## Simulations and experiments of aperiodic and multiplexed gratings in volume holographic imaging systems |

Optics Express, Vol. 18, Issue 18, pp. 19273-19285 (2010)

http://dx.doi.org/10.1364/OE.18.019273

Acrobat PDF (20496 KB)

### Abstract

A new methodology describing the effects of aperiodic and multiplexed gratings in volume holographic imaging systems (VHIS) is presented. The aperiodic gratings are treated as an ensemble of localized planar gratings using coupled wave methods in conjunction with sequential and non-sequential ray-tracing techniques to accurately predict volumetric diffraction effects in VHIS. Our approach can be applied to aperiodic, multiplexed gratings and used to theoretically predict the performance of multiplexed volume holographic gratings within a volume hologram for VHIS. We present simulation and experimental results for the aperiodic and multiplexed imaging gratings formed in PQ-PMMA at 488nm and probed with a spherical wave at 633nm. Simulation results based on our approach that can be easily implemented in ray-tracing packages such as Zemax^{®} are confirmed with experiments and show proof of consistency and usefulness of the proposed models.

© 2010 OSA

## 1. Introduction

1. W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. **27**(10), 854–856 (2002). [CrossRef]

3. Y. Luo, S. B. Oh, and G. Barbastathis, “Wavelength-coded multifocal microscopy,” Opt. Lett. **35**(5), 781–783 (2010). [PubMed]

4. Z. Li, D. Psaltis, W. Liu, W. R. Johson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE **5694**, 33–40 (2005). [CrossRef]

6. Y. Luo, P. J. Gelsinger-Austin, J. M. Watson, G. Barbastathis, J. K. Barton, and R. K. Kostuk, “Laser-induced fluorescence imaging of subsurface tissue structures with a volume holographic spatial-spectral imaging system,” Opt. Lett. **33**(18), 2098–2100 (2008). [CrossRef] [PubMed]

7. A. Sinha and G. Barbastathis, “Volume holographic imaging for surface metrology at long working distances,” Opt. Express **11**(24), 3202–3209 (2003). [CrossRef] [PubMed]

9. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. **73**(9), 1105–1112 (1983). [CrossRef]

10. P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax,” Opt. Express **16**(10), 7516–7524 (2008). [CrossRef] [PubMed]

11. R. R. A. Syms and L. Solymar, “Localized one-dimensional theory for volume holograms,” Opt. Quantum Electron. **13**(5), 415–419 (1981). [CrossRef]

12. R. R. A. Syms and L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. **72**(2), 179–186 (1982). [CrossRef]

13. A. Sinha and G. Barbastathis, “Broadband volume holographic imaging,” Appl. Opt. **43**(27), 5214–5221 (2004). [CrossRef] [PubMed]

16. Y. Luo, P. J. Gelsinger, J. K. Barton, G. Barbastathis, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial imaging filters,” Opt. Lett. **33**(6), 566–568 (2008). [CrossRef] [PubMed]

## 2. Model

### 2.1 Rigorous Coupled Wave Analysis

9. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. **73**(9), 1105–1112 (1983). [CrossRef]

*d*. In Fig. 1, a polarized electromagnetic wave is incident at an angle

*α*. The polarization of the incident beam is given by

*ψ*, which is the angle of the

*β*with respect to the x axis.

*Λ*is the grating period, and

*φ*is the slant angle. The incident medium is region 1, region 2 is the grating recording medium, and region 3 is the substrate. The amplitude of the propagation vector in the

*ℓ*th region

*ℓ*,

*λ*is the free-space wavelength. In the incident medium, the incident normalized electric-field vector iswhere

*n*th reflected wave in region 1 with propagation vector

*n*th transmitted wave in region 3 with propagation vector

*n*th space-harmonic of the electric and magnetic fields so that E

_{2}and H

_{2}satisfy Maxwell’s equations in the region 2: with

*ω*the angular frequency of the incident radiation.

9. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. **73**(9), 1105–1112 (1983). [CrossRef]

### 2.2 Approximate Coupled Wave Theory

^{®}. Moreover, the analytical expressions of ACW provide useful insight of the VHIS operation which is helpful in the design process.

*η*) for a transmission holographic grating with the first order propagation in the positive direction in region 3 is given by where

*ρ*is absorption coefficient,

*ϑ*is the detuning parameter [8,17].

