## Tomographic reconstruction of weak, replicated index structures embedded in a volume

Optics Express, Vol. 15, Issue 21, pp. 14202-14212 (2007)

http://dx.doi.org/10.1364/OE.15.014202

Acrobat PDF (333 KB)

### Abstract

Measurements of weak, embedded index structures are important for material characterization of photopolymers, glass and other optical materials as well as for characterization of fabricated structures such as waveguides. We demonstrate an optical diffraction tomography system capable of measuring deeply-buried, weak, fabricated index structures written in a homogeneous volume. High-fidelity cross sections of these weak index structures are constructed by replicating the structure to be measured to form a diffraction grating. The coherent addition of scattering from each of these objects increases the sensitivity of the imaging system. Measurements are made in the far field, without the use of lenses, eliminating phase aberration errors through thick volumes.

© 2007 Optical Society of America

## 1. Introduction

1. M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, and W. L. Wilson, “Holographic data storage media for practical systems,” Proc. SPIE **5005**, 29–37 (2003). [CrossRef]

8. B. L. Booth, “Low loss channel waveguides in polymers,” J. Lightwave Technol. **7**, 1445–1453 (1989). [CrossRef]

9. A. S. Kewitsch and A. Yariv, “Self-focusing and self-trapping of optical beams upon photopolymerization,” Opt. Lett. **21**, 24–26 (1996). [CrossRef] [PubMed]

10. M. Yonemura, A. Kawasaki, S. Kato, and M. Kagami, “Polymer waveguide module for visible wavelength division multiplexing plastic optical fiber communication,” Opt. Lett. **30**, 2206–2208 (2005). [CrossRef] [PubMed]

11. A. C. Sullivan, M. W. Grabowski, and R. R. McLeod, “Three-dimensional direct-write lithography into photopolymer,” Appl. Opt., **46**, 295–301 (2007). [CrossRef]

12. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. **21**, 1729–1731 (1996). [CrossRef] [PubMed]

19. T. Wilson, Y. Kawata, and S. Kawata, “Readout of three-dimensional optical memories,” Opt. Lett. **21**, 1003–1005 (1996). [CrossRef] [PubMed]

20. C. J. Cogswell and J. W. O’Byrne, “High-resolution confocal transmission microscope, Part I: system design,” Proc. SPIE **1660**, 503–511 (1992). [CrossRef]

21. M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proceedings of the National Academy of Sciences **99**, pp. 5788–5792 (2002). [CrossRef]

22. M. M. Woolfson, *An introduction to X-ray crystallography*, ed. 2 (Cambridge University Press, 1997). [CrossRef]

## 2. Optical diffraction tomography: theory

*n*, as

*E*(

*x*,

*z*) is the electric field of the probe beam,

*n*(

*x*,

*z*) =

*n*+

_{0}*δn*(

*x*,

*z*) is the sum of the background index and the index perturbation, and

*k*=2π/λ

_{0}_{0}is the wave number in free space. This simplification limits the applicable resolution of the tomography system to the order of a wavelength of the probe beam.

*E*(

_{sc}*x*,

*z*) and the incident field,

*E*(

_{inc}*x*,

*z*=

*0*), where the incident field is the solution to the homogeneous wave equation. In this weak-scattering limit, the total field on the right hand side of Eq. (1) can be replaced with the incident field; the wave equation then simplifies to

*δn*and radii of index structures,

*a*, for

*aδn*≤ 0.08λ [25

25. B. Chen and J. J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. **37**, 2996–3006 (1998). [CrossRef]

*E*(

_{sc}*k*,

_{x}*L*) is the one dimensional Fourier transform of the scattered field measured at a distance, L, from the object,

*E*(

_{inc}*k*,

_{x}*z*=

*0*) is the one dimensional Fourier transform of the incident field, and

*k*and

_{x}*k*are the spatial frequency coordinates in Fourier space. The

_{z}*k*values are defined in terms of the

_{z}*k*coordinates and the spatial frequency of the incident wave,

_{x}*k*, such that

*k*= (

_{z}*k*

^{2}-

*k*

_{x}^{2})

^{1/2}. The scattered field measurements are therefore taken on arcs of the

*k*-sphere in Fourier space as shown in Fig. 1. These measurements of the scattered field at various incident angles are used to determine the Fourier transform of the index structure, which is then inverse Fourier transformed to find the index profile in real space.

