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Abstract
In this work, we report the characterization of a Laguerre-Gaussian (LG) beam with given values of topological charge (TC) and radial index in a simple, efficient, and robust experimental diffraction scheme. The beam diffracts from an amplitude parabolic-line linear grating and the resulting diffraction patterns at zero- and first-order reveals the values of the TC, l, and radial index p of the incident LG beam using a simple analysis. The zero-order diffraction pattern consists of p + 1 concentric intensity rings and the first-order diffraction pattern contains an (l + p + 1) by (p + 1) two-dimensional array of intensity spots. The experimental scheme is robust since it is not sensitive to the relative locations of the impinging beam axis and the grating center, and is efficient since most of the energy of the output beam is in the diffraction order of interest for LG beam characterization. The measurement is also simple since the intensity spots of the array are placed exactly over straight and parallel lines. Both experimental and simulation results are presented and are consistent with each other.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laguerre-Gaussian (LG) beams are one the main optical beams carrying orbital angular momentum. An LG beam compared to a Gaussian beam has two additional degrees of freedom called topological charge (TC), $l$, and radial index, $p$. Because of these extra degrees of freedom, LG beams are used in a wide range of applications [1,2] such as optical manipulation [3], microfluidics [4], free-space communication [5] and quantum information processing [6]. The use of LG beams with different radial indices can increase the capacity of data transmission in free-space communication. Also, with the help of an LG beam with a large radial index, it is possible to trap several particles at the same time in co-center circular paths and force them into orbital rotation since it has several intensity rings about the beam axis.
There are many methods for generating and characterizing LG beams including the use of interference methods [7–9], diffraction from different amplitude apertures and phase objects [10–16], and the use of a robust mode converter [17]. Gratings and grating-like structures with varying periods are also widely used for this purpose [18–22].
The TC value and its sign in an LG beam can be determined using other gratings and phase objects having slightly complicated structures, such as Dammann vortex gratings [23,24], composite fork gratings [5], and mode sorters [25]. The Talbot arrangement, consisting of a combination of a pair of linear gratings, is also used to determinate both the wavefront and transverse components of the Poynting vector of the impinging LG beam [26]. The same scheme is also used for determining the TC sign alteration that occurs because of an odd number of reflections [27]. One of the advantages of the use of Talbot arrangement is the improvement of the measurement precision.
Most of the mentioned measurements are done on an LG beam having a zero radial index. Their setups are complicated and are sometimes very expensive. In some of them, the reliability and accuracy of the measurement setups are closely related to the relative position and impinging conditions (such as incident angle) of the beam arriving at the diffracting aperture.
It is worth noting that recently a simple measurement scheme was proposed and experimentally tested using a spiral phase grating to determine the mode indices of LG beams [28], but this method also is very sensitive to the relative position of the beam on the grating and suffers seriously from this limitation. In addition, in this method, the created array of intensity spots are placed over curved paths and formed at a distinct propagation distance in which these features also make the measurements more complex.
We have recently used an amplitude parabolic-line linear grating for the characterization of a single optical vortex [29] and a collinear and on-axis combination of two optical vortices having different winding numbers, where in both experiments the beams’ radial indices were zero [30]. Both of these measurements benefited from simple, efficient, and robust experimental diffraction schemes where the first- and second-order diffraction patterns of the grating were used. The experimental setups used were extremely simple, low cost, and reliable. The measurement schemes were robust since they were not sensitive to the relative locations of the impinging beam axes and the grating center, and they worked even when the incident angle of the beams on the grating plane changed. The method used was also efficient since most of the energy of the output beam was in the diffraction order of interest for vortex beam characterization.
Here we use the same diffraction grating in a modified scheme to produce and characterize an LG beam with given values of TC and radial index. We use a printed parabolic-line linear grating on a transparent plastic sheet and use it simply for the measurement of $l$ and $p$ of the beam in a very simple diffraction setup. We show that the diffraction pattern at the zero-order consists of $p+1$ concentric intensity rings and the first-order is just an $(l+p+1)$ by $(p+1)$ two-dimensional (2D) array of intensity spots. The proposed method was investigated experimentally and by computer simulation.
It is worth noting that transformations of Hermite-Gaussian beams into LG beams using astigmatic transformations have already been made using a cylindrical lens in Ref. [31] and by tilting the optical element that forms an LG beam [32]. However, the diffraction of LG beams only from the parabolic-line linear gratings is insensitive to the off-axis value of the beam and grating centers [29].
