Astigmatic laser beams with a large orbital angular ... - OSA Publishing

Vol. 26, No. 1 | 8 Jan 2018 | OPTICS EXPRESS 141

Astigmatic laser beams with a large orbital angular momentum VICTOR V. KOTLYAR,1,2 ALEXEY A. KOVALEV,1,2,* AND ALEXEY P. PORFIREV1,2 1

Image Processing Systems Institute of RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Samara, Russia, Molodogvardeyskata st., 151, Samara 443001, Russia 2 Samara National Research University, Samara, Russia, Moskovskoe shosse, 34, Samara 443086, Russia *[email protected]

Abstract: We show that an elliptic Gaussian beam, focused by a cylindrical lens, can be represented as a linear combination of a countable number of only even angular harmonics with both positive and negative topological charge. For the orbital angular momentum (OAM) of the astigmatic Gaussian beam, an exact expression is obtained in a form of a converging series of the Legendre functions of the second kind. It is shown that at some conditions only the terms with the positive or negative topological charge are remained in this series. Using a hybrid numeric-experimental approach, we obtained the normalized OAM of the astigmatic beam, equal to 109, which is just 6% different from the exact OAM of 116, calculated by the equation. To generate such laser beams, there is no need in special optical elements such as spiral phase plates. The OAM of such beams can be adjusted by varying the waist radius of the Gaussian beam and the focal length of the cylindrical lens. The OAM of such beams can reach large values. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (050.1960) Diffraction theory; (050.4865) Optical vortices; (350.5500) Propagation.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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#312573 Journal © 2018

https://doi.org/10.1364/OE.26.000141 Received 2 Nov 2017; revised 6 Dec 2017; accepted 8 Dec 2017; published 2 Jan 2018


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1. Introduction Laser beams with an orbital angular momentum (OAM) are currently actively investigated because of wide using of such beams for optical trapping and rotation of microscopic particles [1] and cold atoms [2], as well as in phase contrast microscopy [3], stimulated emission depletion microscopy [4], in optical classical [5] and quantum [6] informatics. Beams carrying the OAM have also potentials in detecting spinning objects due to their rotational Doppler effect [7, 8]. An interesting laser beam which has the OAM but is free of isolated intensity nulls with phase dislocations has been considered in [9]. In [9], the OAM of an elliptic Gaussian beam focused by a cylindrical lens has been calculated. The idea of using the cylindrical lens for assignment of the OAM to a beam has been introduced for the first time in [10]. It has been shown experimentally in [10] that after passing a cylindrical lens at certain propagation distance and at certain conditions an OAM-free Hermite-Gaussian beam transforms to a Laguerre-Gaussian beam which has the OAM. In papers [11–22] there are attempts to obtain as large as possible values of the OAM. In [11], it is proposed to increase the OAM by using an array of Gaussian vortex beams, whose centers lie on a circle and whose axes and the common optical axis are crossed lines. It is shown in [11] that the OAM of such composite beam can reach 204ħ per photon, where ħ is the Planck constant divided by 2π. In [12], instead of the Gaussian beams it is proposed to use small holes in an opaque screen, which act like point sources. If these sources are located along a spiral then together they generate a vortex beam with the OAM. In practice, in [12] a beam has been generated with the topological charge equal to 3. It is shown in [13] that tight focusing of an optical vortex with high topological charge leads to decreased contrast or visibility of sidelobes. In [13], a beam with the OAM per photon equal to 15ħ was practically focused. An interesting method for determining the topological charge of an optical vortex by using an annular diffraction grating has been proposed in [14]. It has been shown experimentally that this way allows to determine the topological charge of ± 25. In [15], by using the three-wave mixing in a nonlinear Kerr medium, vortex harmonics with the OAM up


