phase shift of π/2 between the beams. We analyze generalized HG beams, which change to HG modes and elegant HG beams under certain parameters. It is shown that the modulus of the OAM for two corresponding HG modes is an integer number, whereas the modulus of the OAM for two elegant HG beams is always equal to unity. In the superposition of two corresponding hybrid HG beams, the modulus of the OAM is always a fractional number. A trivial case of the superposition of the two generalized HG modes (0,1) and (1,0) is an exception. In this case, the modulus of the OAM is equal to unity, similar to the LG modes.

2. GENERALIZED HG LASER BEAMS

The HG modes have been known in optics for quite a while [19

19. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966). [CrossRef]

]. The elegant HG beams, which are described by functions of complex argument, were for the first time proposed in 1973 by Siegman [20

20. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical beam eigenfunction,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef]

].

However, there are HG beams that also represent a solution of the paraxial equation of propagation and can be expressed in an explicit analytical form. Under definite parameters, such beams change to HG modes [19

19. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966). [CrossRef]

] and elegant HG beams [20

20. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical beam eigenfunction,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef]

].

Assume that the complex amplitude of light in the initial plane z=0 is given by Enm(x,y)=E1(x)E2(y), where E1(x)=exp[(x/a)2]Hn(x/b) and E2(y)=exp[(y/c)2]Hm(y/d), in which a, b, c, and d are real numbers. Considering that the complex amplitude is a product of two functions dependent on the different Cartesian coordinates, the propagation of the entire 2D beam can be considered as the propagation of a 1D beam along any of the transverse coordinates. For such a 1D light field, the complex amplitude at distance z is calculated in the paraxial approximation using a Fresnel transform [21

21. F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994), pp. 140–148.

], being given by (see Appendix A)
E1(x,z)=(iz0z)1/2in×(1iz0z)(n+1)/2[(ab)21+iz0z]n/2×exp[(xa(z))2+ikx22R(z)]Hn(xb(z)),
(1)
where
z0=ka22,a(z)=a[1+(zz0)2]1/2,R(z)=z[1+(z0z)2],b(z)=b(zz0)(1iz0z)1/2[(ab)21+iz0z]1/2.
(2)

For the complex amplitude E2(y,z), relations similar to Eqs. (1) and (2) can be derived by replacing x, n, a, b with y, m, c, d, respectively.

From Eqs. (1) and (2) at a/b=c/d=2 follows a well-known relationship for the HG modes:
En(x,z)=in[aa(z)]Hn[2xa(z)]×exp[x2a2(z)+ikx22R(z)i(n+1/2)arctg(zz0)].
(3)

In the 2D case, the 1D modes of Eq. (3) are multiplied, producing a 2D HG mode Enm(x,y,z)=En(x,z)Em(y,z).

If the condition a/b=c/d=1 is observed, Eqs. (1) and (2) are reduced to an expression for the elegant HG beams [20

20. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical beam eigenfunction,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef]

]:
Ee(x,z)=(q(z))(n+1)/2×exp[(xaq(z))2]Hn(xaq(z)),
(4)
where q(z)=(1+iz/z0)1/2. Note that the generalized HG beams of Eq. (1) and the elegant HG beam of Eq. (4) are not free-space modes, with the structure of their transverse intensity distribution being changed upon propagation. It stands to reason that the 2D generalized HG beams are generated by multiplying the corresponding functions of Eqs. (1), (3), and (4). It is possible to generate a hybrid HG beam that is described by a HG mode on one coordinate and by an elegant HG beam on the other coordinate:
Eh(x,y,z=0)=exp[(xa)2(yc)2]×Hm(2xa)Hn(yc).
(5)

The HG beams of Eq. (1) are devoid of OAM. A linear combination of the HG beams with real coefficients also has a zero-valued OAM. Only the linear combination of the HG beam with complex coefficients can have a nonzero OAM. In the subsequent sections, we derive the OAM for the superposition of two generalized HG beams characterized by a phase delay of π/2.

