## Abstract

We employ a setup, based on a phase spatial light modulator (SLM), for generation of arbitrary optical fields. The process is based on two sequential phase modulations of a laser beam at two different zones of the SLM. The input beam is transformed in the first contact with the SLM by a phase modulation that includes a diffractive phase element (DPE), which encodes the desired complex field, and a Fourier transforming lens. The Fourier transform of the DPE is projected, using a mirror, to the second SLM modulation area, whose transmittance includes a second Fourier transforming lens and a phase spatial filter, used to eliminate non-desired parts of the DPE Fourier spectrum.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Generation of an arbitrary optical wave field is an important and useful task in optics. In general, its implementation requires the independent control of the spatial distributions of amplitude, phase, and polarization of the field. A particular case is the so called scalar wave field, for which it is assumed a constant polarization at every point of the field. An especially convenient method for generation of an arbitrary scalar wave field, allows the modulation of both the amplitude and the phase by means of a synthetic phase hologram (SPH) [1–17]. Significant advantages of a SPH is that it implements a complex modulation with high efficiency and without necessity of absorbing devices, which can be inconvenient when high power beams are employed. The development of commercial phase-only SLMs in the last two decades has enabled the implementation of SPH approaches in several applications [18–22].

Conventional approaches for generation of a complex field employ a single modulation zone in a phase SLM [4–6,8,10–14]. Sometimes, it is convenient to place the SLM at the input plane of a 4-f double-Fourier transforming optical setup, formed by two Fourier transforming lenses, a spatial filter, and the output plane [6,10,12]. In this setup, the phase SLM that displays a synthetic SPH is placed at the input Fourier plane of the first transforming lens and the spatial filter appears at the output Fourier plane of this lens. If the SPH is appropriately designed to generate a desired complex field, its Fourier spectrum will be formed by a signal term, proportional to the Fourier spectrum of the desired field, and a non-signal term. The desired field is appropriately generated from the SPH when such Fourier spectrum terms present a negligible degree of overlapping. In this case the non-signal term in the Fourier plane can be blocked by a spatial filter; and a subsequent Fourier transform (of the signal section of the SPH Fourier spectrum), will generate the desired field.

In our previous experience using such a double Fourier transforming setup, a critical and difficult issue, in some cases, is the implementation of a precise spatial filter. In particular, this happens when the signal and non-signal terms in the SPH Fourier spectrum occupy small adjacent areas. An especially difficult case occurs when we employ a SPH, designed for the high efficiency generation of a Bessel beam (BB), whose phase modulation is the phase of the beam itself [10]. In this case the signal term in the SPH Fourier spectrum is a bright annular shaped field, surrounded by non-signal adjacent rings.

In this context we considered the option of taking advantage of the high resolution and programmability of available phase SLMs for the implementation of sophisticated spatial filters. Instead of using an additional SLM to implement the spatial filter, we decided to use a single phase SLM to implement both the SPH and the spatial filter by means of two sequential phase modulations of the input beam, at two different zones of the SLM. However, there was necessity of modifying the configuration of the double-Fourier transforming setup described two paragraphs above.

In the modified setup, the input illumination is provided by a laser Gaussian beam, which makes the setup convenient for generation of optical fields with a Gaussian amplitude envelope. The setup can be realized using the first modulation stage to implement a general purpose SPH together with a Fourier transforming lens, and a factor that eliminates the quadratic phase of the input Gaussian beam. The second modulation stage implements another transforming lens, together with a spatial filter that transmits (along a convenient axis, called signal axis) the hologram diffraction order that contains the information of the desired field. This spatial filter is formed by different linear phase carriers, for the signal and non-signal sections of the DPE Fourier spectrum.

As second option, the synthetic phase hologram in the first modulation zone is replaced by the phase modulation of the desired field, which is referred here to as field kinoform. This approach is useful when the Fourier spectrum of the kinoform, is formed by two disjoint parts, one of which is proportional to the Fourier spectrum of the desired field, while the second one is a non-signal or noise term [10,12–13]. For the sake of compactness, in the present report we only discuss this second option, considering the generation of Bessel beams and periodic or quasi-periodic (PQP) fields, both modulated by a Gaussian amplitude envelope.