to 30ħ per photon have been experimentally generated. In [16], a perfect optical vortex with a topological charge of 90 has been generated by using a digital multi-element mirror (with the number of micromirrors 1024 × 768). In [17], an optical vortex with the topological charge of 200 has been generated by using a liquid-crystal spatial light modulator (the number of elements 1900 × 1200). It allowed rotating 1.4-ìm-diameter microparticles at a speed of 500 ìm/sec. In [18], also by using a light modulator (the number of pixels 1920 × 1080), entangled pairs of photons with the OAM per photon of ± 300ħ have been generated. In [19], an optical vortex with a topological charge of 100 was experimentally generated by using a spiral phase mirror fabricated in an aluminum plate by direct machining with a diamond tool. The same authors [20] created a spiral mirror capable to generate optical vortices with the topological charge of 1020. They used an advanced technology on a 75-mm-diameter aluminum substrate with a roughness of 3 nm. In [20], it was also shown interferometrically that an optical vortex that has been generated by a mirror has a topological charge of 5050. In [21], using electronic lithography in a PMMA resist, an 80-μm-diameter hologram with a resolution of 35 nm and a relief height of 25 nm was fabricated. It allowed generating a vortex electron beam with energy of 0.5-1 eV and with a topological charge of 1000. And finally, photons entangled by both the OAM and polarization have been generated by a spiral aluminum mirror with a diameter of about 50 mm for a wavelength of 810 nm [22]. And what is more, the quantum OAM of photons was ± 10010ħ. This is the maximum value of the OAM received so far. In this paper, following [9], we show that an elliptic Gaussian beam after passing a cylindrical lens, whose axis does not coincide with the axes of an elliptical Gaussian profile, rotates upon propagation and at this it is not a vortex laser beam, although its full OAM can reach large values by varying the waist radii of the elliptical Gaussian beam and the focal length of the cylindrical lens. We calculate the coefficients of the angular harmonics expansion of the complex amplitude of such an astigmatic Gaussian beam. An exact formula is also obtained for the OAM in a form of a series of the Legendre functions of the second kind. The experimental results are in consistence with the theoretical predictions. The astigmatic laser beams considered here have complex distribution of the OAM density and the OAM spectrum. Compared with the conventional optical vortices with the annular structure of intensity, the astigmatic laser beams have a shape of an elliptic Gaussian beam with the intensity maximum in the center. Therefore, while the conventional optical vortex with the annular structure of intensity can rotate several dielectric microparticles along the ring simultaneously, the astigmatic beam can rotate only one particle with its mass center placed in the beam center. An undeniable advantage of these beams is that for their generation there is no need in special optical elements and spatial light modulators. To measure the OAM of such beams, a technique suggested in [23] can be used. However, since the astigmatic beam has an infinite number of the angular harmonics, the technique of expansion of the field into OAM modes is not efficient. Therefore, in this work, to measure the OAM, we used two interferograms, obtained with a phase shift of π in the reference beam. It allows to reconstruct unambiguously the phase of the astigmatic beams and then to calculate the OAM. 2. Vortex-free beam with the OAM For the convenience of the reader, the first four formulas in this section coincide with [9]. Vortex laser beams with the orbital angular momentum are usually considered within the paraxial limit. Such beams have singularity points – isolated intensity nulls with undefined phase. Isophase wavefront surface around such point has a spiral shape. However, it turns out that there are simple light fields that have the OAM and do not have the isolated intensity nulls with the vortex phase. We consider a Gaussian elliptic beam with a cylindrical lens [9] in its waist, whose generatrix is rotated in the waist plane by an angle α. The complex amplitude of light immediately after the cylindrical lens reads as


 x2 y 2   ikx 2 cos 2 α iky 2 sin 2 α ikxy sin 2α  − − E ( x, y ) = exp  − 2 − 2  exp  − ,  w w  2f 2f 2f   x y  

(1)

where wx and wy are the waist radii of the Gaussian beam along the Cartesian axes, f is the focal length of the thin cylindrical lens, whose generatrix is rotated by an angle α with respect to the vertical axis y (the lens is rotated counterclockwise), k is the wave number of light. The normalized OAM within the paraxial limit is given by the expressions [9] (up to the constants): J z = Im

 ∂E ( x, y )

∞ ∞

  E ( x, y )  x

−∞ −∞

W=

∂y

−y

∂E ( x, y )   dxdy, ∂x 

(2)

∞ ∞

  E ( x, y ) E ( x, y ) dxdy,

(3)

−∞ −∞

where Jz is the axial component of the OAM vector, W is the energy (power) density of light, Im is the imaginary part, E is the complex conjugate to the amplitude E. Substituting Eq. (1) into Eqs. (2) and (3), we derive a simple expression for the normalized OAM of the light field (1):

J z  k sin 2α  2 2 = (4)  ( wy − wx ) . W  8f  It is seen in Eq. (4) that the OAM equals zero if the Gaussian beam is circular (wx = wy) or the lens is not inclined to the vertical axis (α = 0). For the inclination angle of 45 degrees, the OAM (4) is maximal, all other conditions being the same. It is also seen in Eq. (4) that the OAM of the beam (1) is generally fractional, although it can be integer too. The smaller is the focal length of the cylindrical lens and the larger is the ellipticity of the beam (1), the greater is the OAM. The OAM sign is determined by whether the Gaussian beam is elongated along the x or y axis in its waist. The advantage of the beam (1) is that it can be generated without any additional elements, without a light modulator or a spiral phase plate or a fork hologram. Only two lenses are needed to generate it. The first one generates an elliptic Gaussian beam, while the second one creates its OAM. Now we estimate the OAM for specific values of the variables in Eq. (4). A Gaussian beam is considered to be paraxial if its waist radii exceed its wavelength. Let the waist radii be equal to wx = 2 mm and wy = 1 mm, while the focal length is f = 10 mm, wavelength is λ = 0.5 μm, and the tilt angle of the lens is 45 degrees (α = π/4). Then the OAM in Eq. (4) equals 471.24. It is shown in the Appendix 1 that using of two crossed cylindrical lenses with the focal lengths, equal in modulus but opposite in sign, increases the OAM two times. In Appendix 2, expressions are obtained that describe propagation of the field (1) through an ABCD system. A general formula is also obtained for the OAM density of the astigmatic Gaussian beam at any distance from the waist plane. Appendix 3 contains an expression for the full OAM of an astigmatic Gaussian beam (1) if the cylindrical lens is placed in arbitrary transverse plane (not only in the waist plane). 3. Rotation of the elliptic Gaussian beam after the cylindrical lens

Below, in difference with [9], we show that the elliptic Gaussian beam is rotating after passing the cylindrical lens. Let's derive equations to describe propagation of the beam (1) and show that no isolated intensity nulls appear on propagation, i.e. the beam (1) is not a vortex or a singular beam [11–22]. The Fresnel transform of the complex amplitude (1) reads as