3. ORBITAL ANGULAR MOMENTUM OF A LINEAR COMBINATION OF TWO HG MODES

Assume that the complex amplitude of light in the initial plane is given by
E(x,y,0)=exp[w22(x2+y2)]×[H2p(cx)H2s+1(cy)+iγH2s+1(cx)H2p(cy)],
(6)
where w, c, and γ are real numbers. The OAM can be derived from [22

22. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

]
Jz=Im{R2E*(xEyyEx)dxdy}.
(7)

Strictly speaking, Eq. (7) defines not the entire OAM but its optical-axis projection determined up to a constant and averaged over the transverse plane. In addition, if Eq. (7) were written in the International System of Units, it would have contained the ratio ε0/(2ω) [23

23. J. Humblet, “Sur le moment d’impulsion d’une onde electromagnetique,” Physica 10, 585–603 (1943).

], where ε0 is vacuum permittivity and ω is a cyclic frequency of monochromatic light. This ratio we have put to be equal to unity.

Because the OAM of Eq. (7) is preserved upon the beam propagation [22

22. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

], it can be calculated at an arbitrary plane, for example, at z=0. Substituting Eq. (6) into Eq. (7) yields
Jz=4γc+xexp(w2x2)H2p(cx)H2s+1(cx)dx×[2p+exp(w2y2)H2s+1(cy)H2p1(cy)dy(2s+1)+exp(w2y2)H2p(cy)H2s(cy)dy].
(8)

Considering that in Eq. (8) the integrands are in the form of polynomials, the integrals can be calculated and represented as finite sums:
Jz=4πγ[(2p)!(2s+1)!]2w4(p+s+1)×k=0min(p,s+1)[(s+1)c2kw2](c2w2)p+s2k(2c2)2k(pk)!(s+1k)!(2k)!×[k=0min(s,p1)(2c2)2k+1(c2w2)p+s2k1(p1k)!(sk)!(2k+1)!k=0min(p,s)(2c2)2k(c2w2)p+s2k(pk)!(sk)!(2k)!].
(9)

During the derivation of Eq. (9) we used the following integral (integral 2.20.16.4 from [24

24. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Gordon and Breach Science, 1986).

]):
exp(px2){H2m+1(bx)H2n+1(cx)H2m(bx)H2n(cx)}dx=(2m+δ)!(2n+δ)!πp×k=0min(m,n)(b2pp)mk(c2pp)nk(2bcp)2k+δ(mk)!(nk)!(2k+δ)!,
(10)
where Re(p)>0, δ=1 for upper parts, and δ=0 for lower parts of the expression. We also used the following integral that can be obtained by differentiation of the lower part of Eq. (10) with respect to parameter c (when n is replaced by n+1):
xexp(px2)H2m(bx)H2n+1(cx)dx=π(2m)!(2n+1)!cpm+n+3/2×k=0min(m,n+1)(n+1)c2kp(mk)!(n+1k)!(2k)!×(b2p)mk(c2p)nk(2bc)2k.
(11)

The relationship in Eq. (9) is cumbersome, making it hard to conclude when specifically the OAM will take an integer, fractional, or zero value. It allows some conclusions to be made only under certain conditions. For instance, if p>s, c=w+δ, and δw, then at (ps)(2s+1)1w>δ in Eq. (9), Jz>0. If in Eq. (9) c=w, all sums get cancelled, with only terms with a maximal number k being retained. From Eq. (6) it is seen that at c=w, two HG modes with permuted indices are superimposed. Considering that the HG modes are orthogonal, one can infer that the nonzero OAM can be obtained only for a linear combination of modes with two consecutive indices, i.e., when p=s. In this case, Eq. (9) is replaced with
Jz=24p+2πγ[(2p+1)!]2w2.
(12)

For the OAM to be independent of the laser beam power, let us analyze the OAM normalized with respect to the intensity. The power of the beam in Eq. (6) is described by the relation (c=w, p=s)
I=R2E*Edxdy=π(1+γ2)w221+4p(2p)!(2p+1)!.
(13)

Thus, the normalized OAM (the OAM divided by the beam’s power) for the linear combination of two HG modes with permuted adjacent indices is
JzI=(2γ1+γ2)(2p+1).
(14)

From Eq. (14) it follows that at γ=1, the modulus of the OAM of the linear combination of two HG modes
Em(x,y)=exp[w22(x2+y2)]×[H2p(wx)H2p+1(wy)+iH2p+1(wx)H2p(wy)]
(15)
is an integer number
JzI=(2p+1).
(16)

Note that at some values of p, the OAM will also be an integer number even when γ1. For instance, at γ=1/2, the OAM in Eq. (14) will be integer at p=2(Jz/I=4), at p=7(Jz/I=12), and so on.