## 2. Optical setup

The proposed setup for generation of complex optical fields, depicted in Fig. 1, includes a He-Ne laser (L), which provides the input Gaussian beam. This beam arrives to the first modulation zone (marked as A) of a pixelated reflective phase SLM, where it is modified by the first DPE, whose transmittance is composed by the following phase factors: a compensator for the quadratic phase of the input Gaussian beam, the kinoform of the desired beam, a Fourier transforming lens (of power 1/*f*), and a linear phase carrier. The Gaussian beam modulated by the kinoform will be referred to as kinoform-Gauss (KG) field. The purpose of the first transforming lens is to project, with the aid of a mirror (M), the Fourier transform of the KG field, to the second SLM modulation zone (marked as B). This projection follows the path between points A, M, and B, whose total length corresponds to the focal distance of the first transforming lens (encoded in the modulation zone A). The aim of the linear phase factor (in the modulation zone A) is to obtain a propagation axis (signal axis) that avoids the SLM depolarized light, which propagates along the specular reflection axis, represented by the dashed line that departs from the point A. The axes transmitting the useful beam information are represented by solid lines.

It is expected that the Fourier transform of the KG field includes a section that corresponds to the Fourier spectrum of the desired complex field, called signal term. It will be shown in next section, that for each kinoform to be considered, the signal and non-signal terms in the KG field Fourier spectrum present a negligible overlapping. Thus, it is possible to isolate the signal term with a spatial filter in the second modulation stage (B). The spatial filter is implemented by means of two different linear phase carriers that send the light of the two terms in the KG field Fourier spectrum, along different axes.

Since the first Fourier transforming lens is at the same plane of the KG field (zone A), the Fourier spectrum of this field presents a divergent quadratic phase modulation of power -1/*f*. Considering this fact, one of the factors in the second DPE (at zone B) is a lens with total positive power 2/*f*. Half of this power is required to compensate the quadratic phase in the Fourier spectrum, and the remaining power (1/*f*) performs another Fourier transformation on the signal portion of this spectrum. This transformation generates the desired beam at the distance *f* from the point B, along the output axis. In summary, the optical setup corresponds to a special double Fourier transform array, where both the input object (the KG field) and the first transforming lens are placed at a common plane (modulation zone A). On the other hand, both the Fourier spectrum of the KG field and the second Fourier transforming lens appear at the SLM modulation zone B.

## 3. Generating a complex field using its phase modulation

Now, we describe the theoretical basis for the technique that permits the generation of an optical field, by employing its kinoform. This description is developed in detail for Bessel-Gauss (BG) beams and briefly for PQP fields. The fundamental aspects of the theory of kinoforms, to generate a Bessel beam, have been previously reported in a succinct form [10]. In order to make this manuscript self-contained, we present a detailed review and extension of such a theory. Our discussion includes new analytical expressions, which improve the comprehension of the kinoform, and the optical field that it generates. Some differences in the present analysis are due to the assumption of a Gaussian amplitude envelope instead of the binary envelope considered in the study of Ref. [10].

#### 3.1 Generation of a Bessel-Gauss beam using its kinoform

Let us assume that we want to generate the *n*-th order BG beam, whose transverse complex amplitude, in a plane, is given by

*J*is the first class

_{n}*n*-th order Bessel function, (

*r*, θ) are polar coordinates, ρ

_{0}is the radial frequency, and

*w*is the waist radius in the Gaussian factor. To generate this beam we will display, in the first modulation of the SLM, the Bessel beam kinoform which is as phase function, where

_{0}*ph*[α] is 1 for α>0 and − 1 otherwise. As an example, the amplitude of a first order BG beam

*b*(

_{1}*r*,θ), and its phase modulation (kinoform)

*k*(

_{1}*r*,θ), are displayed in Fig. 2. The waist radius in this example is

*w*=3.5

_{0}*p*, where

*p*=ρ

_{0}

^{−1}is the asymptotic beam period. In the representation of normalized amplitudes (in the range [0, 1]), we employ the gray tones from dark to white. On the other hand, for representation of phase, the same gray tones will represent the phase range [−π,π].