^{2}computations while an FFT requires NlogN computations, so an FFT will be much faster. Another solution to this problem is interpolation onto a regular grid in the Fourier domain [27

27. S. X. Pan and A. C. Kak, “A computation study of reconstruction algorithms for Diffraction Tomography: Interpolation versus Filtered Backpropagation,” IEEE Trans. Acoust. Speech Signal Process. **ASSP-31**, 1262–1275 (1983). [CrossRef]

28. A. J. Devaney, “A Filtered Backpropagation Algorithm for Diffraction Tomography,” Ultrasonic Imaging **4**, 336–350 (1982). [CrossRef] [PubMed]

*δn*, is the convolution of the index change of a single line,

_{tot}*δn*, with a comb spaced at the grating period, Λ,

_{obj}_{x}represents the one dimension convolution in

*x*. The comb(

*x*) and its Fourier transform, comb(

*k*) are defined as

_{x}*A*is the amplitude of the Gaussian field, and

_{inc}*w*is the 1/e electric field radius. Using the expressions for the Fourier transform of the index perturbation and field in Eq. (4), we find

*k*as seen from the result of the convolution integral:

_{x}*m*order of the diffraction grating in terms of the Fourier transform of the index perturbation at the particular value of

^{th}*k*and

_{x}*k*corresponding to that order:

_{z}*m*order is then given by:

^{th}*k*= 2πm/Λ and the corresponding

_{x}*k*value can be given in terms of the ratio of the scattered power measured in the far field and the incident power,

_{z}*dk*= 2π/Λ. The sampling of the Fourier transform defines the length,

_{x}*L*= 2π/

_{x}*dk*= Λ, in real space over which the index can be reconstructed. Therefore, the grating period is the maximum length in

_{x}*x*over which the index can be reconstructed. If the grating period is significantly larger than the size of the feature to be imaged, measurement of these discrete diffracted orders is sufficient to reconstruct the index of a single object.

29. T. C. Wedberg and J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. **4**, 39–54 (1995). [CrossRef]

30. E. Wolf, “Determination of the amplitude and the phase of the scattered field by holography,” J. Opt. Soc. Am. **60**, 18–20 (1970). [CrossRef]

31. W. Singer, B. Dobler, H. Schreiber, K. Brenner, and B. Messerschmidt, “Refractive-index measurement of gradient-index microlenses by diffraction tomography,” Appl. Opt. **35**, 2167–2171 (1996). [CrossRef] [PubMed]

## 3. Direct-write lithography

11. A. C. Sullivan, M. W. Grabowski, and R. R. McLeod, “Three-dimensional direct-write lithography into photopolymer,” Appl. Opt., **46**, 295–301 (2007). [CrossRef]

^{TM}HDS3000 media [1

1. M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, and W. L. Wilson, “Holographic data storage media for practical systems,” Proc. SPIE **5005**, 29–37 (2003). [CrossRef]

32. InPhase Technologies, Tapestry Media, www.inphase-technologies.com.

*y*direction through the beam to draw a 5 mm line, with a constant change in index of refraction along

*y*. A shutter is used to prevent the light from exposing the material in between the writing of each line. The writing powers for the data presented in this report range from 0.5–30 μW focused to a 0.75 μm 1/e

^{2}radius. The size of the beam at the focus is measured using a razor blade knife scan. The speed of the stages during the writing process is 2 mm/s.

## 4. Optical diffraction tomography: experiment and results

*z*dimension, but are thin in the sense of Bragg diffraction and therefore the scattered intensity has many diffracted orders in the

*x*dimension in the far field. Each grating is rotated about the

*y*axis to measure the efficiency of the diffracted orders versus incident angle, as shown in Fig. 3. A beam block is used to block light from all other orders and any scattered light. A high dynamic range power meter and filters measure four orders of magnitude in intensity, which corresponds to 10 or 20 diffracted orders depending on the size of the index structure and the period of the grating. Because the index is real, its Fourier transform is symmetric, so we only need to take data for positive values of

*k*and

_{x}*k*. Data is also taken only in the far field without the use of a lens, avoiding traditional field-of-view constraints.