E ( ξ ,η , z ) =

−ik z p(z)q(z)

exp  A ( z ) ξ 2 + B ( z )η 2 + C ( z ) ξη  ,

(5)

where A( z ) =

ik k2 k 4 sin 2 2α − 2 + , 2 z 4 z p ( z ) 64 f 2 z 2 p 2 ( z )q(z )

B (z) =

ik k2 ik 3 sin 2α − 2 , C ( z) = , 2 z 4 z q( z ) 16 fz 2 p ( z )q ( z )

1 ik 1 ik k 2 sin 2 2α + p( z ) = 2 + , q( z ) = 2 + , wx 2 z x wy 2 z y 16 f 2 p ( z )

(6)

zf zf , zy = . z cos 2 α − f z sin 2 α − f It is seen in Eq. (5) that the Gaussian beam (1) preserves its Gaussian shape on propagation, but changes its scale and rotates. Equation (5) is simplified significantly for α = π/4 and z = 2f, since zx → ∞ and zy → ∞ for these values: zx =

 ik 2 ξ2 η2 ikξη  ξ +η 2 ) − 2 2 − 2 2 + E (ξ ,η , z = 2 f ) = −2iγ −1 exp  , ( 2 f γ 2  wy γ wx γ  4 f

(7)

where 1/ 2

 16 f 2  γ = 1 + 2 2 2  . (8)  k ww  x y   It is seen in Eq. (7) that at the distance z = 2f the elliptic Gaussian beam (1) is rotated by 90 degrees and widened since γ > 1.

4. Orbital angular momentum of a Gaussian beam after passing two cylindrical lenses

In [9], an elliptic Gaussian beam has been generated by using two cylindrical lenses. However, one cylindrical lens is sufficient for generation of converging or diverging elliptic Gaussian beam. Now we consider this in detail. Let a cylindrical lens with the curvature along the x-axis and with focal length f1 be placed into the waist of a conventional circular Gaussian beam with the waist radius w. Then the complex amplitude of the elliptic Gaussian beam at a distance z behind the cylindrical lens reads as E ( x, y , z ) =

 x2 y2  exp  − 2 , − 2  w q ( z ) w q ( z )  q0 ( z ) q1 ( z ) 2 0   1

(9)

where q0 ( z ) = 1 +

iz z , q1 ( z ) = q0 ( z ) − , z0 f1

(10) −1  iz0  kw2 . q2 ( z ) = q1 ( z ) 1 +  , z0 = 2 f1   If a cylindrical lens with the focal length f is placed into the light field (9) and rotated by an angle α, then the complex amplitude immediately behind the lens is


 x2 y2  − 2 exp  − 2 ×  w q2 ( z ) w q0 ( z )   ikx 2 cos 2 α iky 2 sin 2 α ikxy sin 2α  × exp  − − − . 2f 2f 2f   The normalized OAM of the beam (11) reads as E ( x, y, z ) = ( q0 ( z )q1 ( z ) )

−1/ 2

2 2 q2   kw2 sin 2α   z  J z  kw2 sin 2α   q0 z = − =     2 −  . 8f 8f W  f1    Re ( q0 ) Re ( q2 )     f1 

(11)

(12)

It is seen in Eq. (12) that the OAM tends to zero at f1 →∞, since the beam (11) tends to the conventional Gaussian beam. It is also seen in Eq. (12) that increasing of the distance z between the first cylindrical lens with the focal length f1 and the second cylindrical lens with the focal length f allows unlimited increasing of the OAM of a laser beam. At z = 0 and z = 2f1 the OAM (4) is also equal to zero, since the Gaussian beam has a circular shape. The OAM (12) has maximal positive value at z = f1, i.e. when the second cylindrical lens is placed in the focus of the first one. The instead of Eq. (12) we get (z = f1) J z  kw2 sin 2α  = (13) . 8f W   For wy = w and wx = 0, Eq. (13) coincides with the expression (4) for the OAM. At z > 2f1 the OAM (12) changes its sign [24] (becomes negative) and is increasing (in modulus) with increasing distance z.

5. Calculation of the orbital angular momentum of an astigmatic beam by using the angular harmonics expansion

In this section another expression is obtained for the normalized OAM of the beam (1). It is based on the angular harmonics expansion of the amplitude of the field (1). We write the amplitude (1) in the polar coordinates (r, φ) E (r , ϕ ) = exp ( −ar 2 cos 2 ϕ − br 2 sin 2 ϕ − icr 2 sin 2ϕ ) ,

(14)

as a series of the angular harmonics: E (r , ϕ ) =

∞

 C (r ) exp(inϕ ),

n =−∞

n

(15)

where

Cn ( r ) = ( 2π )

−1

2π

 E ( r ,ϕ ) exp ( −inϕ ) dϕ ,

(16)

0

and a = wx−2 + ik ( 2 f ) cos 2 α , −1

b = wy−2 + ik ( 2 f ) sin 2 α , −1

(17)

c = k ( 4 f ) sin 2α . The integral (16) can be written as −1

2π

Cn ( r ) = ( 2π ) e − D  exp ( −inϕ − A cos 2ϕ − iB sin 2ϕ ) dϕ , −1

0

(18)


where A = [(a – b)/2]r2, B = cr2, D = [(a + b)/2]r2. After replacing t = 2φ and for even numbers n = 2m, instead of Eq. (18) we get: Cn ( r ) = (2π ) −1 e − D

