An azimuthally-modified linear phase grating: Generation of varied radial carpet beams over different diffraction orders with controlled intensity sharing among the generated beams

Diffraction gratings are important optical components and are used in many areas of optics such as in spectroscopy. A diffraction grating is a periodic structure that splits and diffracts the impinging light beam into several beams travelling in different directions. The diffracted beams from a grating are commonly called diffraction orders. The directions of the diffraction orders depend on the grating period and the wavelength of the impinging light beam so that a grating can be used as a dispersive element. In the diffraction of a plane wave from a conventional grating, the intensities of diffracted beams decrease with increasing order of diffraction. Here, we introduce a new type of grating where in the diffraction of a plane wave, the intensity of a given higher order diffracted beam can be higher than the intensity of the lower orders. We construct these gratings by adding an azimuthal periodic dependency to the argument of the transmission function of a linear phase grating that has a sinusoidal profile and we call them azimuthally-modified linear phase gratings (AMLPGs). In this work, in addition to introducing AMLPGs, we present the generation of varied radial carpet beams over different diffraction orders of an AMLPG with controlled intensity sharing among the generated beams. A radial carpet beam is generated in the diffraction of a plane wave from a radial phase grating. We show that for a given value of the phase amplitude over the host linear phase grating, one of the diffraction orders is predominant and by increasing the value of the phase amplitude, the intensity sharing changes in favor of the higher orders. The theory of the work and experimental results are presented. In comparison with the diffraction of a plane wave from radial phase gratings, the use of AMLPGs provides high contrast diffraction patterns and presents varied radial carpet beams over the different diffraction orders of the host linear phase grating. The resulting patterns over different diffraction orders are specified and their differences are determined. The diffraction grating introduced with controlled intensity sharing among different diffraction orders might find wide applications in many areas of optics such as optical switches. We show that AMLPG-based radial carpet beams can be engineered in which they acquire sheet-like spokes. This feature nominates them for potential applications in light sheet microscopy. In addition, a detailed analysis of the multiplication of the diffraction pattern of an AMLPG by the 2D structure of a spatial light modulator is presented. The presented theory is confirmed by respective experiments.

gratings, the intensities of diffraction orders decrease with increasing the number of order of diffraction. In the near-field regime, superposition of diffraction orders form self-images and sub-images of the grating's structure at certain propagation distances. This effect is known as Talbot effect 1 . In addition, the intensity pattern over a plane includes the propagation axis and the grating vector (say the longitudinal plane) that is called the Talbot carpet 2 . In the far-field regime, the diffraction orders appear as the impulses of the Fourier transform of the grating's transmission function.
The linear gratings have numerous applications in optics and other areas of sciences and technologies such as in astronomical spectroscopy, moiré fringe technique including moiré deflectometry and moiré topography 3 , interferometry 4-6 , lithography 7,8 , strain and stress analysis 9 , displacement measurement 10 , optical alignment technique, color printing 11 , and 3D displays 12,13 . In recent two to three decades amplitude and phase gratings with topological defects have found serious applications in generating vortex beams and/or characterizing such beams [14][15][16][17][18][19][20][21] . The use of phase hologram gratings instead of the conventional one has some advantages [22][23][24][25] . For instance, a computer generated hologram phase grating provides the implemention of a desired phase map on the incident wave in a controlled way 26 .
Another specific type of grating is radial grating. A radial grating has a periodic structure in the azimuthal direction in polar coordinates. Unlike conventional gratings, there is no diffraction order in diffraction from a radial grating. However, the diffraction of a plane or Gaussian wave from radial gratings shows other interesting features. On illuminating an amplitude radial grating by a spatially coherent light beam the Talbot carpet is formed on transverse planes 27 . The conventional Talbot carpet is formed on the longitudinal plane when a grating is illuminated by a plane beam 2,28,29 . It has also been shown that the plane boundaries between the optical regimes including the geometric shadow and near-field and far-field diffraction regimes have acquired curvature 27 . The diffraction of a vortex beam from an amplitude radial grating presents a simple way for measuring the topological charge alteration 30 . That study also presents a Poisson-Arago spot-like effect on the optical axis when an amplitude grating is illuminated by a vortex beam in which the number of grating spokes is equal to the topological charge of the incident beam. More precisely, a needle-like optical beam forms along the optical axis in far-field diffraction. The diffraction of a plane wave from a radial phase grating was also theoretically and experimentally investigated. It was shown that the diffraction of a plane wave from radial phase gratings is a simple way to generate radial carpet beams that are nondiffracting, accelerating, and shape preserving 31 . These beams have unprecedented patterns that are shape-invariant during propagation. The patterns can be easily turned into tunable 2D optical lattices. We also proposed a new solution for the paraxial Helmholtz equation and introduced combined half-integer Bessel-like beams 32 . It was shown that a huge family of beams can be produced in diffraction from designed radial structures and not necessarily from radial gratings. In that work spatially asymmetric, radial carpet, petallike, and twisted-intensity ringlike vortex beams were introduced through detailed theory and suitable practical approaches were proposed to generate each of them.