_{z}*x*,

*z*plane, as shown in Fig. 4. As expected, the cross section of the index qualitatively resembles the writing beam going through focus. Cross sections of this index are shown in Fig. 5. The cross section of

*δn*in the

*x*plane is taken at the center of the index in

*z*, where

*z*= 0. Similarly, the cross section of

*δn*in the

*z*plane is taken at

*x*= 0. The ripples in the index reconstruction in

*x*are caused by the finite sampling in Fourier space of the scattered field. For this data set, ten orders are measured for a grating with a period of 35 μm, corresponding to a maximum spatial frequency of

*k*= 1.8 μm

_{x max}^{-1}, which gives a minimum sinusoidal period of 3.5 μm in the reconstruction as seen in Fig. 5(a). The noise on the data has an amplitude of approximately 2 × 10

^{-4}. The cross sections show that the index change in this photopolymer is not linearly proportional to the intensity of the focused writing beam, since the index shown is larger in scale than the

*x*,

*z*cross section of the writing beam.

*k*and

_{x}*k*and observing the change in the full width at half maximum (FWHM) of the resulting index structures in

_{z}*x*and

*z*. As shown in Fig. 6, the FWHM’s in both

*x*and

*z*approach a constant value as the number of samples is increased. Therefore, adding more samples, i.e. covering a larger range of spatial frequencies, will not significantly change the reconstructed index. Consequently, the reconstructed images are not limited by the resolution of the measurement system.

33. C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” Proc. SPIE **2984**, 72–81 (1997). [CrossRef]

*x*direction. To compare more directly, the tomographic reconstruction of the index is differentiated along the

*x*direction and shown in Fig. 8(b). There is excellent agreement between the two images. There is also a much higher SNR for the tomography reconstruction due to the coherent addition of scattering from the individual index structures, giving a very clear image of the index change.

## 5. Conclusions

^{-4}. The transverse feature sizes are on the limit of the resolution allowed in the derivation of the expression for the index change in terms of scattered fields due to the approximation that the index is changing slowly on the order of a wavelength. For greater resolution, a more rigorous derivation without this approximation is necessary. The ability to change the period of the gratings also allows us to image a wide variety of index change amplitudes with good SNR. We have shown that for the class of smoothly varying, symmetric objects, optical diffraction tomography can be performed with intensity-only measurements, significantly simplifying the experiment and data analysis. We have fabricated many copies of index structures with the appropriate symmetries, reconstructed the index, and validated the reconstruction. A FFT beam propagation simulation is used to verify both the accuracy of the reconstruction and to provide a quantitative measure of the accuracy of the Born approximation. These images of weak index structures are essential for a variety of volume fabricated index structures where traditional microscopy cannot provide high-resolution, quantitative phase images.

## Acknowledgments

## References and links

1. | M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, and W. L. Wilson, “Holographic data storage media for practical systems,” Proc. SPIE |

2. | R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, “Cationic ring-opening photopolymerization methods for holography,” Proc. SPIE |

3. | W. S. Colburn and K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt. |

4. | G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. |

5. | V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. |

6. | J. T. Sheridan and J. R. Lawrence, “Nonlocal-response diffusion model of holographic recording in photopolymer,” J. Opt. Soc. Am. A |

7. | R. R. McLeod, A. J. Daiber, M. E. McDonald, T. L. Robertson, T. Slagle, S. L. Sochava, and L. Hesselink, “Microholographic multilayer optical disk data storage,” Appl. Opt. |

8. | B. L. Booth, “Low loss channel waveguides in polymers,” J. Lightwave Technol. |

9. | A. S. Kewitsch and A. Yariv, “Self-focusing and self-trapping of optical beams upon photopolymerization,” Opt. Lett. |

10. | M. Yonemura, A. Kawasaki, S. Kato, and M. Kagami, “Polymer waveguide module for visible wavelength division multiplexing plastic optical fiber communication,” Opt. Lett. |

11. | A. C. Sullivan, M. W. Grabowski, and R. R. McLeod, “Three-dimensional direct-write lithography into photopolymer,” Appl. Opt., |