For the linear combination of the HG modes given by
Em(x,y,0)=exp[w22(x2+y2)]×[Hn(wx)Hn+1(wy)+iγHn+1(wx)Hn(wy)],
(17)
the normalized OAM for any integer n can be derived in a similar way to Eq. (16):
JzI=2γ(n+1)1+γ2.
(18)

Note that because for the two modes in Eq. (17) the sum of the indices is the same [with the two modes also having the same Gouy phase, defined as (m+n+1)arctan(z/z0)], the linear combination in Eq. (17) will also form a mode, which will preserve its form upon propagation, changing only in scale. Thus, a remarkable result is arrived at: the mode in Eq. (17) has an OAM, meaning that the Poynting vector is locally describing a spiral about the optical axis in space; in the meantime, the beam is not rotating, preserving its structure upon propagation.

The beam of Eq. (6) or Eq. (17) at c=w can be experimentally generated using a Mach–Zehnder interferometer. The HG mode Enm(x,y) generated at the laser output is divided by a 50% mirror into two identical beams that are coupled into different interferometers’ arms. In one interferometer’s arm, the HG mode is rotated by 90° using a Dove prism, forming the mode Emn(x,y). At the interferometer’s output, the modes are superimposed into a single beam with a relative phase delay of π/2.

4. ORBITAL ANGULAR MOMENTUM OF A LINEAR COMBINATION OF TWO ELEGANT HG BEAMS

Let us calculate the OAM for a linear combination of two elegant HG beams. To this end, we put into Eq. (6) c=w/2 and p=s, obtaining
Ee(x,y)=exp[w22(x2+y2)]×[H2p(wx2)H2p+1(wy2)+iγH2p+1(wx2)H2p(wy2)].
(19)

Then, the normalized OAM similar to that in Eq. (14) takes the form
JzI=2γ1+γ2.
(20)

Equation (20) suggests that the linear combination of two elegant HG beams [Eq. (19)] will always have an OAM equal to unity in modulus (at γ=1) at all possible values of the number p.

This is an extremely unexpected result. It turns out that the OAM in Eq. (16) for two HG modes is determined by the maximal number of the constituent mode of the linear combination. Consequently, the larger the number of the HG mode, the larger the laser beam’s OAM in Eq. (15). For the elegant beams in Eq. (19), it follows from Eq. (20) that the OAM is determined by the difference of the two indices of the constituent beams of the linear combination. Thus, because the difference of the beam indices in Eq. (19) equals unity, hence the OAM also equals unity in modulus (at γ=1).

For the linear combination of elegant modes with different numbers (let k=2l+1 be odd)
Ee(x,y)=exp[w22(x2+y2)]×[H2p(wx2)H2p+k(wy2)+iγH2p+k(wx2)H2p(wy2)],
(21)
the OAM is obtained in the form
JzI=(γ1+γ2)k(4p+k)Γ2(2p+k/2)Γ(2p+1/2)Γ(2p+k+1/2),
(22)
where Γ(x) is the gamma function. Equation (22) is identical to Eq. (20) at k=1. For the subsequent number k=3, from Eq. (22) it follows that
JzI=(2γ1+γ2)3(4p+1)(4p+5).
(23)

At γ=1 and large p, the modulus of the OAM (23) is close to 3, i.e., close to the difference of the numbers of the elegant HG beam of Eq. (21).

5. ORBITAL ANGULAR MOMENTUM OF A LINEAR COMBINATION OF TWO HYBRID HG BEAMS

Below, by “the hybrid HG beam” we mean a beam that is described by a HG mode on one axis and by an elegant HG beam on the other axis. In this case, there are two variants of the linear combination of the hybrid beams.

Let us analyze the sum of two hybrid HG beams in which the HG mode has a larger number than the elegant HG beam:
Eh1(x,y)=exp[w22(x2+y2)]×[H2p(wx2)H2p+1(wy)+iγH2p+1(wx)H2p(wy2)].
(24)

Then the normalized OAM of the beam in Eq. (24) is given by
JzI=(2γ1+γ2)(2p+1)!(4p1)!!.
(25)

Because the numerator of Eq. (25) will always contain both even and odd terms (due to (2p+1)!), whereas the denominator will only contain odd terms (due to (4p1)!!), the OAM of Eq. (25) will never be an integer at γ=1, except in the trivial case of γ=1 and p=0. For instance, at p=2 and γ=1, from Eq. (25) it follows that Jz/I=8/7. This OAM is slightly larger than unity in modulus.