### 2.3 Simulation Procedure

*d*<<

*r*, the aperiodic grating can be decomposed into locally planar sections [12

12. R. R. A. Syms and L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. **72**(2), 179–186 (1982). [CrossRef]

*r*is the radius of curvature of a spherical wavefront. The localized approximation is valid since the recording and reconstruction waves are varying sufficiently slowly within each locally planar region. Therefore, in the locally planar section, the grating can be considered to be formed with two planar waves. The effects of all local planar regions are combined to determine the efficiency and image properties of the full hologram aperture.

*z*

_{con}and adjusting the reference beam angle with Δθ for each exposure. In both RCW and ACW models, cross-coupling between gratings is minimal if the angular difference (∆θ) between reference beams angles is greater than the angular selectivity of each grating. In addition, to simplify the calculation, the objective lens shown in Fig. 3 in the signal arm is assumed to be an aberration-free paraxial lens.

*z*and

*f*is the focal length of the objective lens. 2) The exposed area on the hologram is sampled with a large number N of sampling zones. The local grating periods and diffraction efficiencies are denoted as

*i*= 1,..,N. The spherical wave is decomposed into planar waves. In the individual sampling zone of the hologram, the grating is then considered to be formed by two planar waves, and the resulting grating vector is then computed. 3) The effects of thickness change, polarization state, and reconstruction wavelength on the properties of the grating are considered, and the new grating vector is determined [8,9

**73**(9), 1105–1112 (1983). [CrossRef]

^{®}non-sequential ray tracing which incorporates the zero and first diffracted orders of each grating is implemented to estimate the crosstalk among gratings. To compare with RCW model and experimental results, the first order is more important and interesting since the first diffracted order dominates the volumetric diffraction effects in the VHIS. 5) This procedure is then repeated for each multiplexed grating by moving the constructed point source by Δ

*z*

_{con}and changing the reference beam angle by Δ

*θ*.

## 3. Measurements

### 3.1 Experimental Setup

16. Y. Luo, P. J. Gelsinger, J. K. Barton, G. Barbastathis, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial imaging filters,” Opt. Lett. **33**(6), 566–568 (2008). [CrossRef] [PubMed]

16. Y. Luo, P. J. Gelsinger, J. K. Barton, G. Barbastathis, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial imaging filters,” Opt. Lett. **33**(6), 566–568 (2008). [CrossRef] [PubMed]

*z*

_{con}with each exposure. A second microscope objective lens (M2) with 0.55NA and 3.6mm focal length remains in a fixed position in the signal arm forming the point source. The nominal angle between the two arms is ~68° and is changed by ∆θ = 1° with each exposure to record a hologram with a different reference beam angle and point source location.

18. Y. Luo, J. M. Russo, R. K. Kostuk, and G. Barbastathis, “Silicon oxide nanoparticles doped PQ-PMMA for volume holographic imaging filters,” Opt. Lett. **35**(8), 1269–1271 (2010). [CrossRef] [PubMed]

### 3.2. Depth Separation and Selectivity Measurements

_{z}) at reconstruction wavelength of 488nm. Figure 6(b) shows experimental results of the PSF

_{z}at the reconstruction wavelength of 633nm. Experimental results for the relationship between ∆

*z*

_{con}and ∆

*z*

_{recon}are plotted in Fig. 7 . The slope of ∆

*z*

_{con}and ∆

*z*

_{recon}equals 1 when the recording and probing wavelengths are the same. However, when the probing wavelength changes, so does the slope [19

19. G. Barbastathis and D. Psaltis, “Shift-multiplexed holographic memory using the two-lambda method,” Opt. Lett. **21**(6), 432–434 (1996). [CrossRef] [PubMed]

*η*) for the planar and curved gratings are annotated in the plots.

## 4. Simulation Results

*z*

_{con}= 50µm. N = 1000 zones. Parameters in the simulation, i.e. hologram aperture and thickness, polarization, focal length, angle of probe beam, angle change, and longitudinal displacement between exposures, were chosen to be identical to the experimental values described in Section 3. Additional parameters were: absorption coefficient was 0.045(1/mm) at 488nm and 0.039(1/mm) at 633nm [20,21], the refractive index was 1.49, and the index modulation was 8 × 10

^{−5}for the first grating and 7 × 10

^{−5}for the second grating.

^{®}, and Fig. 8(b) shows the diffraction images on the detector plane of a point source with ∆

*z*

_{recon}= −2µm, −2µm, 0µm, 1µm, and 2µm. Figure 9(a) shows the ACW and RCW simulation modeling results of the point spread function in depth (PSF

_{z}) at the reconstruction wavelength of 488nm, and the average full width of half maximum (FWHM) is ~9µm. Figure 9(b) shows the results of the PSF

_{z}at the reconstruction wavelength of 633nm, and the average FWHM is ~12µm. The diffraction efficiency is also normalized for easy comparison of the width of PSF

_{z}. The experimental data in Fig. 6 (a) and (b) have been duplicated in respective Fig. 9(a) and (b) to easily compare and validate modeling results with experimental results.