2π

 exp ( −imt − A cos t − iB sin t ) dt = 0

m

= ( −i ) e m

− D + imθ

Jm (F ) = e

−D

 F    J m ( F ),  A− B 

(19)

where F = (B2 – A2)1/2, tan θ = iB/A. For the odd numbers of the angular harmonics n = 2m + 1 the coefficients (19) are equal to zero: 2π

2π C2 m +1 ( r ) = e − D  exp  −i ( 2m + 1) ϕ − A cos 2ϕ − iB sin 2ϕ  dϕ = 0

π = e − D   exp  −i ( 2m + 1) ϕ − A cos 2ϕ − iB sin 2ϕ  dϕ +  0 π

 +  exp  −i ( 2m + 1)(ϕ + π ) − A cos 2 (ϕ + π ) − iB sin 2 (ϕ + π )  dϕ  =  0 π

= e − D  1 + ( −1) 

2 m +1

0

(20)

 exp  −i ( 2m + 1) ϕ − A cos 2ϕ − iB sin 2ϕ  dϕ = 0.   

Then, for the coefficients (18) we get:  − D  F n 2 e   J n 2 ( F ) , n = 2m, Cn (r ) =   A− B  0, n = 2m + 1. 

(21)

It is seen in Eq. (21) that the coefficients at the angular harmonics with the positive and negative numbers are different in modulus (n = 2m): C2 m ( r ) = e 2

−2 Re D

A+ B A− B

m

Jm ( F ) , 2

m

2 A− B C−2 m ( r ) = e Jm ( F ) . A+ B The normalized OAM for the expansion (15) can be represented as 2

(22)

−2 Re D

∞

Jz = W

 nC

n =−∞ ∞



n =−∞

n

Cn

∞

, Cn =  Cn (r ) rdr. 2

(23)

0

Further, we suppose α = π/4 to make the difference a – b real and to remove the modulus sign of the squared Bessel function. Coefficients in the sum (23) for the even numbers read as m ∞

2 2  2 2 r 2   (24) exp Re a b r J − + ( ) 0   m  2 4c − ( a − b ) rdr.   In the general case, when α ≠π/4, instead of Eq. (24) a more tedious expression is obtained:

C2 m =

a − b + 2c a − b − 2c


∞

C2 m =  e −2 Re D 0

= = 2

a − b + 2c a − b − 2c

A+ B A− B

m ∞

e

m

J m ( F ) rdr = 2

− Re ( a + b ) r 2

0

1 a − b + 2c 2 a − b − 2c

m ∞

e

J m ( Gr 2 ) J m ( G* r 2 ) rdr =

− Re ( a + b ) u

0

(25)

J m ( Gu ) J m ( G *u ) du.

2 1/2

where G = [4c – (a – b) ] /2. The integral in Eq. (24) can be evaluated by using a reference integral [25] ∞

 p2  −1 2 − pr 2 2 e J qr rdr q Q = 2 π 1 + . (26) ( ) ( )  m m −1/ 2 2  0  2q  To evaluate the integral in Eq. (25), another reference integral can be used [25]: ∞

e

− px

Jν ( bx ) Jν ( cx ) dx =

0

 p2 + b2 + c2  Qν −1 2  , 2bc π bc   1

(27)

where Qv(x) is the spherical Legendre function or the Legendre function of the second kind (x > 1): Qν ( x ) = π ( 2 x )

−ν −1

3 ν + 1 ν + 2  Г ( v + 1) Г −1 ( v + 3 2 )2 F1  , ,ν + , x −2  , 2 2  2 

(28)

where Γ(x) is the Gamma function, 2 F1 (a, b, c, x) is the hypergeometric function [26]. Then, Eq. (24) reads as 2 m  2 (a + b)  2 −1/ 2  c + a −b   2  1 C 2 m = π −1  c a b Q − − +  , ( )  m −1/ 2 2  с−a+b   c 2 − ( a − b ) 

(29)

or in the more general case of the tilt angle α ≠ π/4 instead of Eq. (29) we get: m  Re2 ( a + b ) + 2 Re G 2 a − b + 2c Q m −1 2  2  2π G a − b − 2c 2G 

 . (30)   Let |G|2 ≈0. Then the argument of the Legendre function in Eq. (30) tends to infinity and from the asymptotic of the Legendre function at x >> 1 [26] C2 m =

1

Qν ( x ) ≈ π ( 2 x )

− ν −1

Г ( ν + 1) Г −1 ( ν + 3 2 )

(31)

an expression follows for the expansion coefficients in the formula for the OAM (24): C2 m =

1

Г ( m + 1 2 ) a − b + 2c

m +m

a − b − 2c

m −m

. (32) 2 m +1  Re ( a + b )  We note that the argument of the Legendre function in Eq. (29) is complex. Further, for the certainty, let the cylindrical lens to be rotated again by the angle α equal to 45 degrees. When the Gaussian beam is circular (wy = wx), then a – b = 0 in Eq. (29) and the coefficients (29) at the angular harmonics are equal for the numbers m and –m, i.e. the OAM (24) equals zero. If 2c = b – a, i.e. f −1 = z0−1y − z0−x1 , all the coefficients (29) are equal to zero at m > 0, 2