In the other case, when the HG mode has a smaller number than the elegant HG beam in the superposition of two hybrid HG beams,
Eh2(x,y)=exp[w22(x2+y2)]×[H2p(wx)H2p+1(wy2)+iγH2p+1(wx2)H2p(wy)],
(26)
Eq. (25) for the OAM is replaced with
JzI=(2γ1+γ2)(2p+1)!(2p+1)(p21)(4p+1)!!.
(27)

Note that the OAM of Eq. (27) has the opposite sign with respect to all above-calculated OAMs. From Eq. (27) it also follows that the OAM for the beam in Eq. (26) will always be a fractional number, except in the trivial case of γ=1 and p=0. For example, at p=2 and γ=1, Eq. (27) yields Jz/I=40/21. This OAM is slightly smaller than 2 in modulus. It should be noted that in the above-considered cases the change of sign of the parameter γ results in a change of sign of the OAM. Note also that the common factor in Eqs. (14), (20), (23), (25), and (26), 2γ/(1+γ2), is always smaller than or equal to unity.

6. NUMERICAL SIMULATION

For simulation, a linear combination of two generalized HG beams will be considered. Assume that in the initial plane (z=0) the light field has the complex amplitude
E(x,y,0)=exp(x2+y2w2)×[H2p(bx)H2s+1(cy)+iγH2s+1(cx)H2p(by)].
(28)

The total intensity of the beam [Eq. (28)] for wavelength λ and p=s=2 (i.e., for the Hermite polynomials of fourth and fifth degree) in the initial plane (at 7λx7λ, 7λy7λ) at some values of the scaling coefficients b and c is shown in Figs. 14.

Fig. 1. Intensity of the beam [Eq. (28)] (a) without and (b) with the carrier frequency at w=2λ, b=c=2/w (two HG modes).
Fig. 2. Intensity of the beam [Eq. (28)] (a) without and (b) with the carrier frequency at w=2λ, b=c=1/w (two elegant HG beams).
Fig. 3. Intensity of the beam [Eq. (28)] (a) without and (b) with the carrier frequency at w=2λ, b=1/(7λ), c=1/(3λ) (two generalized HG beams with different widths on the axes).
Fig. 4. Intensity of the beam [Eq. (28)] (a) without and (b) with the carrier frequency at w=2λ, b=1/(5λ), c=1/(5λ) (two generalized HG beams with equal width on the axes).

Figures 1(b)4(b) depict the intensity of the beams of Eq. (28) coherently superimposed with an oblique plane wave
I(x,y,z=0)=|E(x,y,0)+Cexp(iαx)|2,
(29)
where the amplitude C and the spatial frequency α were chosen so as to ensure a better visualization of Figs. 14. Characteristic “forks” that can be observed in Figs. 1(b)4(b) in the interference fringes show the locations of isolated intensity zeroes and phase singularities. The modules of the OAM for the beams in question are equal to 5 [Fig. 1(a)], 1 [Fig. 2(a)], 0.95 [Fig. 3(a)], and 0.92 [Fig. 4(a)].

7. DISCUSSION OF RESULTS

Putting p=s in Eq. (6) yields that the light field has (2p+1)2 isolated intensity zeros with topological charge n=1 found at the intersections of the horizontal lines formed by zeros of the polynomial H2p+1(y) and the vertical lines of zeros of the polynomial H2p+1(x), also having (2p)2 isolated zeros with topological charge n=1 found at the intersections of the horizontal lines of zeros of the polynomial H2p(y) and the vertical lines of zeros of the polynomial H2p(x). Thus, if all the said zeros were grouped very near to the optical axis, the maximum OAM of the field [Eq. (6)] would have been equal to the difference (2p+1)2(2p)2=4p+1. However, in practice the OAM of the field of Eq. (6) is defined by Eqs. (14), (20), (23), (25), and (27).