Now we show analytically how the BG beam is generated using its kinoform. For this purpose, the factor *ph*[*J _{n}*(2πρ

_{0}r)] is expressed by an orthogonal Bessel series [23] for

*r*∈[0,

*L*], with the limit

*L*that will be determined below. Considering this series, the kinoform transmittance is expressed by

_{q}=λ

*/(2π*

_{q}*L*) is the radial frequency in the

*q*-th term, λ

*denote the*

_{q}*q*-th positive root of

*J*(

_{n}*x*), and the coefficients

*A*are [23]

_{q}*r*

^{2}/

*w*

_{0}^{2}). Thus, the field transmitted by the kinoform is given by

*f*(which corresponds to the KG field) is formed by multiple

_{n}*n*-th order BG beams with radial frequencies β

_{q}(

*q*= 1, 2…).

Let us assume now that we want to obtain the BG beam with radial frequency ρ_{0}, in the *M*-th order of the series in Eq. (5). This is achieved assuming that the identity β_{M}=λ_{M}(2πL)^{−1}=ρ_{0} is fulfilled, and then we determine the radius *L*=λ_{M}(2πρ_{0})^{−1}, of the orthogonal series domain. Employing this result, Eqs. (5) and (4) take the form

*f*(

_{n}*r*,θ), and their coefficients

*A*. First of all it is noted that the

_{q}*q*-th term in the sum of Eq. (6) is the

*n*-th order BG beam with radial frequency β

_{q}=(λ

_{q}/λ

_{M})ρ

_{0}. The Fourier transform of the KG field

*f*(

_{n}*r*,θ) can be expressed (omitting a constant) by

*F*(ρ,ϕ) is formed by concentric rings of radii β

_{n}_{q}=(λ

_{q}/λ

_{M})ρ

_{0}(

*q*= 1, 2, etc). The transverse profiles of such rings is determined by the convolution of each annular Dirac delta with the Gaussian exp(−π

^{2}

*w*

_{0}^{2}ρ

^{2}).

An important issue is related to the optical powers of the different beams that appear in Eq. (6). The *q*-th order beam power is proportional to the squared modulus of coefficient *A _{q}*. As a particular case, it is noticed that for the

*M*-th beam order, which corresponds to the desired beam (of radial frequency ρ

_{0}), the integrand in Eq. (7) is transformed into the non-negative function

*r*|

*J*(2πρ

_{n}_{0}r)|. This fact propitiates a relatively high value for the constant |

*A*|

_{M}^{2}and for the power of the

*M*-th order beam obtained from the kinoform. For any other beam order

*q*≠

*M*, such integrand is a real function with many sign oscillations (in the integration domain) and the constant |

*A*|

_{q}^{2}acquires a relatively low value. The Fourier spectrum of the already considered first order BG beam [Fig. 2] is shown in Fig. 3(a); and a section of the Fourier transform of the corresponding KG field

*f*(

_{n}*r*,θ) is displayed in Fig. 3(b). In these plots, where (

*u*,

*v*) denote the rectangular frequency coordinates; we displayed the square root of the amplitudes to enhance the low intensity non-signal rings, in Fig. 3(b). Previous assertions, regarding the dominance of the spectrum ring of radius ρ

_{0}, are confirmed in Fig. 3(b).

The formation of the Fourier spectrum *F _{n}*(ρ,ϕ) by concentric rings of different radii opens the possibility of applying a spatial filter that only transmits one of the rings. E. g. if the spatial filter only transmits the light from the brightest ring in the Fourier spectrum of Fig. 3(b), the application of an additional Fourier transformation to the transmitted field generates the field whose amplitude is displayed in Fig. 3(c), which is a good approximation to the desired BG beam.

The rings in *F _{n}*(ρ,ϕ) have finite transverse profiles, determined by the convolution of the annular deltas [in Eq. (8)] with a Gaussian function. Thus, it may happen that adjacent rings present some degree of overlapping. We note that in the considered case, a consequence of this overlapping is that the signal bright ring in Fig. 3(b) is thinner than the BG beam Fourier spectrum ring in Fig. 3(a). In consequence, the generated BG beam [Fig. 3(c)] present an increased waist radius, respect to original BG beam [Fig. 2(a)]. The detailed features of the Fourier spectrum

*F*(ρ,ϕ), and the BG beams that it generates, may deserve an extended discussion, which is not developed in this report. However, a brief analysis of such issues is developed in the Appendix.