In this work, we integrate the features of linear and radial gratings to introduce a new grating that, in addition to the properties of linear and radial gratings, has additional properties. We construct this type of grating by adding an azimuthal periodic dependency to the argument of the transmission function of a linear phase grating that has a sinusoidal profile and we call them azimuthally-modified linear phase gratings (AMLPGs). In the diffraction of a plane wave from an AMLPG, we observe different diffraction orders in which the pattern of each order is very similar (but not exactly equal) to the diffraction pattern of a radial grating. Unlike the conventional gratings, here the intensity of a given higher order diffracted beam can be higher than the intensities of the lower orders. We show that the intensity sharing among different diffraction orders of an AMLPG can be adjusted by the value of the phase amplitude of the host linear grating. This kind of grating might find wide applications in many areas of optics such as optical switches. The diffraction of a plane wave from AMLPGs is formulated and using a spatial light modulator (SLM) the respective experimental works are presented. In comparing with the diffraction of a plane wave from radial phase gratings 31 the use of AMLPGs provides a set of high contrast and varied radial carpet patterns over the different diffraction orders of the host linear phase grating. Since an SLM has a two dimensional periodic structure it multiples the diffraction pattern of the AMLPG. A detailed formulation is also presented and the theoretical predictions are verified by experiments.
It is worth noting that, as the transmittance of an AMLPG is not the product of the transmittances of the radial and linear gratings, the diffraction pattern from an AMLPG is not the convolution of the diffraction patterns of radial and linear gratings. However, the transmittance of an AMLPG imposed on an SLM is the product of an ideal AMLPG and the two-dimensional periodic structure of the SLM, therefore the resulting diffraction pattern is the convolution of the diffraction pattern of the ideal AMLPG and the two dimensional impulses of the SLM.

Results
Formulation of plane wave diffraction from an AMLPG. Here, we formulate the diffraction of a plane wave from an AMLPG. When the phase structure of a linear phase grating with a sinusoidal profile hosting an additional radial phase grating, it can be considered as "azimuthally-modified linear phase grating (AMLPG)". Some AMLPGs with almost sinusoidal phase profiles are shown in Fig. 1. Since a radial phase grating is periodic in the azimuthal direction and has a phase singularity at the origin, an AMLPG has also the same property.
We introduce an AMLPG with the following transmission function: where γ, l, d, and ρ θ ( , ) are the amplitude of the phase variation, the number of spokes of the radial part of the grating, the period of the linear part of the grating, and the polar coordinates in the input plane, respectively.
As is apparent from Eq. 1, the transmission function of the AMLPG is not the product of the transmission functions of two distinct radial and linear gratings 33 . Therefore the diffraction pattern from an AMLPG is not the convolution of the diffraction patterns of the radial and linear gratings. Below we formulate the diffraction of a plane wave from an AMLPG.
Using the Jacobi-Anger identity 34 where J s is the s-th Bessel function of the first kind, the grating's transmission function, Eq. 1, can be rewritten in the following form: By illuminating this phase structure with a plane wave, the complex amplitude of the light field after the structure is given by 2 is the wave-number, and ϕ r ( , ) are the polar coordinates on the output plane.