A visual picture of how the isolated zeros of the field [Eq. (6)] are arranged can be illustrated by a specific example. Figure 5(a) depicts an interference pattern similar to that shown in Fig. 1(b), except for the 2.5 times increased carrier frequency, and Fig. 5(b) the phase distribution for the intensity pattern in Fig. 1(a).

Fig. 5. Patterns of (a) intensity and (b) phase of the beam [Eq. (28)] with a carrier frequency for the parameters w=2λ, b=c=2/w (mode).

Substituting Eq. (30) into Eq. (31) yields
JzI=11+(r0σ)2.
(32)

From Eq. (32) it follows that if the isolated zero with topological charge n=+1 is found on the optical axis at r0=0, the OAM equals 1, but if the optical vortex is displaced from the optical axis onto the Gaussian beam’s periphery (r0>0), the OAM decreases in inverse proportion to the square of the distance from the optical axis. The OAM is reduced by half if the intensity zero is offset by a distance of the Gaussian beam’s waist r0=σ from the optical axis.

Let us analyze a similar but more complex example: two isolated identical zeros with n=+1 introduced into the Gaussian beam and found on the horizontal axis symmetrically with respect to the center. The complex amplitude of the field is
E(x,y)=exp(x2+y22σ2)[(xr0)+iy]×[(x+r0)+iy]=exp(r22σ2)×[r2exp(2iϕ)r02].
(33)

Substituting Eq. (33) into Eq. (31) yields
JzI=42+(r0σ)4.
(34)

From Eq. (34) it follows that when both zeros with n=+1 are simultaneously found on the optical axis, the OAM equals 2. If both zeros are symmetrically moving away from the optical axis, r0>0, the OAM [Eq. (34)] decreases as the fourth degree of the distance from the optical axis. The OAM will be equal to 1 at r0/σ=21/41.19. Thus, the OAM is reduced by half if both zeros are offset from the optical axis by a distance somewhat larger than the Gaussian beam’s waist radius.

Therefore, we can infer that the contribution of the optical vortices in Figs. 1 and 5 will be smaller than 2516=9, because all of them, except for the central zero, are found at different distances from the Gaussian beam’s center.

Equations (32) and (34) also explain why the linear combination of the elegant HG beams has a smaller OAM. From comparison of Eq. (6) with Eq. (19) and Fig. 1 with Fig. 2, the Gaussian beam’s waist of the elegant HG beams is seen to be half as large as that of the HG modes. This is the reason why the isolated zeros (optical vortices) in Fig. 2, while found at the same distance from the optical axis as those in Fig. 5, turn out to be located on the periphery of the Gaussian beam with the two-times-smaller waist radius. Because of this, the OAM of the beam in Fig. 2 equals just 1 rather than 5, as is the case in Fig. 1.

What remains to be discovered is why the beam of Eq. (15) or Eq. (17), which has an OAM and has no radial symmetry, does not rotate upon propagation.

An insight into this question can be gained using simple examples. Let us return to the simplest example of Eq. (30), when an optical vortex (i.e., an isolated intensity zero) is offset from the optical axis and introduced into the Gaussian beam. Such a field in Eq. (30) has an OAM and has no radial symmetry. By calculating the Fresnel transform of the complex amplitude (30), the intensity zero can be shown to rotate and move away from the optical axis upon propagation in accordance with the relation
{tanθ=zz0,ρ=r01+(zz0)2,
(35)
where z0=kσ2 is the Rayleigh range and (ρ,θ) are the polar coordinates in a plane perpendicular to the optical axis at distance z from the waist. From Eq. (35), the isolated intensity zero [Eq. (30)] is seen to rotate during propagation. At z=z0, the zero will rotate counterclockwise by the angle θ=π/4, rotating by the angle θ=π/2 at z. The second equation in Eq. (35) shows that as the beam propagates, the intensity zero will be moving away from the optical axis as fast as the Gaussian beam is diverging.

Let us analyze the more complex example in Eq. (33), when two isolated intensity zeros with the same topological charge are introduced into the Gaussian beam. The light field in Eq. (33) also has the OAM [Eq. (34)], being devoid of radial symmetry. Similar to Eq. (35), the formula to describe the rotation and moving away from the optical axis of two intensity zeros in Eq. (33) during propagation is given by
{tan(2θ)=(2zz0)[1(zz0)2]1/2,ρ=r01+(zz0)2.
(36)

From Eq. (36), both intensity zeros are seen to rotate counterclockwise as a whole during propagation. As previously, at z=z0, the zeros will rotate by the angle θ=π/4, rotating by the angle θ=π/2 at z.

