## Abstract

In this work we present the theoretical and experimental results of the investigation of the diffraction of a spatially coherent and collimated white light beam from radial amplitude gratings. Theoretical part of the work is resolved with the Fresnel-Kirchhoff integral. In the experimental part, a collimated wave-front of a white light beam emitting from an LED lamp is transmitted through a radial amplitude grating. We digitally record the diffraction pattern in various distances from the grating using a CCD camera. The resulted diffraction pattern that we call it “Colorful radial Talbot carpet at the transverse plane” has a shape-invariant form under propagation. The other significant aspects of this pattern are the existence of a quite patternless dark area located around the optical axis and an intense rainbow-like ring in the vicinity of the patternless area. The rainbow color changes radially from the violet in the vicinity of the patternless area to red by increasing the radius in which the purity of the colors in the inner side is dominant. We call this phenomena “diffraction-based rainbow”. In addition, the transverse plane Talbot carpet pattern consists colorful self-images of the grating’s spokes at the larger radii. The theoretical calculations and experimental results verify each other completely. The introduced diffraction-based rainbow can be utilized in spectrometry.

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

## 1. Introduction

The periodic structures, commonly named the diffraction gratings, are the most widely used devices in optics. When a light beam passes through a grating, a fantastic behavior can be observed at the near-field distances that is known as the Talbot effect or self- imaging phenomenon [1]. The term “self-imaging” is used because, in the diffraction of a highly spatially coherent plane wave from a grating, the light field intensity profile immediately after the grating reproduces itself at certain propagation distances without utilizing any imaging systems. Formation and specifics of the self-imaging phenomenon depend strongly on the gratings structure [2]. The common types of the gratings are one dimensional (1D) periodic structures. The amplitude and/or phase of the incident beam can be modulated by a grating when the beam transmits or reflects through it. Therefore, the gratings are categorized into amplitude and/or phase and transmissive or reflective types. Furthermore, the gratings are determined by the spatial profile changes on the wavefront. For instance, the transmission or reflection profile of the gratings can be divided into the sinusoidal, binary, triangular, sawteeth, etc [2]. Based on the fine structure of the gratings, their self-images and far-field diffraction patterns change. The far-field diffraction pattern of a grating reveals its spatial frequency spectrum. The direct superimposition of two gratings and both near-and far-field diffraction patterns of the gratings have found many applications in sciences and technology; including in the displacement sensing and moiré deflectometry [3], interferometry [4], lithography, and spectrometry [5]. Another type of gratings is the radial gratings that consists of lines that are emerged from a single point and the angle between all adjacent lines are equal. Due to the geometry of the radial gratings, their spatial period boosts with increasing the radius. The monochromatic light diffraction over radial gratings using the Fresnel-Kirchhoff integral reveals very surprising results [6]. Unlike the diffraction from conventional gratings, the geometric shadow, near-field and far-field diffraction regimes are mixed at various propagation distances and the boundary of three mentioned regimes gets curved surfaces instead of parallel planes. In fact, for each transverse plane and any arbitrary distance from the radial grating, the far-field diffraction regime is always located at the vicinity of the optical axis. While the intermediate and large radial distances belong to the near-field and geometric shadow regimes, respectively. In other words, in the diffraction of a plane wave from a radial grating, a structureless and roughly dark volume surrounded by a paraboloid forms the far-field diffraction region, in which the vertex of the paraboloid coincides the singular point of the grating and its axis lies on the optical axis (see Ref. [6]). It has also been shown that in contrast of the diffraction pattern for the conventional gratings with Talbot carpet extension along the longitudinal plane [7–9], the Talbot carpet appears in the transverse planes for the diffraction fromradial gratings. Investigation of the diffraction of monochromatic light from radial phase gratings using Fresnel-Kirchhoff integral and with the direct solution of the wave equation was also reported [10, 11]. The first work introduces a new class of non-diffracting, self-healing and accelerating beams that are called “radial carpet beams” [10]. In the second work, in addition to the previously proposed radial carpet beams, the petal-like, intensity asymmetric, and ring-like vortex beams that may carry optical angular momentum are introduced. This huge variety of beams are defined as “combined half integer Bessel-like beams” [11]. In another work, the diffraction of vortex beams from radial amplitude gratings was considered for determining magnitude and sign of the topological charge of an incident beam. That work illustrates a beautiful effect that is resemble to the Arago-Poisson spot [12] and we call it “Arago-Poisson-like light-bar”. The effect shows a maximum intensity on the optical axis over the far-field diffraction pattern of a doughnut-shaped beam with a topological charge equal to the spoke number of the grating, in which the intensity on the optical axis is zero before the diffraction [13]. This feature that appears in a sequence of passing of a plane wave through a pair of linear forked and radial gratings, presents a new intensity-based way for information transmission using combined half integer Bessel-like beams [13, 14].