2 m +1

π Г ( m + 1)

while at m < 0 they are not equal to zero:


0, m > 0,  −2 m −1 C2 m =  f Г ( m + 1 2 )  wy2 + wx2  (33) , m ≤ 0.  2  2    k π Г ( m + 1)  wx − wy  The OAM (23) in this case is non-zero. In the opposite case, when 2c = a – b or f −1 = z0−x1 − z0−1y , all the coefficients at m < 0 are equal to zero, while at m > 0 they are non-

zero: −2 m −1  Г ( m + 1 2 )  wx2 + wy2   f , m ≥ 0,   (34) C2 m =  k π Г ( m + 1)  wy2 − wx2     0, m < 0. In this case, the OAM (23) is also non-zero. It is seen in Eq. (34) that with increasing number the coefficients Cn of the series (23) decrease to zero. For large values of m, instead

of Eq. (34) it can be written (m > 0) that: 2 m +1

 wy2 − wx2  (35) . C2 m ≈  2 2  k π  wy + wx  Let for the certainty a – b > 0 or wy > wx, and let c > a – b, i.e. f −1 > z0−1y − z0−х1 . Then, the f

coefficients (29) with the positive numbers m > 0 will exceed the coefficients with the negative numbers: C2 m > C−2 m . This means that the angular harmonics exp(i2mφ) with the positive numbers give the larger contribution into the full OAM (23), which is therefore positive. In the opposite case, when a – b < 0 or wy < wx and f −1 < z0−1y − z0−х1 , the OAM is negative since C2 m < C−2 m . These conclusions are in concordance with the expression (4). So, we have shown that an elliptical Gaussian beam with an astigmatism (1) has the OAM (23), which consists of contributions of only even angular harmonics with both positive (2m) and negative (–2m) topological charge, although these contributions are not equal ( C2 m ≠ C−2 m ). For m = 0, the coefficient C0 is non-zero and equals  2(a + b) 2  (36) Q−1/ 2 1 + 2 . 2  π c 2 − ( a − b) 2  c − ( a − b)  The value of the Legendre function in Eq. (36) can be obtained via the complete elliptic integral [26]: C0 =

Q−1/ 2 ( x) =

1

1  −1/ 2 2 2  2 2 K   , K (t ) =  (1 − x )(1 − tx )  dx. 1+ x  1+ x  0

(37)

Simulation 1

Using the Fresnel transform, intensity and phase distributions of the field (1) were computed for several propagation distances from the cylindrical lens. The following parameters were used: wavelength λ = 532 nm, Gaussian beam waist radii wx = 20λ and wy = 10λ, cylindrical lens focal length f = 100λ, inclination angle of the lens from the Cartesian coordinates α = π/4, computation area –R ≤ x, y ≤ R. Normalized OAM density of the field (1) was computed by using the expression: jz = kI ( x, y )(2 f ) −1 ( y 2 − x 2 ) .

(38)


Fig. 1. Distributions of intensity (left column), phase (middle column) and OAM density (right column) of the field (1) at different distances from the initial plane: (a) z = λ (R = 40λ); (b) z = f (R = 40λ); (c) z = 2f (R = 80λ). Positive and negative OAM density is shown by the red and blue color respectively.

Figure 1 shows distributions of intensity, phase and OAM density of the elliptical Gaussian beam (1) for the different distances after the cylindrical lens. It is seen in Fig. 1 that the elliptic Gaussian beam rotates on propagation after passing the cylindrical lens. The OAM density rotates with the beam synchronously, while the total OAM is certainly preserved. It is also seen that at the double focal distance the Gaussian beam is turned by 90 degrees [Fig. 1(c), left column] with respect to its initial position [Fig. 1(a)], as predicted by Eq. (7). The OAM of this beam [Fig. 1], computed by Eq. (4), is Jz/W = 3π/4. Simulation 2

Now we verify numerically the above obtained expressions. We calculate the OAM by using the initial expression (4) and by using the angular harmonics expansion (23). The calculation parameters are the following: the wavelength λ = 532 nm, tilt angle of the cylindrical lens α = π/4, waist radii of the elliptic Gaussian beam wx = 20λ and wy = 400λ, focal length of the cylindrical lens f = 1 / (1/z0x – 1/z0y) ≈1260λ. Figure 2 shows the distribution of the absolute values of the coefficients Cn (continuous curve). For the other waist radii wx = 10λ, wy = 100λ and for the other focal length of the cylindrical lens f = 1 / (1/z0x – 1/z0y) ≈317λ, the distribution of the absolute values of the coefficients Cn in Eq. (23) is shown also in Fig. 2 (dashed line).


Fig. 2. Distribution of the coefficients

Cn

(only even numbers, since for the odd numbers

they are equal to zero) for two different beams and focal lengths of the cylindrical lens: wx = 20λ, wy = 400λ, f = 1260λ (continuous curve); wx = 10λ, wy = 100λ, f = 317λ (dashed curve). The inset shows a magnified fragment for n = −10, ..., 50 (each dot is depicted for integer value of n).