*z*

_{con}and ∆

*z*

_{recon}using the same wavelength of 488nm. Figure 10(b) plots ∆

*z*

_{con}vs ∆

*z*

_{recon}when different wavelengths of 488nm and 633nm were used for recording and reconstruction, respectively.

## 5. Conclusions

^{®}were confirmed with experiments and showed good agreement with the measurements.

11. R. R. A. Syms and L. Solymar, “Localized one-dimensional theory for volume holograms,” Opt. Quantum Electron. **13**(5), 415–419 (1981). [CrossRef]

12. R. R. A. Syms and L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. **72**(2), 179–186 (1982). [CrossRef]

*i.e.*∆

*z*

_{recon}) can be increased using longer wavelengths to probe multiplexed gratings, which are formed at shorter wavelengths. Therefore, the depth separation in the VHIS can be controlled by adjusting the operation wavelength of light sources. However, the diffraction efficiency of either aperiodic or periodic gratings in the VHIS is lower using a longer probe wavelength of 633nm that is different with the construction wavelength of 488nm. In addition, the depth selectivity is degraded and the FWHM becomes wider when the longer wavelength of 633nm is used for reconstruction.

^{®}. Our model can be integrated with other ray-tracing software with proper modifications to theoretically predict the performance of multiplexed gratings in the VHIS. However, there is one limitation: localized grating approximation assumes that grating vectors are slowly varying along the lateral directions. Currently, we have verified the model for aperiodic and multiplexed imaging gratings formed in PQ-PMMA at 488nm and probed at 488nm/633nm. It is left for future research to extend the model for holograms recorded with aberrated or fast varying waves, investigation of optics with aberrations in VHIS to reach comparable depth selectivity with other imaging systems such as digital holographic microscopy [22

22. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics **2**(3), 190–195 (2008). [CrossRef]

## Acknowledgements

## References and links

1. | W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. |

2. | P. J. Gelsinger-Austin, Y. Luo, J. M. Watson, R. K. Kostuk, G. Barbastathis, J. K. Barton, and J. M. Castro, “Optical design for a spatial-spectral volume holographic imaging system,” Opt. Eng. |

3. | Y. Luo, S. B. Oh, and G. Barbastathis, “Wavelength-coded multifocal microscopy,” Opt. Lett. |

4. | Z. Li, D. Psaltis, W. Liu, W. R. Johson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE |

5. | A. V. Veniaminov, V. G. Goncharov, and A. P. Popov, “Hologram amplification by diffusion destruction of out-of phase periodic structures,” Opt. Spectrosc. |

6. | Y. Luo, P. J. Gelsinger-Austin, J. M. Watson, G. Barbastathis, J. K. Barton, and R. K. Kostuk, “Laser-induced fluorescence imaging of subsurface tissue structures with a volume holographic spatial-spectral imaging system,” Opt. Lett. |

7. | A. Sinha and G. Barbastathis, “Volume holographic imaging for surface metrology at long working distances,” Opt. Express |

8. | H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. |

9. | G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. |

10. | P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax,” Opt. Express |

11. | R. R. A. Syms and L. Solymar, “Localized one-dimensional theory for volume holograms,” Opt. Quantum Electron. |

12. | R. R. A. Syms and L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. |

13. | A. Sinha and G. Barbastathis, “Broadband volume holographic imaging,” Appl. Opt. |

14. | A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. |

15. | G. Barbastathis, and D. Psaltis, “Volume holographic multiplexing methods,” in |

16. | Y. Luo, P. J. Gelsinger, J. K. Barton, G. Barbastathis, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial imaging filters,” Opt. Lett. |

17. | R. K. Kostuk, Multiple grating reflection volume holograms with application to optical interconnects, Ph. D. Thesis at Stanford University, 1986. |

18. | Y. Luo, J. M. Russo, R. K. Kostuk, and G. Barbastathis, “Silicon oxide nanoparticles doped PQ-PMMA for volume holographic imaging filters,” Opt. Lett. |

19. | G. Barbastathis and D. Psaltis, “Shift-multiplexed holographic memory using the two-lambda method,” Opt. Lett. |

20. | W. K. Maeda, “Edge-illumination gratings in PQ-doped PMMA for OCDMA applications,” The University of Arizona, ECE Department, Thesis, 2005. |