Here in this work, for the first time, we investigate the diffraction of spatially coherent and collimated white light from the radial amplitude gratings. We theoretically and experimentally show that the diffraction pattern under multichromatic illumination has a colorful form, and just like the monochromatic illumination, the resulted colorful diffracted pattern is fully shape-invariant under propagation. The central part of the pattern is roughly dark and there is an intense “rainbow-like ring” at the vicinity of the dark area. As in a rainbow the spectrum of the sun light appears in the sky by reflection, refraction, and dispersion of the light beam in water droplets [15], we call this phenomenon “diffraction-based rainbow”. Also, remarkable colorful self-images of the grating’s spokes appear at the larger radii. The formation of rainbow-like ring in the central area of the diffraction pattern makes the phenomenon an appropriate candidate for the spectrometry.

## 2. Theory

#### 2.1. Diffraction of a plane wave from a radial grating

First the theory of diffraction of a monochromatic spatially coherent plane wave from radial structures is briefly reviewed, then the case of multichromatic illumination of the structures will be presented. A structure is defined as a radial structure when there is not radial dependence in its transmittance [6]. Considering $\left(r,\theta \right)$ as polar coordinates, a radial structure’s transmittance can be written as follows:

*t*is the

_{n}*n*th Fourier series coefficient. It needs to be remarked that, in the theory, for the radial gratings and illuminating beams we assumed unlimited lateral extensions. It is worth noting that in a similar study, the far-field diffraction of a plane wave from circular sectors and some related apertures by considering limited lateral extension for the apertures were done in Ref. [16]. Because of the limited lateral extension of the apertures in Ref. [16], the results of Refs. [6] and [16] are different, especially on the optical axis.

By passing a monochromatic coherent plane light beam through this structure, using the Fresnel-Kirchhoff integral, the complex amplitude of the diffracted light after a propagation distance of *z* can be written as [6]

*λ*is the wavelength, and is a dimensionless parameter. For convenience the phase factor of ${e}^{ikz}$ is ignored in Eq. (2). As a special form of Eq. (1), we consider ${t}_{n}={t}_{-n}$ which is commonly encountered, and Eq. (1) reduces to where

*t*in this equation is equal to $2{t}_{n}$ in Eq. (1). Accordingly, Eq. (2) can be rewritten as

_{n}*z*and

*λ*and explicitly depend on $\mathcal{R}$. This fact shows that the diffracted pattern from a given radial structure is shape-invariant under propagation and is also shape-invariant with the variation of wavelength of the illuminating beam (see also [10] and [11]). Hereafter, as $\mathcal{R}$ is a dimensionless parameter, we call the wavelength- and propagation-independent intensity profile of $I\left(\mathcal{R},\theta \right)$ as “dimensionless shape-invariant diffraction pattern”. We will omit the term “dimensionless” when we deal with $I\left(r,\theta ,z;\lambda \right)$ instead of $I\left(\mathcal{R},\theta \right)$. A given ordered pair $\left(\mathcal{R},\theta \right)$ corresponds to a unique point on the shape-invariant diffracted pattern. As $\mathcal{R}$ is a function of

*z*, here we show that the shape-invariant diffraction intensity pattern, $I\left(r,\theta ,z;\lambda \right)$, expands under propagation. Trajectory of a given point on the diffracted pattern, having a given value of $\mathcal{R}$, under propagation can be determined by in which we used the definition of $\mathcal{R}$ presented in Eq. (3). This equation shows that the resulted shape-invariant diffracted pattern expands by a factor $\sqrt{z}$ under propagation.