12. | K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. |

13. | M. Will, S. Nolte, B. N. Chichkov, and A. Tünnermann, “Optical properties of waveguides fabricated in fused silica by femtosecond laser pulses,” Appl. Opt. |

14. | A. M. Kowalevicz, V. Sharma, E. P. Ippen, J. G. Fujimoto, and K. Minoshima, “Three-dimensional photonic devices fabricated in glass by use of a femtosecond laser oscillator,” Opt. Lett. |

15. | M. Pluta, |

16. | C. J. Cogswell and C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. |

17. | S. V. King , Department of Electrical and Computer Engineering, University of Colorado, Campus Box 425, Boulder, CO 80309, USA, A. Libertun, C. Preza, R. Piestun, and C. J. Cogswell are preparing a manuscript to be called “Quantitative phase microscopy through differential interference imaging.” |

18. | C. J. R. Sheppard and D. M. Shotton, |

19. | T. Wilson, Y. Kawata, and S. Kawata, “Readout of three-dimensional optical memories,” Opt. Lett. |

20. | C. J. Cogswell and J. W. O’Byrne, “High-resolution confocal transmission microscope, Part I: system design,” Proc. SPIE |

21. | M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proceedings of the National Academy of Sciences |

22. | M. M. Woolfson, |

23. | M. Born and E. Wolf, |

24. | A. C. Kak and M. Slaney, |

25. | B. Chen and J. J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. |

26. | R. T. Weverka, K. Wagner, R. R. Mcleod, K. Wu, and C. Garvin, “Low-Loss Acousto-Optic Photonic Switch,” in |

27. | S. X. Pan and A. C. Kak, “A computation study of reconstruction algorithms for Diffraction Tomography: Interpolation versus Filtered Backpropagation,” IEEE Trans. Acoust. Speech Signal Process. |

28. | A. J. Devaney, “A Filtered Backpropagation Algorithm for Diffraction Tomography,” Ultrasonic Imaging |

29. | T. C. Wedberg and J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. |

30. | E. Wolf, “Determination of the amplitude and the phase of the scattered field by holography,” J. Opt. Soc. Am. |

31. | W. Singer, B. Dobler, H. Schreiber, K. Brenner, and B. Messerschmidt, “Refractive-index measurement of gradient-index microlenses by diffraction tomography,” Appl. Opt. |

32. | InPhase Technologies, Tapestry Media, www.inphase-technologies.com. |

33. | C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” Proc. SPIE |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(110.6960) Imaging systems : Tomography

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: August 9, 2007

Revised Manuscript: October 5, 2007

Manuscript Accepted: October 10, 2007

Published: October 12, 2007

**Citation**

Amy C. Sullivan and Robert R. McLeod, "Tomographic reconstruction of weak, replicated index structures embedded in a volume," Opt. Express **15**, 14202-14212 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-14202