2. Governing equations of the diffraction of an LG beam from an amplitude parabolic-line linear grating
In Cartesian coordinates, the complex amplitude of an LG beam with the radial index $p \neq 0$ over the $x-y$ plane can be written as [2]
The transmission function of a parabolic-line linear grating having an amplitude transmission profile can be written as [29,30]
in which all of the grating lines have an equal second degree curvatures along the $y$ direction with a curvature coefficient of $\gamma$, and $d$ shows the spatial period of the grating along the $x$ direction.We let the LG beam pass through the grating with a transmission function $t(x,y)$. The light beam complex amplitude immediately after passing through the grating is
where $-0$ and $+0$ signify immediately before and after the grating, respectively.After a propagation distance $z$, the diffracted complex amplitude $U(x, y, z)$ can be calculated using the Fresnel integral as
Figures 1(a) and 1(b), the first and second patterns, show the intensity and phase profiles of two LG beams with $l$ = 3 and in (a) $p$ = 0 and in (b) $p$ = 2. The third patterns in Fig. 1(a) and 1(b) show the transmission profiles of an amplitude parabolic-line linear grating.
Fig. 1. (a) First line, simulated intensity and phase profiles of an LG beam with $\lambda =$ 532 nm, $w$ = 0.5 mm, $l$ = 3, and $p$ = 0, and the transmission profile of an amplitude parabolic-line linear grating having a sinusoidal profile and parameters $d$ = 0.2 mm and $\gamma \ = \ 0.25 \ mm^{-1}$. The second line shows the diffraction pattern of the beams from the grating at a propagation distance of $z$ = 2 m. (b) The same patterns as in (a) with $p=2$.
Using Eq. (5) in Eq. (4) and with the aid of the transfer function of the free space, we simulated the corresponding intensity and phase profiles of the diffracted patterns at a propagation distance of $z$ = 2 m. The second lines of Figs. 1(a) and (b) show such diffracted patterns.
As it is seen from the diffracted patterns, over the zero diffraction order, a set of $p+1$ concentric intensity rings appears, and over each of the first-order diffraction patterns we see an $(l+p+1)$ by $(p+1)$ 2D array of intensity spots, in which these patterns reveal both $l$ and $p$ values, simply and visually clear. The $p$ value is determined by the number of intensity rings observed over the zero-order diffraction pattern. The $l$ value can be determined by the size of the array of intensity spots formed over the first-order diffraction pattern.
Figure 2 shows simulated intensity and phase profiles of the diffraction patterns of an LG beam with $l=3$ and $p=2$ from an amplitude parabolic-line linear grating having a sinusoidal profile at different propagation distances. In the right column, insets of the phase profiles are enlarged for better illustration. With a bit of calculation, the same results can be derived using the two general theoretical approaches presented in Ref. [10]. In the background Visualization 1, one can see the evolution of the simulated zero- and first-order diffraction patterns of an LG beam having a non-zero radial index from an amplitude parabolic-line linear grating under propagation.
Fig. 2. Simulated intensity and phase profiles of the diffraction pattern of an LG beam with $\lambda$ = 532 nm, $w$ = 0.5 mm, $l$ = 3, and $p$ = 2 from an amplitude parabolic-line linear grating having a sinusoidal profile and parameters $d$ = 0.2 mm and $\gamma \ =0.25 \ mm^{-1}$, at different propagation distances (see also Visualization 1).
Figure 3(a) shows the simulated zero-order and first-order intensity patterns when an LG beam with $l=4$ and $p=3$ is diffracted from an amplitude parabolic-line linear grating having a sinusoidal profile at a propagation distance of $z$=3 m. In Fig. 3(b) the same patterns are shown for $l=-4$. In the second column, high contrast patterns of the intensity spots are shown. We see that, here again, the diffraction pattern at the zero-order consists of $p+1=3$ concentric intensity rings and the first-order diffraction pattern shows an $(l+p+1=8)$ by $(p+1=4)$ 2D array of intensity spots.
Fig. 3. Simulated intensity patterns when an LG beam diffracted from an amplitude parabolic-line linear grating has a sinusoidal profile. The TC of the beam in (a) is $l=4$ and in (b) $l=-4$ where for both of cases $p=3$ and the propagation distance is $z$ = 3 m.
In Fig. 4, we examined the diffraction of the LG beams having different TC values and radial indices from an amplitude parabolic-line linear grating having a sinusoidal profile. Patterns show the resulting first-order diffracted intensity profiles at a distance of $z=$2 m. The other parameters are the same as in Fig. 2.
Fig. 4. Simulated intensity profiles of the diffracted field over the first-order at a propagation distance of $z$=2 m, when different LG beams with different $l$ and $p$ values pass from an amplitude parabolic-line linear grating. The other parameters of the beam and grating are the same as in Fig. 2.