As an important example of the radial structures we consider an amplitude radial grating with a sinusoidal transmission function

where*m*is the number of spokes of the grating. Comparing this transmittance with Eq. (4) we see that ${t}_{m}={t}_{0}=\frac{1}{2}$ and all other coefficients vanish, then Eq. (5) reduces to

Using Eq. (8), in the first row of Fig. 1, dimensionless and wavelength- and propagation-independent intensity profile of the diffracted light, $I\left(\mathcal{R},\theta \right)$, from a sinusoidal amplitude radial grating with $m=50$ is illustrated. Because $\mathcal{R}$ is a dimensionless parameter, in the resulted shape-invariant intensity pattern $I\left(\mathcal{R},\theta \right)$, both *x* and *y* coordinates are dimension-free.

In the second row of Fig. 1, using Eq. (6) expansion of three definite rings determined on the shape-invariant Talbot carpet pattern of the first row of Fig. 1 are plotted in terms of propagation distance, *z*, for *λ* = 470 nm.

Plots of the third row of Fig. 1 show expansion of the inner ring of the shape-invariant Talbot carpet pattern in terms of propagation distance, *z*, for three different wavelengths. In the following we present the signifying reasons on the use of indices of $\mathcal{R}$ on Fig. 1.

Now, considering the dependence of $\mathcal{R}$ on *z* and *λ* and using Eq. (6), in Fig. 2 we illustrate the calculated intensity patterns of the diffracted light beam from two sinusoidal radial gratings with spoke numbers of $m=20$ and $m=50$ for three typical illuminating wavelengths of 470 nm, 530nm, and 700 nm at a propagation distance of *z* = 1 m.

In subsection 2.3 we will present formulation of the diffraction of a multichromatic plane wave from radial gratings, and for a given spectral density of the incident beam we will present an analytical approach to predict colorful diffraction pattern at the transverse plane. But, here we simply simulate the diffraction of a multichromatic light beam from a radial grating by combination of intensity profiles of diffraction patterns calculated for three main monochromatic wavelengths of 470 nm, 530 nm, and 700 nm. In the last row of Fig. 2, the intensity profiles of three main colors illustrated in the first three columns were combined with equal weights using MATLAB program and the corresponding resulted colorful Talbot carpet patterns at the transverse planes are illustrated. In the background Visualization 1, Visualization 2, and Visualization 3 shape-invariant feature of the colourful Talbot carpet pattern under propagation are illustrated for three amplitude gratings having different spoke numbers.

For better illustration of the details of the diffraction patterns at the vicinity of the optical axis, *r* = 0, the patterns of the determined slices in Fig. 2 are projected from the polar to Cartesian coordinates and are re-illustrated in the first to forth rows of Fig. 3. The last row of Fig. 3 shows the intensity profiles over the diffracted patterns, along a line in front of a bright spoke of the grating, for all three monochrome diffracted patterns. As is seen, there are patternless areas at the vicinity of the optical axis and the main lobes of the intensity for lower wavelengths are located at the inner radii.

#### 2.2. Characterization of the dimensionless shape-invariant Talbot carpet pattern

It was recently reported that in the diffraction of a monochromatic and spatially coherent light beam from a radial grating the shadow and near- and far-field diffraction areas all appear over a given transverse plane in which the optical axis is surrounded by the far-field regime, the shadow area is located at the farther radii, and the near-field area fills the median area [6]. In fact, due to the central singularity of the radial grating, the plane boundaries between the optical regimes have acquired curvature. The same aspects are seen in the multichromatic illumination, see Figs. 2 and 3, and
Visualization 1,
Visualization 2, and
Visualization 3. As is apparent from the illustrated diffracted patterns, at the shadow area (larger radii), the diffracted patterns have the same structure of the gratings. Over a given diffraction pattern recorded at a distance *z* from the grating, the transition from the shadow area to the near-field starts with the duplication of azimuthal spatial frequency that is defined by

By getting closer to the optical axis, the first self-imaging effect occurs at the following radius [6]:

*n*th self-imaging occurs at

Here again ${\mathcal{R}}_{{T}_{1}}$ and ${\mathcal{R}}_{{T}_{n}}$ show the radii of the first and *n*th self-images of the grating’s spokes on the dimensionless shape-invariant diffracted pattern, respectively.