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### References

- M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003). [CrossRef]
- R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996). [CrossRef]
- W. S. Colburn and K. A. Haines, ‘‘Volume hologram formation in photopolymer materials,’’Appl. Opt. 10, 1636-1641 (1971). [CrossRef] [PubMed]
- G. Zhao and P. Mouroulis, ‘‘Diffusion model of hologram formation in dry photopolymer materials,’’J. Mod. Opt. 41, 1929-1939 (1994). [CrossRef]
- V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, "Quantitative model of volume hologram formation in photopolymers," J. Appl. Phys. 81, 5913-5923 (1997). [CrossRef]
- J. T. Sheridan and J. R. Lawrence, "Nonlocal-response diffusion model of holographic recording in photopolymer," J. Opt. Soc. Am. A 17, 1108-1114 (2000). [CrossRef]
- R. R. McLeod, A. J. Daiber, M. E. McDonald, T. L. Robertson, T. Slagle, S. L. Sochava, and L. Hesselink, "Microholographic multilayer optical disk data storage," Appl. Opt. 44, 3197-3207 (2005). [CrossRef] [PubMed]
- B. L. Booth, "Low loss channel waveguides in polymers," J. Lightwave Technol. 7, 1445-1453 (1989). [CrossRef]
- A. S. Kewitsch and A. Yariv, "Self-focusing and self-trapping of optical beams upon photopolymerization," Opt. Lett. 21, 24-26 (1996). [CrossRef] [PubMed]
- M. Yonemura, A. Kawasaki, S. Kato, and M. Kagami, "Polymer waveguide module for visible wavelength division multiplexing plastic optical fiber communication," Opt. Lett. 30, 2206-2208 (2005). [CrossRef] [PubMed]
- A. C. Sullivan, M. W. Grabowski, R. R. McLeod, "Three-dimensional direct-write lithography into photopolymer," Appl. Opt. 46, 295-301 (2007). [CrossRef]
- K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, "Writing waveguides in glass with a femtosecond laser," Opt. Lett. 21, 1729-1731 (1996). [CrossRef] [PubMed]
- M. Will, S. Nolte, B. N. Chichkov, and A. Tünnermann, "Optical properties of waveguides fabricated in fused silica by femtosecond laser pulses," Appl. Opt. 41, 4360-4364 (2002). [CrossRef] [PubMed]
- A. M. Kowalevicz, V. Sharma, E. P. Ippen, J. G. Fujimoto, and K. Minoshima, "Three-dimensional photonic devices fabricated in glass by use of a femtosecond laser oscillator," Opt. Lett. 30, 1060-1062 (2005). [CrossRef] [PubMed]
- M. Pluta, Advanced Light Microscopy Vol. 2. Specialized Methods, (Elsevier, NY 1989), pp. 146-197.
- C. J. Cogswell and C. J. R. Sheppard, "Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992). [CrossRef]
- S. V. King, Department of Electrical and Computer Engineering, University of Colorado, Campus Box 425, Boulder, CO 80309, USA, A. Libertun, C. Preza, R. Piestun, and C. J. Cogswell are preparing a manuscript to be called "Quantitative phase microscopy through differential interference imaging."
- C. J. R. Sheppard and D. M. Shotton, Confocal laser scanning microscopy, (BIOS Scientific, 1997).
- T. Wilson, Y. Kawata, and S. Kawata, "Readout of three-dimensional optical memories," Opt. Lett. 21, 1003-1005 (1996). [CrossRef] [PubMed]
- C. J. Cogswell, and J. W. O’Byrne, "High-resolution confocal transmission microscope, Part I: system design," Proc. SPIE 1660, 503-511 (1992). [CrossRef]
- M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, "Adaptive aberration correction in a confocal microscope," Proceedings of the National Academy of Sciences 99, 5788-5792 (2002). [CrossRef]
- M. M. Woolfson, An introduction to X-ray crystallography, ed. 2 (Cambridge University Press, 1997). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, ed. 7 (Cambridge University Press, 1999).
- A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).
- B. Chen and J. J. Stamnes, "Validity of diffraction tomography based on the first Born and the first Rytov approximations," Appl. Opt. 37, 2996-3006 (1998). [CrossRef]
- R. T. Weverka, K. Wagner, R. R. Mcleod, K. Wu, and C. Garvin, "Low-Loss Acousto-Optic Photonic Switch," in Acousto-Optic Signal Processing Theory and Implementation, N. J. Berg and J. M. Pellegrino, eds. (Marcel Dekker, 1996), pp. 479 - 573.
- S. X. Pan and A. C. Kak, "A computation study of reconstruction algorithms for Diffraction Tomography: Interpolation versus Filtered Backpropagation," IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262-1275 (1983). [CrossRef]
- A. J. Devaney, "A Filtered Backpropagation Algorithm for Diffraction Tomography," Ultrasonic Imaging 4, 336-350 (1982). [CrossRef] [PubMed]
- T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods for optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995). [CrossRef]
- E. Wolf, "Determination of the amplitude and the phase of the scattered field by holography," J. Opt. Soc. Am. 60, 18-20 (1970). [CrossRef]
- W. Singer, B. Dobler, H. Schreiber, K. Brenner, B. Messerschmidt, "Refractive-index measurement of gradient-index microlenses by diffraction tomography," Appl. Opt. 35, 2167-2171 (1996). [CrossRef] [PubMed]
- InPhase Technologies, Tapestry Media, www.inphase-technologies.com.
- C. J. Cogswell, N. I. Smith, K. G. Larkin, P. Hariharan, "Quantitative DIC microscopy using a geometric phase shifter," Proc. SPIE 2984, 72-81 (1997). [CrossRef]

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