By substituting Eq. 3 in Eq. 4, we have Equation 5 can be solved respect to the azimuthal variable. First we use the following variable transformations in the output plane: www.nature.com/scientificreports www.nature.com/scientificreports/ and rewrite Eq. 5 in the following form: s s Now using Eqs 2 and 9 reduces to the following form:   Now using the following integral identity 35 : the resulting light field can be written in the following form: Here s shows the number of the diffraction order of the AMLPG, and the corresponding light field is  This is the main result, showing the complex amplitude of the diffracted beam from an AMLPG and indicates that all the diffraction patterns forming over the individual diffraction orders are different.
It is worth noting that unlike the case of diffraction from the product of a given function and a linear grating in which the spectrum of the function is multiplied over different diffraction orders of the grating, here such behavior does not occur. The reason is that the structure of an AMLPG is not separable into a linear grating structure and another definite function.
Equation 17 can be considered as a summation over individual diffraction orders s s s s Similar to the diffraction of a plane wave from a radial phase grating 31 , here again we see that each of individual diffraction patterns has a radial form with a considerable structural complexity. All of the patterns are shape invariant under propagation. Therefore we consider each of them a "radial carpet beam". In one of the next subsections we will show that for a specific value of the phase amplitude of an AMLPG having a given value of l, one of the individual diffraction patterns will get the same intensity distribution of the radial carpet beam produced directly in the diffraction of a plane wave from a radial phase grating with the same spokes number, l.
Relative rotation of the resulting patterns over ±s diffraction orders. We show that the diffraction patterns generated over a pair of diffraction orders ±s are similar in form but have a relative rotation with respect to each other.
For a given positive value of s, substituting s with −s in Eq. 17 we have the following equation: and using Eq. 7 we have   www.nature.com/scientificreports www.nature.com/scientificreports/ Controlled intensity sharing among different diffraction orders: Effect of γ on the resulting patterns. Here we show that the intensity of an incident beam on an AMLPG can be divided among different diffraction orders with desired proportions. For an AMLPG, unlike the conventional gratings, the intensity share of the higher diffraction orders may be larger than the intensity on the lower diffraction orders, when given values are chosen for the phase amplitude of the grating, γ. Figure 2 shows the calculated diffracted intensity patterns for an AMLPG with = l 10 spokes and different values of γ at a distance = z 555 cm. By selecting the values of 2.4048, 3.8317, and 5.1356 for the arguments of J 0 , J 1 , and J 2 their first zeros appear, respectively, and 5.5201 leads to the second zero of J 0 . As is apparent, the visibility of the patterns and intensity sharing among different diffraction orders depend to the value of γ.
2 4048 the diffraction orders of = ± s 1 are more visible and they have the maximum values of the intensities between the diffraction orders.
3 8317 and γ = π 3 2 the diffraction orders for = ± s 3 are the dominant patterns and have the maximum share of the intensity.
This means that by increasing the value of γ the energy of the incident light transfers to the higher diffraction orders. As is apparent, at larger radii, all individual diffraction patterns have the same number of spokes equal to the number of spokes of the AMLPG, but at smaller radii, the fine structure of the patterns strongly depends on the value of s. Also, it is seen that for larger values of s, the spoke patterns become needle-like and therefore the path of each needle-like pattern under propagation becomes a light sheet. These features might find some application in light sheet microscopy. Now, we estimate the diffraction coefficients of an AMLPG by calculating the percentage of the incident power flows among different diffraction orders. This is done by calculating the ratios of the mean values of the intensities over the diffracted patterns to the mean value of the incident beam intensity. The mean value of the intensity over www.nature.com/scientificreports www.nature.com/scientificreports/ each of the individual patterns, separated by the dashed lines in the first row of Fig. 2, was calculated. Table 1 shows the results of the intensity sharing between different orders for some selected values of γ.
Using Eq. 17, we can directly calculate the diffraction coefficients of an AMLPG as the ratio of the power of each order of diffraction P s to the power of the incident beam P i  Table 1 were derived using Eq. 23.
The following results in Table 1 are worth noting: • For γ = π 2 and γ = .
2 4048, the diffraction orders with = ± s 1 have the maximum values of intensity sharing of 32% and 27%, respectively. • For γ π = the diffraction orders with = ± s 2, have the maximum values of intensity sharing of 23.5%. • For γ = .
3 8317 and γ = 2 4048 and γ = . 5 5201 the intensity sharing of the DC terms, with = s 0, are zero. These values of γ correspond to the first and second zeros of J 0 .
• For the values of γ = .