It is worth noting that half-Talbot images of the grating’s spokes form over the transverse plane when a half-period shift in the location of the spokes occurs. In Fig. 3 location of the first half-Talbot images are determined by *r _{hT}*.

The near-field regime begins with the radial colorful ring that appears after the patternless dark area. Therefore the inner-border of the near-field regime is given by [6]

where ${\mathcal{R}}_{in}$ corresponds to the same ring on the dimensionless shape-invariant diffracted pattern. As it is shown all parameters ${\mathcal{R}}_{out}$, ${\mathcal{R}}_{{T}_{1}}$, ${\mathcal{R}}_{{T}_{n}}$, and ${\mathcal{R}}_{in}$ are*λ*and

*z*independent. Over the dimensionless shape-invariant Talbot carpet pattern of Fig. 1 (first row), three rings with radii of ${\mathcal{R}}_{out}$, ${\mathcal{R}}_{{T}_{1}}$, and ${\mathcal{R}}_{in}$ are illustrated by red line circles.

In Fig. 3 locations of the corresponding ${r}_{out}$, *r _{T}*, and

*r*for each of the main colors are determined by the vertical dashed lines.

_{in}The total number of Talbot images, *N _{T}*, over a given diffraction plane, is limited by the value of the grating’s spokes number,

*m*, and it can be determined from ${r}_{{T}_{n}}={r}_{in}$, where ${r}_{{T}_{n}}$ shows the radius of the

*n*th Talbot image. Therefore

In the patterns of the forth row of Fig. 3, it is seen that the intensity of the white color of the shadow image in front of bright spokes ($\theta =2q\frac{\pi}{m},q=0,1,2,\mathrm{...}$) tends to zero by getting close to the small radii and at the location of the half-Talbot image it reaches to zero. As the last disappearing color has the lowest wavelength, the bright spokes disappear with the blue color at the vicinity of half-Talbot radius. After the dark area of the half-Talbot image, at the radii closer to the optical axis, again self-image of the bright spokes appear with red color and gets a white pattern at the first self-image radius. At the same time, in front of the dark spokes of the grating ($\theta =\left(2q+1\right)\frac{\pi}{m},q=0,1,2,\mathrm{...}$), the intensity of the black area of the shadow image increases by getting close to the small radii and it reaches to a maximum value at the locationof the first half-Talbot image. The first appearing color is the highest wavelength, therefore bright patterns appear in front of black spokes with the red color at the vicinity of the first half-Talbot radius. Here, after the bright area of the half-Talbot image, at the radii closer to the optical axis, again image of the dark spokes appear with blue color and gets a fully dark pattern at the first self-image radius. Similar behaviors occur around the next self-images of the spokes. In a brief statement, over the diffraction pattern, by closing to the optical axis, successive discrete colorful radial fringes are formed, so that each fringe begins with the red color and ends with the blue color.

Now we show that under multichromatic illumination of a radial structure, as same as the monochromatic case, the resulted colorful diffracted pattern is fully shape-invariant under propagation. For this purpose, we simply consider monochromatic illumination of a radial structure with two different wavelengths *λ*_{1} and *λ*_{2}. We show that despite existence of initial differences between the structures of individual diffraction patterns, their differences remain unchanged under propagation. This fact guarantees that the resulted colorful diffracted pattern is shape-invariant under propagation. For a given point with a radius *r*_{0} on the resulted individual diffraction patterns of the illuminating wavelengths of *λ*_{1} and *λ*_{2} at a given propagation distance of *z*_{0} we have

*z*. This means that the resulted colorful Talbot carpet pattern at the transverse plane is also shape-invariant.