_{n}#### 3.2 Periodic and quasi-periodic fields

An interesting type of non-diffractive field results from the superposition of multiple plane waves, whose propagation vectors have a common axial component. If the transverse projections (of amplitude *k _{t}*) of the propagation vectors have uniformly distributed azimuth angles, the superposition corresponds to a PQP field. Using polar coordinates, the superposition of

*Q*of such plane waves have been expressed [12] as

*C*is a normalization constant, and ρ

_{0}=

*k*/2π is the field spatial frequency. The added plane waves in Eq. (9) are modulated by phase delays exp[

_{t}*itq*(2π/

*Q*)], for

*q*= 0 to

*Q*− 1, where

*t*is an integer number that represents a discrete topological charge. Of course the option

*t*= 0 causes the elimination of such phase delays.

A convenient method to generate the field in Eq. (9) is based on the phase modulation (or kinoform) of this field [12,13]. If the complex amplitude of the PQP field is *f*(*r*,θ)= |*f*(*r*,θ)|exp[*iϕ*(*r*,θ)], then its kinoform has complex amplitude exp[*iϕ*(*r*,θ)]. We will consider the physically realizable versions of *f*(*r*,θ) and its kinoform, respectively given by

*k*(

_{G}*r*,θ) in Eq. (11) is generated by illuminating the kinoform exp[

*iϕ*(

*r*,θ)] by the Gaussian beam exp(−

*r*

^{2}/

*w*

_{0}^{2}). As examples, the field amplitudes (|

*f*(

_{G}*r*,θ)|), for parameters (

*Q*= 5,

*t*= 0) and (

*Q*= 6,

*t*= 1), are shown in Fig. 4. In these examples we considered again the waist radius

*w*=3.5ρ

_{0}_{0}

^{−1}.

The Fourier spectra of the fields *f _{G}* and

*k*, are denoted as

_{G}*F*and

_{G}*K*, respectively. The square roots of the modules of such spectra, corresponding to the field in Fig. 4(b), are displayed in Figs. 5(a) and 5(b). We displayed the square root of the modules to enhance the visibility of high diffraction orders in the kinoform Fourier spectrum

_{G}*K*. It is noted that the 6 brightest spots in this Fourier spectrum, are quite similar to the spots in the field Fourier spectrum

_{G}*F*. Although we don’t prove it here, this correspondence occurs also in the phase distribution of the Fourier spectra [12,13]. Therefore, a spatial filtering that only transmits the 6 brightest spots in

_{G}*K*followed by a Fourier transformation, will produce a good approximation of the desired field

_{G}*f*. In the example under consideration the field obtained from the kinoform Fourier spectrum, displayed in Fig. 5(c), is quite similar to the desired field [Fig. 4(b)]. As spatial filter, in this context, we employed an array of circular pupils to transmit the 6 signal spots in

_{G}*K*. The radius of such pupils was 2 times the spot waist radius.

_{G}## 4. Numerical simulations and experimental demonstration

We develop numerical simulations regarding the operation of the different phase factors that appear in the two SLM modulation zones, according to the setup description in section 2. The different parameters of the optical setup, employed in the numerical simulation, are also used in the experimental demonstration. To make the numerical simulations realistic we considered the pixelated nature of the SLM, in particular we employed the sampling resolution δ_{x}=8 μ*m*, which is the pixel pitch of the SLM used in the experimental demonstration [24]. The SLM active area, formed by 1080 × 1920 pixels, has dimensions of 8.64 mm × 15.3 mm. The input beam is obtained from a He-Ne laser, with power of 20 mW and wavelength of 633 nm. The beam waist radius, measured at the first SLM modulation zone, is *w _{0}*=1200 μ

*m*.

We consider first the generation of BG beams, using their kinoforms. The asymptotic radial period of the different beams to be generated is *p *= *w _{0}*/3.5 (≅343 μ

*m*), and the radial frequency is ρ

_{0}=1/

*p*= 3.5/

*w*(≅2.9

_{0}*mm*

^{−1}). In all the cases to discuss, we assume that the factors of the DPE, at the first SLM modulation zone, are the phase of the kinoform, the first lens with focal length

*f*= 1

*m*, and a linear phase with first order deviation angle of − 0.5°, respect to the specular reflection axis. In Fig. 6 we show (a) the amplitude of the input Gaussian beam and (b) the central portion of the DPE, for generation of the first order BG beam. Since the DPE includes the beam kinoform as a factor, such a phase element will be different for different BG (and PQP) beams.