For the first case (continuous curve in Fig. 2), the OAM, calculated by Eq. (4), is equal to 99.500625, why the OAM calculated by Eq. (23) equals 99.48454. For the second case (dashed curve in Fig. 2), the OAM calculated by Eq. (4) equals 24.5025, while the OAM calculated by Eq. (23) is equal to 24.5025. In the second case there is a coincidence up to four decimal points. In Eq. (15), 2000 coefficients Cn were used for the calculation (Fig. 2 shows only 200 such coefficients). It is seen in Fig. 2 that the less is the OAM, the less is the value of the non-zero angular harmonics expansion coefficients Cn (the dashed curve is below the continuous curve). 6. Experiment

Figure 3 shows an experimental setup for generation and analysis of the elliptical Gaussian beams. The radiation from a solid-state laser was collimated by a system consisting of a pinhole PH and a spherical lens L (f1 = 150 mm). The collimated Gaussian beam had the radius of w = 3.5 mm. To implement the Mach-Zehnder interferometric setup, the collimated laser beam was split into two beams by using a beam splitting cube BS1. In one of the interferometer arms, a cylindrical lens CL1 (f1 = 500 mm) was used to generate an elliptical Gaussian beam in the plane of the second cylindrical lens CL2 (f2 = 100 mm). The distance between the lenses CL1 and CL2 was 200 mm. The analyzed laser beam, focused by the lens CL2, and the reference laser beam with a plane wavefront were combined by the second beam splitting cube BS2. The neutral filter F was used to equalize the intensities of the beams when registering the interferograms. Interference patterns were registered by using a CCD camera Toup-Cam U3CMOS08500KPA (pixel size 1.67 μm, resolution 3328 × 2548 pixels).

Fig. 3. Experimental setup for generation and analysis of the elliptical Gaussian beams: Laser is a solid-state laser (λ = 532 nm), PH is a pinhole (hole size 40 μm), L is a spherical lens (f = 150 mm), BS1 and BS2 are beam splitting cubes, F is a neutral density filter, M1 and M2 are mirrors, CL1 and CL2 are cylindrical lenses (f1 = 500 mm, f2 = 100 mm), CCD is a videocamera.


Figure 4(a) shows the intensity distribution of a converging elliptical Gaussian beam in the plane of a cylindrical lens with a focal length f = 100 mm, rotated by an angle of 45 degrees in the plane of the beam. Figure 4(b) shows the intensity cross section of the same beam at a distance z = 2f from the cylindrical lens. It is seen that the beam is turned by almost 90 degrees. Figure 4(c) shows the fringe pattern of an elliptical Gaussian beam at a distance z = 2f [Fig. 4(b)] with an almost plane wave. Shown in Fig. 4(d) is the intensity of the beam at a distance z = 3f. It is seen that he is turned by 135 degrees relative to the original cross section [Fig. 4(a)]. Shown in Fig. 4(e) is the fringe pattern of this cross section [Fig. 4(d)] with a plane wave.

Fig. 4. Intensity distributions of the astigmatic Gaussian beam, registered at a distance z from the cylindrical lens, equal to 0 (a), 2f (b), 3f (d), as well as fringe patterns of this beam with a plane wave at the distances 2f (c) and 3f (e). The size of all frames is 5608 × 4255 μm.

Shown in Fig. 5 are two interferograms [Fig. 5(a, b)] similar to those in Fig. 4(c), but different with each other in that the reference beam had a phase delay of π. Using these two interferograms, it is possible to unambiguously reconstruct the phase of the astigmatic Gaussian beam at a distance 2f from the cylindrical lens [Fig. 4(b)]. The reconstructed phase of the astigmatic Gaussian beam [Fig. 4(b)] is shown in Fig. 5(c). Using the reconstructed phase [Fig. 5(c)] and intensity [Fig. 4(b)] in the general expressions (2) and (3), the normalized OAM was computed: J z / W ≈ 63 . This value is almost two times less than the theoretical one. This is because of low signal-to-noise ratio in the reconstructed phase [Fig. 5(c)].

Fig. 5. Two interferograms (a, b), registered in the optical setup from Fig. 3 at a distance of 2f from the second cylindrical lens. The second interferogram was registered by using a reference beam with the phase delayed by π. Phase of the astigmatic Gaussian beam reconstructed by using the interferograms (c). The size of all frames is 5608 × 4255 μm.

Further, to make the experimental OAM value ( J z / W ≈ 63 ) more precise, we applied a hybrid numerical-experimental method. Shown in Fig. 6(a, b) are two interferograms similar to those in Fig. 5(a, b), but obtained by numerical simulation. Figure 6(c) shows the


reconstructed phase distribution at a distance z = 2f, which is consistent with the experimental phase [Fig. 5(c)]. The phase distribution is shown in Fig. 6(c) not for the whole calculation area (the whole area is 14 × 14 mm), and it is not limited by the beam intensity, in contrast to the interferograms shown in Fig. 6(a, b). All the frames in Fig. 6(a, b) show the interferograms limited by the beam intensity.

Fig. 6. Computed intensity distributions (a, b) of the interferograms, registered in the optical setup from Fig. 3 at a distance 2f. The second interferogram was calculated by using a reference beam with the phase delayed by π. Phase of the astigmatic Gaussian beam, reconstructed by using the computed interferograms (c). The size of all frames is 5608 × 4255 μm.