3 8317 corresponds to the first zero of J 1 and the value of γ = . 5 1356 corresponds to the first zero of J 2 .

Comparison of the diffraction patterns of an AMLPG and a radial phase grating.
Here we compare the diffraction pattern of a plane wave diffracted directly from a radial phase grating 31 with the diffraction pattern of an AMLPG. The diffracted light field in the diffraction of a plane wave from a radial phase grating having a sinusoidal transmission function can be written as 31 the two equations will be the same. It is apparent that this is the case for γ = s. This means that the diffraction pattern from a radial phase grating, known as the "radial carpet beam" 31 , and the s-th diffracted order of an   Table 1. Calculated ratios of the mean intensity of the diffraction order to the mean intensity of the incident beam for the diffraction patterns illustrated in Fig. 2. The red numbers show the maximum values of the intensity sharing in each row. By increasing the value of γ, the maximum values shift toward the higher orders.
www.nature.com/scientificreports www.nature.com/scientificreports/ AMLPG are similar when the phase amplitude and spokes number of the AMLPG are equal to the spokes number of the radial phase grating.
In Fig. 3 the calculated diffraction pattern (or equally the whole spectrum) of an AMLPG (left column) and the diffraction pattern of a plane wave diffracted directly from a radial phase grating (right column) are illustrated. As it is seen for UU s s and the s-th diffraction order has the maximum value of intensity between all the diffraction orders. From Figs 2 and 3, it is also seen that by increasing the value of γ, the higher diffraction orders get a larger fraction of the energy of the incident beam.
Since the value of intensity at different diffraction orders can be adjusted with the value of the grating phase amplitude, γ, this feature can be used in optical switching.

Replication of the spectrum of an AMLPG by an SLM structure.
Here we investigate the replication of the spectrum of an AMLPG by the SLM structure under diffraction. First we consider the diffraction of a plane wave from an SLM when a secondary structure is not embedded in it.
Assume that the structure of the SLM is a rectilinear 2D grating with the following transmission: x y where f x and f y show two one dimensional (1D) linear gratings placed together at right angles. The Fourier transform of f x y ( , ), apart from a multiplicative factor, is given by 36 Now suppose that the structure of an AMLPG with a transmission function of ρ θ T( , ) is imposed on the SLM and a plane wave is diffracted by it. In this case the light field immediately after the SLM is given by the multiplication of f x y ( , ) and ρ θ T( , ), and the resulting diffraction pattern or equally the corresponding spatial spectrum is obtained as total The diffracted light field distribution or equally the spectrum of the AMLPG is replicated by the impulses of the SLM. The replicated spectrum by the m n ( , ) impulse of the SLM can be written as m n m n , , This means that the spectrum of an AMLPG is replicated by each of the diffraction impulses of the SLM. Using Eqs 18 and 27 in Eq. 29, the distribution of the diffracted light field corresponding to the m n ( , ) impulse of the SLM after a propagation distance of z is given by Figure 3. Comparing the diffraction pattens of an AMLPG and a radial phase grating having the same number of spokes. Left column, calculated diffracted intensity patterns obtained in the diffraction of a plane wave from an AMLPG with = l 10 at = z 555 cm for γ equal to 1, 2, 3, 4, and 5. Right column, calculated diffracted intensity patterns obtained in the diffraction of a plane wave from a radial grating with = l 10 at = z 555 cm for γ equal to 1, 2, 3, 4, and 5 31 . In each row, for γ = s , the s-th diffraction order of the AMLPG has the maximum value of intensity between all the diffraction orders and its diffraction pattern and the illustrated radial carpet pattern at the right column has the same form. The intensity over the patterns is normalized to the intensity of the incident beam.
where s shows the order of diffraction of the AMLPG imposed on the SLM.

Discussion
We show that by adding an azimuthally periodic term into the argument of a linear phase grating, say θ l cos( ) in Eq. 1, and by adjusting the value of γ, one can control the intensity sharing between different diffracted beams (see Figs 2 and 3). The theoretical perditions show that, in order to have maximum share of energy on a higher order diffraction pattern, we need to use an SLM with a large value of phase variation. The proposed method for controlling the intensity sharing between different diffraction orders can be implemented with the aid of other additional phase terms in Eq. 1. For example, by replacing θ l cos( ) with the phase function of a zone plate