After that the input Gaussian beam is modulated by the first DPE in the SLM, the modified field freely propagates to a distance of 1 meter, along the trajectory SLM-mirror-SLM (see Fig. 1). The square root of the amplitude of the computed propagated field, displayed in Fig. 7(a), corresponds to the KG field of the first order BG beam. The square root is employed to enhance the low intensity sections in this field. In addition to the brightest ring in this Fourier spectrum, there are other inner and outer rings, with relatively lower intensity. The above Fourier spectrum is modulated (in the second SLM modulation zone) by a phase DPE whose factors are: a second lens with focal length of 50 cm, and a composed linear phase carrier. This linear phase carrier produces a horizontal deviation angle of + 0.5°, for the bright ring (i. e. the Fourier spectrum of the signal), and a vertical deviation angle of + 0.5°, for the non-signal section of the Fourier spectrum. A half of the power of the lens compensate the quadratic divergent phase in the Fourier spectrum; and the second half of that power corresponds to a second Fourier transforming lens of focal distance *f *= 100 *cm*. The DPE phase modulation, within the square area marked in Fig. 7(a), is displayed in Fig. 7(b). The Fourier spectra patterns in Fig. 7 are horizontally decentered 4.75ρ_{0} respect to the origin, due to the linear grating used in the DPE at the first modulation zone.

After that the kinoform Fourier spectrum is modulated by the second phase DPE, the light propagated to a distance *f*, along the output signal axis (upper line in Fig. 1), produce the desired BG beam. We applied such a procedure for generation of BG beams of orders 0, 1, and 2. The amplitudes of the BG beams in Fig. 8 correspond to the numerically simulated (top) and the experimental results (bottom). The output field, in each case, is modulated by a divergent quadratic phase (of curvature radius -*f*) that is (optionally) eliminated by transmitting the field through a positive lens of power 1/*f*.

Similar simulations were performed to generate PQP fields. In this case, the annular shape of the spatial filter that was previously employed in the second SLM modulation, for generation of BG beams, is replaced by a set of appropriate circles, which only transmit the signal Fourier spectrum spots [e. g. the brightest spots in Fig. 5(b)]. The radius of the employed pupil circles is twice the waist radius of the Fourier spectrum spots. In this case, we assumed the same parameters *w _{0}* and ρ

_{0}that were employed for BG beams. Both, the numerically and experimentally generated PQP fields, with parameters (

*Q*= 5,

*t*= 0), (

*Q*= 6,

*t*= 0), and (

*Q*= 6,

*t*= 1) are displayed in Fig. 9. The top and bottom images correspond to the numerical simulations and the experimental results, respectively.

The experimentally generated BG and PQP fields show a remarkable similitude respect to the numerically simulated ones. This fact is appreciated in the displayed field amplitudes in Figs. 8 and 9. A slightly reduced contrast in the experimental amplitudes is due to the background noise in the medium quality CCD, used to capture the experimental fields. Although a quantitative assessment of the optical fields generated by the optical setup is not considered in this report, in the next section we include a brief discussion about the accuracy and efficiency of the discussed technique.

## 5. Discussion and conclusions

We evaluated the efficiency of the setup, implemented with the kinoform approach. This efficiency is given by the fraction *P _{B}*/

*P*where

_{G}*P*and

_{B}*P*are the optical powers of the generated beam and the incident Gaussian beam, respectively. The computed efficiencies, in the considered numerical simulations for generation of BG beams are in the range of 0.6 to 0.65. On the other hand, the setup efficiencies in the generation of PQP fields are in the range of 0.56 to 0.66, for the considered cases. These efficiencies are dependent on the following factors: the kinoform efficiency, the phase carrier first-order efficiency, and the zero-order SLM efficiency, due to the finite SLM pixel size.