If two intensity zeros have topological charges of opposite signs, they show entirely different behavior during propagation. In this case, it is impossible to derive a relationship similar to Eq. (35) or Eq. (36). The reason is that depending on where the intensity zeros are located in the Gaussian beam, the OAM can be positive, negative, or zero. For instance, assume that there are two isolated intensity zeros with the topological charges of n=1 and n=1 introduced into the Gaussian beam’s waist and located on the horizontal axis at points r0 and r1. At z=0, the complex amplitude of such a field is
E(x,y)=exp(x2+y22σ2)[(xr0)+iy]×[(x+r1)iy]=exp(r22σ2)×[r2r0r1+rr1exp(iϕ)rr0exp(iϕ)].
(37)

The OAM of the field [Eq. (37)] takes the form
JzI=(r12r02)(r1r0)2+2σ2+(r0r1σ)2.
(38)

From Eq. (38) it follows that the OAM of the field [Eq. (37)] is equal to zero at r1=r0, to a positive number at r1>r0, and to a negative number at r1<r0. At r1, the OAM [Eq. (38)] is coincident with the OAM of a single isolated intensity zero in Eq. (32).

From the above, one may infer that two intensity zeros with topological charges of opposite signs can rotate clockwise or counterclockwise, or not rotate at all during propagation. With a larger number of optical vortices, as in Fig. 5, the number of differently arranged combinations of the intensity zeros will essentially increase, consequently increasing the number of feasible behavioral patterns of the zeros during propagation. Therefore, the question of why the light beam of Eq. (15) or Eq. (17), while having the OAM and being devoid of radial symmetry, does not rotate during propagation can be addressed only in general.

A feasible explanation can be found in Ref. [25

25. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007). [CrossRef]

], where, using the expansion of the light field in terms of the LG modes, the rotation condition of the laser beam intensity cross section during propagation (Eq. 3 in [25

25. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007). [CrossRef]

]) was shown to be essentially different from the nonzero OAM condition of the beam (Eq. (4) in [25

25. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007). [CrossRef]

]). One can always find laser beams (as a combination of a finite number of LG modes) that rotate during propagation, while being devoid of the OAM, and, vice versa, there are laser beams that have the OAM but do not rotate during propagation. The light fields in Eqs. (15) and (17) are the illustration of such beams. It should be noted that in Ref. [17

17. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004). [CrossRef]

] the Hermite–Laguerre–Gaussian beams have the same properties as the beams [Eqs. (15) and (17)], i.e., these beams are spatial modes that do not rotate upon propagation but have the OAM.

Let us also note that if instead of Eq. (28) at s=p, we add the HG beams with permuted indices:
E˜(x,y,0)=exp[w22(x2+y2)]×[H2p+1(bx)H2p(cy)+iγH2p(cx)H2p+1(by)],
(39)
then for the superposition of modes (b=c=w), the normalized OAM [Eq. (14)] only changes sign:
JzI=(2γ1+γ2)(2p+1),
(40)
for the superposition of elegant beams (b=c=w/2), the normalized OAM [Eq. (20)] only changes sign as well:
JzI=2γ1+γ2,
(41)
while for two hybrid HG beams the normalized OAMs not only change sign but also swap. For b=w/2, c=w, instead of Eq. (25) we have
JzI=(2γ1+γ2)(2p+1)!(2p+1)(p21)(4p+1)!!,
(42)
and for b=w, c=w/2, instead of Eq. (27) we obtain
JzI=(2γ1+γ2)(2p+1)!(4p1)!!.
(43)

8. CONCLUSION

Thus, the following results have been obtained in this work:
  • An expression for the complex amplitude of the generalized paraxial HG beams of Eq. (1) has been derived. These beams have been shown to change to the familiar HG modes and elegant HG beams under certain parameters.
  • The OAM of a linear combination of two generalized HG beams with double indices composed of two adjacent integer numbers taken in direct and inverse order and a phase delay of π/2 has been calculated.
  • The modulus of the OAM has been shown to be an integer number for the HG mode for the superposition of two HG modes, always equal to unity for the superposition of two elegant HG beams and a fractional number for two hybrid HG beams.