21. | J. M. Russo, “Temperature dependence of holographic filers in phenanthrenquinone-doped poly(methyl methacrylate),” The University of Arizona, ECE Department, Thesis, 2007. |

22. | J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics |

**OCIS Codes**

(090.2890) Holography : Holographic optical elements

(090.4220) Holography : Multiplex holography

(090.7330) Holography : Volume gratings

(110.0110) Imaging systems : Imaging systems

**ToC Category:**

Holography

**History**

Original Manuscript: June 14, 2010

Revised Manuscript: July 18, 2010

Manuscript Accepted: July 19, 2010

Published: August 26, 2010

**Citation**

Yuan Luo, Jose Castro, Jennifer K. Barton, Raymond K. Kostuk, and George Barbastathis, "Simulations and experiments of aperiodic and multiplexed gratings in volume holographic imaging systems," Opt. Express **18**, 19273-19285 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19273

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### References

- W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27(10), 854–856 (2002). [CrossRef]
- P. J. Gelsinger-Austin, Y. Luo, J. M. Watson, R. K. Kostuk, G. Barbastathis, J. K. Barton, and J. M. Castro, “Optical design for a spatial-spectral volume holographic imaging system,” Opt. Eng. 49(4), 043001–043005 (2010). [CrossRef]
- Y. Luo, S. B. Oh, and G. Barbastathis, “Wavelength-coded multifocal microscopy,” Opt. Lett. 35(5), 781–783 (2010). [PubMed]
- Z. Li, D. Psaltis, W. Liu, W. R. Johson, and G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005). [CrossRef]
- A. V. Veniaminov, V. G. Goncharov, and A. P. Popov, “Hologram amplification by diffusion destruction of out-of phase periodic structures,” Opt. Spectrosc. 70(4), 505–508 (1991).
- Y. Luo, P. J. Gelsinger-Austin, J. M. Watson, G. Barbastathis, J. K. Barton, and R. K. Kostuk, “Laser-induced fluorescence imaging of subsurface tissue structures with a volume holographic spatial-spectral imaging system,” Opt. Lett. 33(18), 2098–2100 (2008). [CrossRef] [PubMed]
- A. Sinha and G. Barbastathis, “Volume holographic imaging for surface metrology at long working distances,” Opt. Express 11(24), 3202–3209 (2003). [CrossRef] [PubMed]
- H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2946 (1969).
- G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73(9), 1105–1112 (1983). [CrossRef]
- P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax,” Opt. Express 16(10), 7516–7524 (2008). [CrossRef] [PubMed]
- R. R. A. Syms and L. Solymar, “Localized one-dimensional theory for volume holograms,” Opt. Quantum Electron. 13(5), 415–419 (1981). [CrossRef]
- R. R. A. Syms and L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. 72(2), 179–186 (1982). [CrossRef]
- A. Sinha and G. Barbastathis, “Broadband volume holographic imaging,” Appl. Opt. 43(27), 5214–5221 (2004). [CrossRef] [PubMed]
- A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43(7), 1533–1551 (2004). [CrossRef] [PubMed]
- G. Barbastathis, and D. Psaltis, “Volume holographic multiplexing methods,” in Holographic Data Storage (Springer, 2000).
- Y. Luo, P. J. Gelsinger, J. K. Barton, G. Barbastathis, and R. K. Kostuk, “Optimization of multiplexed holographic gratings in PQ-PMMA for spectral-spatial imaging filters,” Opt. Lett. 33(6), 566–568 (2008). [CrossRef] [PubMed]
- R. K. Kostuk, Multiple grating reflection volume holograms with application to optical interconnects, Ph. D. Thesis at Stanford University, 1986.
- Y. Luo, J. M. Russo, R. K. Kostuk, and G. Barbastathis, “Silicon oxide nanoparticles doped PQ-PMMA for volume holographic imaging filters,” Opt. Lett. 35(8), 1269–1271 (2010). [CrossRef] [PubMed]
- G. Barbastathis and D. Psaltis, “Shift-multiplexed holographic memory using the two-lambda method,” Opt. Lett. 21(6), 432–434 (1996). [CrossRef] [PubMed]
- W. K. Maeda, “Edge-illumination gratings in PQ-doped PMMA for OCDMA applications,” The University of Arizona, ECE Department, Thesis, 2005.
- J. M. Russo, “Temperature dependence of holographic filers in phenanthrenquinone-doped poly(methyl methacrylate),” The University of Arizona, ECE Department, Thesis, 2007.
- J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008). [CrossRef]

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