The size of the interferograms in Fig. 6(a, b) coincides with that in Fig. 5(a, b). It is seen from comparison of Figs. 5(a, b) and 6(a, b) that the frames of the same sizes contain the same number of fringes. Thus, the phases of the fields in Figs. 5(c) and 6(c) are in qualitative agreement with each other. However, outside the area with the intensity higher than 0.1 from its maximum value the phase of the astigmatic beam [Fig. 5(c)] is reconstructed with a large error. Therefore, the OAM computed by using the experimental phase [Fig. 5(c)], taken only in the area with the intensity higher than 10% of the maximum value, gives a much smaller OAM value: Jz/W = 63. For a more accurate calculation of the OAM, we used the phase and intensity distributions calculated for the larger area than that in the experiment [Fig. 6(c)]: 14 × 14 mm (4000 × 4000 pixels). Using the numerically reconstructed phase φ(x, y) [Fig. 6(c)] and the computed intensity I(x, y) (not shown in Fig. 6) in the general formula (2) and (3), the OAM of the beam in Fig. 4 was obtained, which was equal to: Jz W

∞ ∞   ∂ ∂  =    I ( x, y )  x − y  ϕ ( x, y ) dxdy  × ∂x   ∂y −∞ −∞ 

(39) −1 ∞ ∞   ×    I ( x, y )dxdy  ≈ 109.63. −∞ −∞  The theoretical value of the OAM of the astigmatic Gaussian beam in Fig. 4, calculated by the formula (12), is equal to (w = 3.5 mm, k = 2π/λ = 11810 mm–1, f1 = 500 mm, f = 100 mm, z = 200 mm, α = π/4) J z  kw2 sin 2α   z   z = (40)     2 −  ≈ 116. 8f W  f1    f1   Thus, using the interferograms and phase distribution in Fig. 5, we obtained the experimental value of the normalized OAM ( J z / W ≈ 63 ) of the beam from Fig. 4, which

differs from the theoretical value ( J z / W ≈ 116 ) by 50%. In case of using the simulated interferograms and phase [Fig. 6], which are consistent with the experimental ones [Fig. 5], but are free of noise, then the computed OAM ( J z / W ≈ 109.63 ) differs from the theoretical value (40) by only 6%.


7. Conclusion

In this paper we obtained the following results. For an astigmatic Gaussian beam generated by focusing of an elliptical Gaussian beam with a cylindrical lens, an expression is obtained for the normalized OAM in the form of a series of Legendre functions of the second kind with half-integer orders. If the parameters of the elliptic Gaussian beams and of the cylindrical lens are such that f −1 = z0−1y − z0−x1 then only the angular harmonics with the negative numbers (2m < 0) make a contribution to the OAM. And vice versa, if f −1 = z0−x1 − z0−1y then only the angular harmonics with the positive numbers (2m > 0) contribute to the OAM. In all cases, the largest coefficient in the angular harmonics expansion of the astigmatic field is the zero coefficient. However, since this coefficient is multiplied by zero (zero topological charge), it does not contribute to the OAM. Because of the large contribution of the zero angular harmonic to the amplitude of the astigmatic field, there are no isolated intensity nulls, and therefore there are no points with phase singularities. Such astigmatic beam looks like a vortex-free beam, although it has the non-zero OAM. Using two cylindrical lenses, we experimentally generated an astigmatic Gaussian beam. Using two interferograms, we reconstructed the beam phase function at some propagation distance. Using the phase and intensity in general expressions, we computed the experimental normalized OAM value, which is different from the theoretical one by 50%. Further, an astigmatic beam with the same parameters was simulated on a computer, two interferograms were calculated (the reference beam of the second interferogram had a phase delay of π), and the phase of the model astigmatic beam was reconstructed by using these interferograms. Both phase distributions (experimental and simulated) are in good agreement with each other. The number of fringes in the same area of the experimental and simulated phase distributions is the same. The OAM, computed by the general equations by using the simulated data, which are consistent with the experiment, coincided with the value calculated by the analytical formula with an error of about 6%. In three Appendices we have obtained: 1) an explicit expression for the normalized OAM of an astigmatic Gaussian beam focused by two crossed cylindrical lenses (converging and diverging); this beam has a larger OAM than the one focused only by the converging lens; 2) an explicit expression for the normalized OAM density of an astigmatic Gaussian beam that has passed through an ABCD system; 3) an explicit expression for the normalized OAM of an astigmatic beam if the cylindrical lens is located at an arbitrary distance from the waist of the elliptical Gaussian beam. Appendix 1

Now instead of the field (1) we consider a more general light field, when an elliptical Gaussian beam passes through two crossed cylindrical lenses with different focal lengths fx, fy, whose axes are inclined at an angle α to the horizontal axis. The complex amplitude of such a field in the initial plane (z = 0) reads as  x 2 y 2 ikx′2 iky ′2 − E ( x, y ) = exp  − 2 − 2 −  w w 2 fx 2 fy x y 

  , 

(41)

where  x′ = x cos α + y sin α , (42)   y ′ = y cos α − x sin α , Then, substitution of the field (41) into Eqs. (2) and (3) gives the following normalized OAM:


Jz k  1 1  2 =  −  ( wx − wy2 ) sin 2α . (43) W 8  f y f x  Comparison of Eq. (43) with Eq. (4) shows that if the focal lengths of the cylindrical lenses have different signs (one lens is converging and the other is diverging) then the OAM can be increased two times compared to using only one cylindrical lens.

