Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pure-amplitude holograms for high-efficiency generation of phase radial grating based radial carpet beams: Theory and experiments under plane-wave and Gaussian illumination

Saifollah Rasouli, Somaye Fathollazade, Mohammad Mohammadi

Abstract

This study introduces a pure-amplitude hologram (PAH) for generating radial carpet beams (RCBs), which are conventionally produced using pure phase radial gratings (PRGs). The hologram is designed by embedding the transmission function of a binary PRG into the phase argument of the cosine term(s) of an amplitude linear grating. When illuminated with a plane wave, this hologram generates RCBs in the non-zero diffraction orders, and when illuminated with a Gaussian beam, it generates RCB-like patterns at specific propagation distances. This method entirely eliminates the need for complex and expensive spatial light modulators (SLMs). The study presents a theory of diffraction for plane and Gaussian beams from such holograms, including a specific theoretical treatment of Gaussian beam diffraction from PRGs. Through theoretical analysis and experiments, we demonstrate how different RCBs can be generated at different diffraction orders due to the phase-amplitude enhancement resulting from multiplying the diffraction order number by the phase amplitude of the embedded base PRG, when the illuminating beam is a plane wave. For the Gaussian beam case, we show how different RCB-like patterns can be generated at different diffraction orders for the same reason, though only at specific propagation distances. Experimental and numerical results indicate that this technique yields RCBs and RCB-like patterns with approximately five times the useful power of their SLM-generated counterparts, demonstrating significantly higher power efficiency. This advantage renders the proposed method highly suitable for applications such as multiple optical trapping and free-space optical communication.

Pure-amplitude holograms for high-efficiency generation of phase radial grating based radial carpet beams: Theory and experiments under plane-wave and Gaussian illumination

Abstract

This study introduces a pure-amplitude hologram (PAH) for generating radial carpet beams (RCBs), which are conventionally produced using pure phase radial gratings (PRGs). The hologram is designed by embedding the transmission function of a binary PRG into the phase argument of the cosine term(s) of an amplitude linear grating. When illuminated with a plane wave, this hologram generates RCBs in the non-zero diffraction orders, and when illuminated with a Gaussian beam, it generates RCB-like patterns at specific propagation distances. This method entirely eliminates the need for complex and expensive spatial light modulators (SLMs). The study presents a theory of diffraction for plane and Gaussian beams from such holograms, including a specific theoretical treatment of Gaussian beam diffraction from PRGs. Through theoretical analysis and experiments, we demonstrate how different RCBs can be generated at different diffraction orders due to the phase-amplitude enhancement resulting from multiplying the diffraction order number by the phase amplitude of the embedded base PRG, when the illuminating beam is a plane wave. For the Gaussian beam case, we show how different RCB-like patterns can be generated at different diffraction orders for the same reason, though only at specific propagation distances. Experimental and numerical results indicate that this technique yields RCBs and RCB-like patterns with approximately five times the useful power of their SLM-generated counterparts, demonstrating significantly higher power efficiency. This advantage renders the proposed method highly suitable for applications such as multiple optical trapping and free-space optical communication.
Paper Structure (10 sections, 32 equations, 7 figures)

This paper contains 10 sections, 32 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Theory and simulations
  3. Amplitude hologram diffraction: Generating RCBs/RCBLPs with embedded PRGs
  4. Calculation of $U_0(r,\theta,z)$:
  5. Calculation of $U_n(r_n,\theta_n,z)$
  6. Propagation Analysis and Acceptable Invariance Range for RCBLPs from a PAH
  7. Determination of the propagation distance $z_1$ for separation of diffraction orders
  8. Assessing the Near-Invariance of PAH-Generated RCBLPs with a Similarity Function
  9. Experimental results and comparison with theory and simulation
  10. Conclusion

Figures (7)