Combining Eqs. (14) and (6), the radius *r* at an arbitrary propagation distance of *z* can be calculated, and the trajectory of the point is given by

This equation determines the position of a given point on the beam under propagation, and can be considered as the ray equation. The main feature of this result is that the trajectory of the point is independent of the wavelength.

Now we investigate theoretically the formation of colorful radial Talbot carpet.

#### 2.3. Diffraction of a multichromatic plane wave from a radial grating

Here we consider multichromatic illumination of the same structures. For a position independent, statically stationary, and spatially fully coherent white light field illumination given with a spectrum ${V}^{\left(0\right)}\left(\lambda \right)$ at *z* = 0, the diffracted light field after a propagation distance of *z* at a given point over the transverse plane can be written by [17]

Using $\mathcal{R}=\mathcal{R}\left(r,z,\lambda \right)=r\sqrt{\frac{\pi}{2\lambda z}}$ as a dimensionless parameter for the diffraction from a radial structure which is defined in the previous section; Eq. (17) can be rewritten as follows:

The spectral density is defined by [18]

Substituting Eq. (17) in Eq. (19) we get the spectral density of the diffracted light field as

For an incident beam having Gaussian spectrum we have

*λ*is the central wavelength and

_{c}*σ*is the width of its Gaussian profile. Using Eq. (21) in Eq. (20) one can determine spectrum of the diffracted light from a given radial structure at any point.

_{λ}Here, we use Eq. (20) to investigate the spectral behavior of the diffracted pattern when a collimated and spatially coherent multichromatic beam with a Gaussian spectrum ${S}^{\left(0\right)}$ of Eq. (21) illuminates a sinusoidal radial grating having *m*spokes. Figure 4 illustrates the calculated normalized spectrum distributions on various locations over the diffraction pattern. It shows the spectra at different radial distances $r={r}_{in,\lambda},{r}_{T1,\lambda},{r}_{out,\lambda},10{r}_{out,\lambda}$ defined for three main colors of $\lambda =470,530,700\phantom{\rule{0.2em}{0ex}}nm$, and particular azimuthal coordinates of $\theta =2\pi /m$ (in the center of bright), $\theta =\pi /2m$ (in the middle of bright and dark), and $\theta =\pi /m$ (in the center of dark) spokes. In all plots of Fig. 4 with $\theta =\pi /2m$ (the image of the middle area of bright and dark spokes), the spectra remain invariant under propagation. This effect is expected by considering $\theta =\frac{\pi}{2m}$ in Eqs. (8) and (20). A similar effect has been reported for the diffraction of spatially coherent white light from conventional 1D and 2D gratings [18, 19].

The radial coordinate of the observation points in Fig. 4(a) are $r={r}_{in,\lambda}$ with $\lambda =470,530,700\phantom{\rule{0.2em}{0ex}}nm$. We see that, over image area of the bright spokes $\theta =2\pi /m$ at the radial distance $r={r}_{in,\lambda}$, each spectrum maximizes at the same wavelength *λ*. Over the image area of the dark spokes $\theta =\pi /m$, the spectra have a local minimum at the same wavelengths.

The radial coordinate of the observation points in Fig. 4(b) are chosen over the first self-Talbot images $r={r}_{T1,\lambda}$, where $\lambda =470,530,700\phantom{\rule{0.2em}{0ex}}nm$. It is apparent that, over image area of the bright spokes $\theta =2\pi /m$ at the radial distance $r={r}_{T1,\lambda}$, the values of the spectra are locally maxima at the same wavelengths. Over the image area of the dark spokes $\theta =\pi /m$, the spectrum has a local minimum at the same wavelength.

The radial coordinate of the observation points in Fig. 4(c) are chosen from the boundary of the geometric shadow and near-field areas $r={r}_{out,\lambda}$, where $\lambda =470,530,700\phantom{\rule{0.2em}{0ex}}nm$. We see that, over image area of the bright spokes $\theta =2\pi /m$ at the radial distance $r={r}_{out,\lambda}$, the spectra experience a blue shift, while over the image area of the dark spokes $\theta =\pi /m$, the spectra experience a red shift.

