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

Abstract

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.

Figure 1

Illustration of eight typical AMLPGs with $d=3\mathrm{mm}$, $\gamma =\pi /2$, and different values of l.

Figure 2

Controlled intensity sharing among the different diffraction orders of an AMLPG by adjusting the value of the phase amplitude, γ. Calculated intensity patterns of different diffracted beams in the diffraction of a plane wave from an AMLPG having $l=10$ and different values of γ: π/2, 2.4048, π, 3.8317, 3π/2, 5.1356, 5.5201, and 2π at $z=555\mathrm{cm}$. The intensity over the patterns is normalized to the intensity of the incident beam (for details see the color bars). The dashed white lines in the first row show the boundaries of the different diffraction orders.

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 $z=555\mathrm{cm}$ 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 $z=555\mathrm{cm}$ for γ equal to 1, 2, 3, 4, and 5. In each row, for $s=\gamma$, 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.

Figure 4

Experimentally recorded diffraction pattern at a propagation distance of (a) $z=77\mathrm{cm}$ and (b) $z=350\mathrm{cm}$, in the diffraction of a plane wave from an SLM when it experiences a uniform phase map. The real size of the recorded rectangular patterns is 11 mm × 15 mm. The pairs of numbers correspond to the diffraction orders of the SLM’s impulses. The dashed white lines are used to distinguish the areas of different diffraction orders.

Figure 5

Diffraction pattern of a plane wave from an SLM at $z=350\mathrm{cm}$ when a 1D linear phase grating with a sinusoidal profile in the x direction and a period of 0.11 mm is imposed on the SLM. Here $\gamma =\pi /2$. This pattern corresponds to the (0,0) diffraction order of the SLM and was formed on a diffuser and imaged by camera.

Figure 6

Central pattern: The diffraction pattern of a plane wave from an SLM with $\gamma =\pi /2$ when an AMLPG is imposed on the SLM. Here a diffuser is placed at $z=350\mathrm{cm}$ and the diffraction pattern is imaged by camera. For the radial phase structure $l=10$ and for the linear phase grating $d=0.11\mathrm{mm}$. The diffraction patterns of four typical diffraction orders are enlarged in the first and third columns.

Figure 7

(a, b) are experimental diffraction patterns of two AMLPGs with $l=10$ and $l=15$ at $z=350\mathrm{cm}$, respectively. These patterns are generated over the (0,0) diffraction order of the SLM. Here $\gamma =\pi /2$. (c, d) are the corresponding theoretical patterns. The intensity over the simulated patterns is normalized to the intensity of the incident beam, and for this reason a color bar is used for (c, d).

Figure 8

Experimentally generated diffraction patterns of four AMLPGs with $l=5$, $l=10$, $l=15$, $l=20$, and $\gamma =\pi /2$. Each of the individual patterns was formed directly on the active area of the camera at a distance of $z=555\mathrm{cm}$. A set of low contrast crossed linear fringes appear over the central patterns (0,0,0). These fringes are the edge diffraction patterns of the SLM window.