APPENDIX A: DERIVATION OF HG BEAM COMPLEX AMPLITUDE [EQS. (1) AND (2)]

Using the reference integral (integral 7.374.8 in [26

26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

])
+exp[(xy)2]Hn(αx)dx=π(1α2)n/2Hn(αy1α2)
(A4)
allows us to obtain the complex amplitude [Eqs. (1) and (2)].

ACKNOWLEDGMENTS

The authors wish to thank Professor V. Volostnikov for the useful discussion of the work. The work was supported by the Ministry of Education and Science of the Russian Federation (proposal 2014-14-576-0012-3053), Russian Federation presidential grants for Support of Leading Scientific Schools (NSh-3970.2014.9), grants for Young Candidate of Science (MK-3912.2012.2) and Young Doctor of Science (MD-1929.2013.2), and the RFBR grant (13-07-97008).

REFERENCES

1.

L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992). [CrossRef]

2.

M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979). [CrossRef]

3.

J. M. Vaughan and D. Willetts, “Interference properties of a light-beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979). [CrossRef]

4.

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989). [CrossRef]

5.

V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).

6.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992). [CrossRef]

7.

E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]

8.

M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

9.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). [CrossRef]

10.

V. V. Kotlyar and A. A. Kovalev, Vortex Laser Beams (Novaya Tekhnika, 2012) [in Russian].

11.

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (2007). [CrossRef]

12.

E. Fraczek and G. Budzyn, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

13.

B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012). [CrossRef]

14.

P. Vaity, A. Aadhi, and R. Singh, “Formation of optical vortices through superposition of two Gaussian beams,” Appl. Opt. 52, 6652–6656 (2013). [CrossRef]

15.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013). [CrossRef]

16.

J. B. Gotte, K. O’Holleran, D. Precce, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008). [CrossRef]

17.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004). [CrossRef]

18.

D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. C. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010). [CrossRef]

19.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966). [CrossRef]

20.

A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical beam eigenfunction,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef]

21.

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994), pp. 140–148.

22.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

23.

J. Humblet, “Sur le moment d’impulsion d’une onde electromagnetique,” Physica 10, 585–603 (1943).

24.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Gordon and Breach Science, 1986).

25.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007). [CrossRef]

26.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(350.5500) Other areas of optics : Propagation

ToC Category:
Diffraction and Gratings

History
Original Manuscript: October 22, 2013
Revised Manuscript: December 5, 2013
Manuscript Accepted: December 8, 2013
Published: January 17, 2014

Citation
V. V. Kotlyar and A. A. Kovalev, "Hermite–Gaussian modal laser beams with orbital angular momentum," J. Opt. Soc. Am. A 31, 274-282 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-2-274


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References

  1. L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992). [CrossRef]
  2. M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979). [CrossRef]
  3. J. M. Vaughan and D. Willetts, “Interference properties of a light-beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979). [CrossRef]
  4. P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989). [CrossRef]
  5. V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).
  6. S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992). [CrossRef]
  7. E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]
  8. M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]
  9. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). [CrossRef]
  10. V. V. Kotlyar and A. A. Kovalev, Vortex Laser Beams (Novaya Tekhnika, 2012) [in Russian].
  11. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (2007). [CrossRef]
  12. E. Fraczek and G. Budzyn, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).
  13. B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012). [CrossRef]
  14. P. Vaity, A. Aadhi, and R. Singh, “Formation of optical vortices through superposition of two Gaussian beams,” Appl. Opt. 52, 6652–6656 (2013). [CrossRef]
  15. Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013). [CrossRef]
  16. J. B. Gotte, K. O’Holleran, D. Precce, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008). [CrossRef]
  17. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004). [CrossRef]
  18. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. C. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010). [CrossRef]
  19. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966). [CrossRef]
  20. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical beam eigenfunction,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef]
  21. F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994), pp. 140–148.
  22. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
  23. J. Humblet, “Sur le moment d’impulsion d’une onde electromagnetique,” Physica 10, 585–603 (1943).
  24. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Gordon and Breach Science, 1986).
  25. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007). [CrossRef]
  26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

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