For the plots of Fig. 4(d) the observation points are sufficiently faraway from the near field regime and plots show that the spectrum is invariant over shadow area.

## 3. Experiments

In our experiment, we use an almost white light source (MD6039 220V 4W LED Lamp) which is collimated using a pair of doublet lenses where at the focal point of the first lens we use a pinhole to increase the spatial coherency of the beam (see Fig. 5). A radial grating is placed on a holder in which the grating’s plane is perpendicular to the optical axis of the beam. We use a digital camera (NIKON D7200) to record the diffraction patterns at different distances from the grating. The lens of thecamera is removed and the diffracted pattern is directly imaged on its sensitive area.

Figure 6 shows the radiation spectrum of the used MD6039 220V 4W LED Lamp determined using a hand operated monochromator of Oriel and a power meter of Newport. As is apparent from the radiation spectrum of the source, the dominant wavelengths are 447 nm, 540 nm, and 680 nm. Therefore we used these wavelengths for the numerical simulations in Figs. 7 and 8 to compare with the experimentally recorded patterns.

Figure 7(a) illustrates the simulated diffraction patterns of multichromatic light from radial sinusoidal gratings that are generated by combination of three calculated diffraction patterns with monochrome wavelengths: 447 nm, 540 nm, and 680 nm.. The simulation is realized at a distance of *z* = 1 m from the gratings having different spoke numbers. Figure 7(b) shows the experimental results for the diffraction patterns of white light beam from the radial sinusoidal gratings with the same spoke numbers of the gratings in Fig. 7(a). The third rows show two insets of the last patterns for clear demonstration. By increasing the number of spokes, a patternless dark area (far-field regime) located around the optical axis is clearly observed and the radius of an intense rainbow-like ring (*r _{in}*) at the vicinity of the patternless area is also increased. In addition, for small values of

*m*at the large radial distances the shadow images are formed.

Similar to the case of diffraction from radial sinusoidal gratings in Fig. 7, we also calculate the theoretical diffraction patterns of multichromatic light from radial binary gratings and the corresponding experimental results are shown in Figs. 8(a) and 8(b), respectively. Contrast-enhanced image of an inset of the recorded pattern is shown at the last row of Fig. 8(b).

As shown in Figs. 7(b) and 8(b), a dark spot is observed in the center of colorful Talbot carpet patterns, meanwhile, in the vicinity of dark spot, a faint bright ring also observed. It seems that this effect is inconsistent with the theoretical calculation shown in Figs. 7(a) and 8(a). We think that it occurs due to the imperfect structure of the radial gratings in the vicinity of singular point. The radial gratings were prepared by printing their structures over plastic sheets. Since the printing system had a limited spatial resolution, it failed to print the singular point perfectly.

## 4. Conclusion

A novel study on colorful radial Talbot carpet at the transverse plane based on the theoretical calculations and experimental demonstration was introduced. Formation of the colorful radial Talbot carpet in the diffraction of a multichromatic and spatially coherent plane wave from an amplitude radial grating was investigated. We proved that as same as the case of monochromatic illumination, in the multichromatic illumination the produced radial Talbot carpet pattern is also shape-invariant under propagation. Two main aspects of the produced colorful radial Talbot carpet are: A quite patternless dark area forms around the optical axis and an intense rainbow-like ring appears in the vicinity of the patternless area. The colors of the rainbow change radially from the violet in the vicinity of the patternless area to red by increasing the radius. In this work we named the second feature as “diffraction-based rainbow”. The resulted diffraction pattern also consists colorful self-images of the grating’s spokes at the larger radii. It is shown that, over the near-field diffraction pattern, by getting distant from the optical axis, successive discrete colorful radial fringes are formed. The color of each of the created colorful fringes begins with the blue color and ends with the red color. A detailed analytical study on the spectral properties of the colorful self-images was presented. We think that the diffraction-based rainbow might find applications in spectrometry.

## Funding

Institute for Advanced Studies in Basic Sciences (IASBS) (G2018IASBS12632 and G2019IASBS12632)

## Acknowledgements

This work was supported by Iran’s National Elites Foundation. Also the work was supported in part by the IASBS Research Council.

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