Appendix 2

Now we consider propagation of the field (41) through an ABCD-system. Then the complex amplitude in the output plane reads as: E2 ( ξ , η ) =

2 2  ikD 2 k 2 Pxxη + Pyyξ − Pxyξη  2 ξ + η − exp  , ( ) B2 G B G  2 B 

−ik

(44)

where Pxx =

1 ik ik ikA cos 2 α + sin 2 α − , + 2 2 fy 2B wx 2 f x

Pyy =

1 ik ik ikA sin 2 α + cos 2 α − , + 2 wy 2 f x 2 fy 2B

(45)

ik  1 1  2  −  sin 2α , G = 4 Pxx Pyy − Pxy . 2  fx f y  The intensity distribution of the field (41) in the output plane of the ABCD-system is as follows: Pxy =

I 2 ( ξ ,η ) =

 2k 2  k2 exp  − 2 2 Ψ (ξ ,η )  , 2 B G  B G 

(46)

where Ψ (ξ ,η ) = Re { Pyy G * } ξ 2 + Re { Pxx G * }η 2 − Re { Pxy G* } ξη .

It is seen in Eq. (46) that the intensity distribution looks like an elliptic Gaussian beam, as in the initial plane (41). If there is only one cylindrical lens with the focal length fx and fy →∞, α = π/4, A = 1, B = z, then, using Eq. (44), an equation can be obtained for the OAM density in the polar coordinates (r, φ) at an arbitrary propagation distance z: jz E2

2 z0 x z0 y  f x z 2 + z0 x z0 y ( z − f x )  cos 2ϕ + f x ( z02y − z02x ) z ( z − 2 f x ) sin 2ϕ , (47) = kr 2 2 2 4  f x z 2 + z0 x z0 y ( z − f x )  + ( z0 x + z0 y ) z 2 ( z − 2 f x ) 2

2

where z0 x = kwx2 2 , z0 y = kwy2 2 are the Rayleigh distances. Particularly, for the double focal length (z = 2fx) the following simple expression for the OAM density can be derived from Eq. (47): jz

=

z0 x z 0 y k r 2 cos 2ϕ . 2 2 f x 4 f x + z0 x z0 y

(48) E2 From Eq. (47) it follows that the normalized OAM density depends on the distance to the lens z. The numerator is proportional to z2, while the denominator is proportional to z4. This means that with increasing distance the OAM density (47) decreases parabolically, similarly 2


to the decreasing intensity of the Gaussian beam due to its divergence. We note that the OAM density (48) has the same form as for one cylindrical lens (38), since y 2 − x 2 = r 2 cos 2ϕ . Appendix 3

Now we consider an elliptic Gaussian beam not in the waist plane, but in any other plane at a distance z from the waist. The complex amplitude at a distance z from the waist of the Gaussian beam reads as E ( x, y, z ) =  qx ( z ) q y ( z ) 

−1 2

 x2 y2  exp  − 2 − 2 ,  w q ( z ) w q ( z )  x x y y  

(49)

where qx ( z ) = 1 +

iz iz , qy ( z ) = 1 + , z0 x z0 y

(50)

and the Rayleigh distances z0x and z0y are the same as in Eq. (47). If in the field (49) a cylindrical lens is placed with a focal length f, whose generatrix is rotated in the transverse plane by an angle α, then the complex amplitude of light immediately after the cylindrical lens is as follows  x2 y2  − 2 × exp  − 2  w q ( z ) w q ( z )  x x y y    ikx 2 cos 2 α iky 2 sin 2 α ikxy sin 2α  × exp  − − − . 2f 2f 2f   The normalized OAM of the beam (51) reads as E ( x, y, z ) =  qx ( z ) q y ( z ) 

−1/ 2

(

(51)

)

2 J z  k sin 2α  2 2 2 = (52)  wy q y − wx qx . W  8f  It is seen in Eq. (52) that the sign and the magnitude of the OAM depend on the distance z from the waist of the elliptical Gaussian beam to the plane of the cylindrical lens. This is more clearly seen if we write Eq. (52) in a different form:

 J z  k sin 2α  2 4z2  2 (53) =  ( wy − wx ) 1 − 2 2 2  . W  8f   k wx wy  It is seen in Eq. (53) that if the cylindrical lens is placed at the distance z = kwxwy/2, then the OAM equals zero, since at this distance the elliptical Gaussian beam has a circular transverse section. If z < kwxwy/2 then the OAM is positive and if z > kwxwy/2 then the OAM is negative. At large values of z the modulus of the OAM increases with z parabolically. We note, however, that for the above used parameters of the Gaussian beam (wx = 2 mm, wy = 1 mm, λ = 0.5 μm) the distance, along which the OAM drops from its maximum value to zero, is z = 4π meters. Therefore, to achieve the maximal OAM the cylindrical lens should be placed into the waist of the elliptic Gaussian beam. For z = 0, Eq. (53) coincides with Eq. (4).

Funding

Russian Science Foundation project #17-19-01186.