2.1 Diffraction of LG beams with non-zero radial indices from a pure phase parabolic-line linear grating
Now let us examine the diffraction of LG beams having different non-zero radial indices from a pure phase parabolic-line linear grating. Here, we will also compare the diffraction of the same beams from amplitude and phase spiral gratings and show that the use of amplitude and phase parabolic-line linear gratings has considerable advantages. The transmission function of a pure phase parabolic-line linear grating is given by [29]
where here we consider $x_0$ as a given lateral shear between the grating and beam centers, we call it an off-axis value.Figure 5 shows the simulated diffraction patterns of an LG beam passing through amplitude and phase parabolic-line linear gratings (first and second columns), amplitude and phase spiral gratings (third and forth columns) for different off-axis values of the beam $x_0$ at a given propagation distance. Parameters of the transmission functions are the same for all schemes. The diffraction patterns in the first and second columns indicate that in the use of parabolic-line linear gratings the forms of the array of spots used for measuring the values of the beam’s TC and radial index are not sensitive to the off-axis values. In the third and forth columns of Fig. 5, we see that in the diffraction of an LG beam from amplitude and a phase spiral phase gratings, the formation of an array of intensity spots is very sensitive to the relative position of the beam on the grating. Therefore, the method suffers seriously from this limitation. It is also seen that the array of intensity spots is formed over curved paths, which also makes the measurement more complex. The transmission functions of the amplitude and phase spiral gratings are written over the corresponding columns in Fig. 5.
Fig. 5. Diffraction of an LG beam with $\lambda$ = 532 nm, $w$ = 0.5 mm, $l=4$, and $p=3$ from an amplitude parabolic-line linear grating (first column), a phase parabolic-line linear grating (second column), an amplitude spiral grating (third column), and a phase spiral grating (forth column) for different off-axis values of the beam, $x_0$, at a propagation distance of $z$ = 3 m. The parameters of the transmission functions for the first and second columns are $d$ = 0.2 mm and $\gamma = 0.25 \ mm^{-1}$, and for the third and forth columns we used $q=80$.
It is worth noting that the measurements based on the parabolic-line linear gratings are also remarkably insensitive to the incident angle of the beams on the gratings [29].
3. Measurements
Figure 6(a) shows the experimental setup used for characterizing an LG beam having different $l$ and $p$ parameters using a pure amplitude grating. Figure 6(b) is proposed and can be used for generating, on axis superposing, and characterizing of a pair of LG beams having different $l$ and $p$ parameters again using a pure amplitude parabolic-line linear grating.
Fig. 6. (a) Experimental setup used for characterizing LG beams with non-zero $l$ and $p$ parameters in diffraction from a pure amplitude grating and (b) proposed setup for generating and on axis superposing, and characterizing a pair of LG beams with different non-zero $l$ and $p$ parameters.
3.1 Diffraction of an LG beam from a pure amplitude parabolic-line linear grating
Using the setup shown in Fig. 6(a), we demonstrated the performance of the proposed method, experimentally. The plane wave of the second harmonic of a neodymium-doped yttrium aluminum garnet (ND:YAG) diode-pumped laser beam having a wavelength of $\lambda$= 532 nm and a Gaussian profile, $\exp \bigg (-\frac {x^2+y^2}{w^2}\bigg )$, was passed through the first spatial light modulator (SLM). To produce an LG beam with $l$ and $p$ indices, we needed to add a phase profile of $\varphi (r,\phi )=l\phi -\pi \textrm {sgn}\bigg (L_P^{|l|}(\frac {2r^2}{w^{2}})\bigg )$ on the incident Gaussian beam. Therefore, the following transmission function was imposed on the first SLM:
The resulting LG beam passed through a printed amplitude parabolic-line linear grating. The diffracted pattern was directly recorded on the sensitive area of a camera (Nikon D7200). Figure 7 shows simulated (red) and the corresponding experimentally recorded (green) intensity patterns formed over the first-order when the LG beams having different TC values and radial indices were diffracted from a pure amplitude parabolic-line linear grating.
Fig. 7. Simulated (red) and experimentally recorded (green) diffraction patterns of LG beams with $l$ = 3 and $p$ = 0, 1, 2 from an amplitude parabolic-line linear grating having a sinusoidal profile. The zero-order patterns are shown in the first and second columns and the first-order patterns are shown in the third and forth columns. For simulation patterns the propagation distance was $z$ = 3 m and for the experimental work it was $z$ = 4.5 m. Other parameters were $\lambda$ = 532 nm, $w$ = 1.5 mm, $d$ = 0.1 mm, and $\gamma = \ 0.040 \ mm^{-1}$.