_{G}The kinoform type SPHs employed in this report for the generation of BG beams and PQP fields present the advantage of a relatively high efficiency in comparison to other general purpose SPHs [10,12]. On the other hand, an attractive feature of the modified double Fourier transform setup (Fig. 1), it that not only the required spatial filter, but also the Fourier transforming lenses (and other components, e. g. phase compensators) can be implemented in the plane of a the phase SLM, used in the setup. The precise alignment and positioning of such optical elements is easily achieved using the computer programmability of the SLM. A minor problem of the reported setup is the requirement of a high precision positioning and angular alignment of the SLM respect to the input beam and the external mirror. These adjustments require a high quality mechanical system with translation and angular adjustments.

It was pointed out in section 2 that the beam generated by the kinoform of a BG beam of waist radius *w _{0}*, presents an approximate Gaussian modulation with an extended waist radius larger than

*w*. In the discussed examples, the BG beams generated by the kinoforms show a waist radius approximately equal to 1.5

_{0}*w*. To assess the quality of the generated beams, in Fig. 10 we display simultaneously the normalized transverse amplitudes of beams generated by the discussed kinoforms (for BG beams of orders

_{0}*n*= 0, 1 and 2) and for the theoretical BG beams of waist radius 1.5

*w*. In each plot the beam generated by the kinoform is displayed in red line and the exact BG beam is displayed in blue trace. In each case, the red trace is almost perfectly mounted on the blue trace, except at the marginal beam oscillations (i. e., for

_{0}*r*/

*p*> 5), indicating a good fitting of the two beams. If one desires a beam generated by the kinoform with an effective waist radius equal to

*w*it is necessary to employ the kinoform of a BG beam with a waist radius smaller than

_{0}*w*.

_{0}The quality of the PQP fields generated (in numerical simulations) with the kinoforms of such fields, using the proposed setup, is also remarkable high. In this case, the high similitude of the Gaussian signal spots in the KG Fourier spectrum, respect to the spots in the PQP field spectrum [see e. g. Figure 5(a) and Fig. 5(b)], propitiates a low error in the beams generated by the kinoforms, without an alteration in the waist of the Gaussian envelope.

In the numerical simulations and the experimental demonstration we employed linear phase carriers that deviate the signal axes (in both SLM modulation zones) by 0.5 degrees, respect to the specular reflection axes (dashed lines of Fig. 1). Such small angles, required to obtain high diffraction efficiencies of the linear phase carriers, imposed restrictions in the optical setup, e.g., the necessity of using a large focal length (*f *= 100 *cm*.) in the Fourier transforming lenses.

The Optical setup implements two Fourier transform operations with transforming lenses that are displayed at the same planes of the DPEs that are Fourier transformed. Thus, the generated Fourier transforms are modulated by quadratic phases in the radial coordinate. In particular the quadratic phase at the Fourier domain of the first DPE is compensated at the second SLM modulation zone.

There is a practical limit to the order of the BG beam that can be generated in the discussed setup, due to the SLM finite pixel size δx. Denoting the maximum radius of the ring shaped BG beam Fourier domain as *R _{M}*, the length of a section in this ring covering a phase modulation of 2π radians is 2π

*R*/

_{M}*q*(where

*q*is the BG beam order). Since this section must be sampled at least by two pixels of the SLM, the maximum allowable order

*q*is limited by the integer part of πR

_{M}/δx, which is much larger than 1 since

*R*>>δx.

_{M}Our conclusion is that the use of a pixelated phase SLM to implement multiple phase elements, in a double Fourier transforming optical setup, as the one discussed here, is quite feasible and useful. Using the same setup it is possible to implement alternate approaches for generation of optical fields, complementary to the kinoform approach. E. g. it is possible to replace the kinoform, as a device that encodes the desired optical field, by other DPEs, for example a general purpose synthetic phase hologram [1–17].

## Appendix

In the theoretical analysis of the kinoform technique for generation of BG beams, it was noticed that the signal ring in the Fourier spectrum of the KG field showed a smaller thickness than the BG Fourier spectrum ring. This fact has an influence on the effective width of the generated BG beam. Here we extend our analysis to understand this feature of such Fourier spectra.

We can represent the KG field and the *q*-th term in the series of Eq. (6), by means of the algebraic expressions

*G*(ρ) and

*G*(ρ) are the n-th order Hankel transforms of the radial functions

_{q}*g*(

*r*) and

*g*(

_{q}*r*), respectively.