  • Figure 1: Design steps of a PAH and the generation of an/a RCB/RCBLP using it. (a) Transmission function of a linear sinusoidal amplitude grating with a spatial frequency of 10 lines/mm. (b) Addition of the phase of a PRG with $m=10$ spokes and a phase modulation depth of $\gamma=\pi/2$ to the phase function of the linear grating, forming the desired hologram phase. (c) Amplitude hologram for generating an RCB. (d) Illumination of the hologram with a Gaussian beam of width $w_0 = 8$ mm, producing RCBs in the $+1$ and $-1$ diffraction orders. The propagation distance between the hologram and the observation plane is $z = 2.5$ m, ensuring sufficient separation of the diffraction orders. (e) and (f) Intensity and phase distributions, respectively, of the RCB generated in the $+1$ diffraction order. (g) and (h) present the intensity and phase profiles, respectively, of the RCB generated in the $+1$ diffraction order when the PAH is illuminated by a plane wave. (i) and (j) Intensity distributions of RCBLPs and RCBs generated during propagation in the $+1$ diffraction order, respectively.
  • Figure 2: Investigation of the influence of the phase modulation depth $\gamma$ on the structure of the PAH and on the RCBLP generated in the first diffraction order. (a) Transmission function of the amplitude hologram for $\gamma = \pi/2$ with $m = 10$ spokes, along with the corresponding intensity distribution of RCBLP. (b) Same as (a), but for $\gamma = \pi/4$. (c) and (e) Three-dimensional representations of the RCBLP produced by the holograms with $\gamma = \pi/2$ and $\gamma = \pi/4$, respectively. (d) and (f) 1D intensity profiles extracted along circles passing through the MISs of the RCBLPs for $\gamma = \pi/2$ and $\gamma = \pi/4$. These circles are indicated by blue dashed lines in panels (a) and (b). In both cases, the underlying linear grating has a spatial frequency of 10 lines/mm.
  • Figure 3: Investigation of the propagation and invariance of the RCB generated in the $+1$ diffraction order using a sinusoidal amplitude hologram. (a) Comparison of the intensity distribution of the RCB generated in the $+1$ diffraction order using the amplitude hologram with that obtained directly from a binary PRG with $m=20$. The waist of the incident Gaussian beam illuminating both structures is $8~\mathrm{mm}$. (b) 1D radial intensity profiles extracted from the beam centers in panel (a). The two profiles show excellent agreement. (c), (e) Intensity distributions of RCBs with $m=20$ and $m=10$, respectively, at three different propagation distances from a hologram with a base grating period of $8~\mathrm{lines/mm}$. The insets show magnified views of the regions marked by the yellow dashed boxes in panels (c) and (e). (d), (f) Corresponding 1D radial intensity profiles extracted from the enlarged regions in panels (c) and (e), respectively, at the same three propagation distances. The radial extent of the profiles at each propagation distance is re-scaled with respect to that at the reference distance. (g), (h) 1D similarity function as a function of the propagation distance $z$ for beams generated using the amplitude hologram, illustrating the effect of the number of spokes ($m$) and the waist of the incident Gaussian beam on the propagation-invariant length of the RCB.
  • Figure 4: (a) Longitudinal cross-section of an RCBLP under Gaussian beam illumination ($w_0 = 6$ mm) after passing through a sinusoidal amplitude hologram with parameters $\gamma = \pi/2$, $m = 20$, and a spatial frequency of 8 lines/mm. The white dashed lines indicate three key propagation distances: $z = 1.5$ m (diffraction orders not yet separated), $z = z_1 = 2.8$ m (complete separation of orders), and $z = 4$ m. (b) Transverse intensity distributions at these three propagation distances. (c) Intensity distribution of the RCBLP in the first diffraction order at $z = z_1$. (d) Magnified view of the region marked by the green box in part (c). (e) Azimuthal intensity profiles over a $30^\circ$ arc along two distinct lines: the blue line A--B traversing the MISs and the green line C--D passing through a region where the number of spots has doubled. (f) Azimuthal intensity profiles along the C--D line at various propagation distances, illustrating the evolution of the intensity distribution in this region.
  • Figure 5: (a) Experimental setup for generating RCBs using an amplitude-only hologram. (b) Image of the amplitude hologram used, with dimensions of $3\times3$ cm, a period of $d=0.1$ mm, and a phase modulation coefficient of $\gamma=\pi/4$. (c) Microscopic image of the magnified region of the hologram shown in part (b). (d) Binarized image of the microscopic image. (e) Recorded pattern of different beams generated by the amplitude hologram at a distance of 1.5 m, captured using a camera equipped with a lens. (f) Theoretical intensity logarithm calculated at a distance of $z=1.5$ m from the hologram based on Eq. \ref{['eq:t20']} for the diffraction of a Gaussian beam with a width of $w_0=3.5$ mm, using a hologram with a phase modulation coefficient of $\gamma=\pi/4$ and a period of 0.1 mm. While the actual dimensions of the pattern in (e) on the diffuse paper screen were comparable to those in (f), the dimensions captured on the optical sensor differ after imaging with the camera lens, as illustrated in the figure.
  • ...and 2 more figures