Figure 8 shows the first-order intensity diffraction patterns of the LG beams with different values of $l$ and $p$ after passing through an amplitude parabolic-line linear grating having a sinusoidal profile.
Fig. 8. First-order intensity diffraction patterns of LG beams with given values of $l$ and $p$ after passing through an amplitude parabolic-line linear grating having a sinusoidal profile. The propagation distances for the simulated patterns (red) were $z$ = 3 m and for the experimentally recorded patterns (green) were 1.5 m. Other parameters were $\lambda$ = 532 nm, $w$ = 1.5 mm, $d$ = 0.1 mm, and $\gamma \ = \ 0.040 \ mm^{-1}$.
Figure 9 shows the first-order intensity diffraction patterns of two LG beams after passing through an amplitude parabolic-line linear grating having a sinusoidal profile. For the right column we have $l$ = 1 and for the left column $l$ = 5; $p$ = 2 for both beams.
Fig. 9. Simulated (red) and experimentally recorded (green) first-order diffracted intensity patterns of two LG beams after passing through an amplitude parabolic-line linear grating having a sinusoidal profile, where for the right column $l$ = 1 and for the left column $l$ = 5, and in both columns $p$ = 2. For the simulation patterns, the used parameters were $\lambda$ = 532 nm, $w$ = 0.5 mm, $d$ = 0.125 mm, $\gamma =0.25 \ mm^{-1}$, and $z$ = 3 m, and for the experimentally recorded results we had $\lambda$= 532 nm, $w$ = 1.5 mm, $d$ = 0.1 mm, $\gamma \, = \, 0.040 \, mm^{-1}$, and $z$ = 1.5 m.
4. Investigating diffraction of two on-axis and collinear LG beams with different values of $l$ and $p$ from an amplitude parabolic-line linear grating
Here, we simulated the diffraction of a pair of LG beams having different values of $l$ and $p$ and superimposed on-axis and collinear from an amplitude parabolic-line linear grating at a proper propagation distance. Figure 10 shows the simulated intensity patterns over the zero-order. These patterns are similar to the interference intensity patterns of the corresponding combined LG beams before the grating. Figure 11 shows the simulated first-order diffraction patterns, when two on-axis and collinear LG beams with different values of TCs and radial indices were diffracted from an amplitude parabolic-line grating.
Fig. 10. Simulated interference patterns of two LG beams with different $l$ and $p$. The parameters used were $\lambda$ = 532 nm and $w$ = 0.5 mm.
Fig. 11. Simulated first-order diffraction patterns of combined LG beams presented in Fig. 10, when they pass through an amplitude parabolic-line linear grating having a sinusoidal profile and parameters $d$ = 0.2 mm and $\gamma \ = 0.25 \ mm^{-1},$ at a propagation distance of $z$ = 3 m.
We see that in each case two 2D arrays of intensity spots are formed over the first-order diffraction patterns, in which in most of the superimposed states, the determination of the beams’ parameters is not an easy task.
5. Conclusion
An amplitude parabolic-line linear grating printed on a transparent plastic sheet in a simple and inexpensive diffraction setup was used for the characterization of the TC and radial index of an LG beam. The LG beam passed through the grating and the resulting zero- and first-order diffraction patterns were used simply for the determination of the beam’s TC and radial index. It was shown that the zero-order diffraction pattern consists of $p+1$ concentric intensity rings and the first-order diffraction pattern contains an $(l+p+1)$ by $(p+1)$ two-dimensional array of intensity spots. The proposed method was investigated experimentally and by simulation and their results completely confirmed each other.
It is worth mentioning that the method proposed here can be used on other classes of beams carrying orbital angular momenta, characterized with radial and azimuthal indices, and having radial symmetric intensity. For example, the diffraction of Bessel-Gaussian beams from an amplitude parabolic-line linear grating is being studies, and the results are satisfactory and will appear elsewhere. However, the proposed method does not benefit on the beams such as HermitGaussian beams that have symmetry in the Cartesian coordinate system. This is because these beams themselves can be converted into LG beams under astigmatic transformations [31].
Funding
Iran National Science Foundation (98019152).
Acknowledgments
Author Saifollah Rasouli would like to acknowledge Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy for Senior Associate Fellowship. He would also like to express his gratitude to Prof. Juan P. Torres for making some useful suggestions in the Spotlight Summary that appeared in Spotlight on Optics, Highlighted Articles from OSA Journals, October 2020 (see the following link: https://opg.optica.org/spotlight/summary.cfm?URI=josab-37-9-2668).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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