It is noticed that the radial factor *g _{q}*(

*r*) [and therefore its Hankel transform

*G*(ρ)] is dependent on the radius (

_{q}*L*) of the domain where the series representation [Eq. (6)] is valid. This dependence is observed explicitly in the expressions for spatial frequencies β

_{q}and the coefficients

*A*[see Eqs. (3) and (4) and related text]. Although

_{q}*L*could be freely determined, we adopt the reasonable criterion

*L*≥ 2

*w*. Recalling that

_{0}*L*=λ

_{M}(2πρ

_{0})

^{−1}, and adopting the restriction

*w*=3.5ρ

_{0}_{0}

^{−1}, and the order

*n*= 1 for the desired BG beam, the above restriction for

*L*is fulfilled for

*M*≥ 14. In agreement with this condition, in the following steps we assume

*M*= 15, for which we obtain the real-valued coefficients

*A*[computed with Eq. (7)] that are displayed in Fig. 11(a).

_{q}For the first-order BG beam under discussion, the computed radial modulation *G*(ρ) of the KG field Fourier spectrum, is displayed in Fig. 11(b). On the other hand, the radial modulations *G _{q}*(ρ), for

*q*= 14, 15 and 16, are displayed in Fig. 11(c). The Fourier spectra in Fig. 11 have been normalized respect to the peak value of

*G*(ρ). It is noted that the separated functions

*G*(ρ) have Gaussian shapes, in agreement with Eq. (8). But this is not the case for the shape of the radial factor

_{q}*G*(ρ) in Fig. 11(b). The reason is that

*G*(ρ) is the superposition of the multiple Fourier spectra

*G*(ρ). In the case under discussion, the form and height of

_{q}*G*(ρ) can be essentially obtained by the superposition of the spectra

*G*(ρ),

_{14}*G*(ρ), and

_{15}*G*(ρ), displayed in Fig. 11(c), which are the more significant among all the

_{16}*G*. The reduced width in the Fourier spectrum

_{q}^{’}s*G*(ρ), can be explained by the sign inversion of the functions

*G*(ρ) and

_{14}*G*(ρ).

_{16}Employing the kinoform Fourier spectrum, with radial modulation *G*(ρ), it is not possible to obtain exactly the initially desired beam [Eq. (1)]. But we can obtain a good approximation of this beam, by transmitting (through a spatial filter) only the central section of this Fourier spectrum, limited by the zeros of *G*(ρ) that are closest to frequency ρ=ρ_{0}. The radial modulation, *G _{f}*(ρ), of the filtered kinoform Fourier spectrum, in the discussed case, is displayed in Fig. 12(a). The approximate BG beam is obtained by the inverse Fourier transformation of the filtered Fourier spectrum. The radial modulation of this generated beam,

*g*(

_{r}*r*); computed as the n-th order inverse Hankel transform of the filtered function

*G*(ρ), is displayed in Fig. 12(b). In this computation we employed the non-normalized version

_{f}*G*(ρ), which generates a field

_{f}*g*(

_{r}*r*) with a peak amplitude that is approximately 2.22 times the peak amplitude (=1) of the Gaussian beam used to illuminate the beam kinoform. This relatively high amplitude of the generated beam is one of the remarkable features of the kinoform-based method. We have noted that the amplitude envelope in the generated beam have an approximate Gaussian shape, whose waist is larger than the waist (

*w*) of the considered input Gaussian beam. For comparison, the radial profile of the initially specified BG beam (of waist

_{0}*w*) is displayed in Fig. 12(c). This result is explained by the narrowing of the spectrum

_{0}*G*(ρ) [and

*G*(ρ)] caused by the influence of the Fourier components

_{f}*G*(ρ), adjacent to

_{q}*G*(ρ), whose coefficients present the sign inversion that is observed in Fig. 11.

_{M}The Fourier spectra of KG fields, for different *n*-th order BG beams, present similar features that the first order case, discussed above.

## Funding

Consejo Nacional de Ciencia y Tecnología (719190).

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**24. **We employed the electrically addressable phase SLM, model “Pluto”(HOLOEYE Photonics AG), with format of 1080 × 1920 pixels and pixel size of 8 microns.