## Homodyne scanning holography

Optics Express, Vol. 14, Issue 10, pp. 4280-4285 (2006)

http://dx.doi.org/10.1364/OE.14.004280

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### Abstract

We have developed a modified version of a scanning holography microscope in which the Fresnel Zone Plates (FZP) are created by a homodyne rather than a heterodyne interferometer. Therefore, during the scanning the projected pattern on the specimen is frozen rather than varied as previously. In each scanning period the system produces an on-axis Fresnel hologram. The twin image problem is solved by a linear combination of at least three holograms taken with three FZPs with different phase values.

© 2006 Optical Society of America

## 1. Introduction

1. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. **22**, 1268–1269 (1997). [CrossRef] [PubMed]

2. B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Threedimensional holographic fluorescence microscopy,” Opt. Lett. **22**, 1506–1508 (1997). [CrossRef]

4. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A **17**, 380–390 (2000). [CrossRef]

## 2. Principles of homodyne scanning holography

*p*(

*x,y*) is a disk function with the diameter

*D*that indicates the limiting aperture on the projected FZP,

*A*is a constant,

*θ*is the phase difference between the two spherical waves and

*λ*is the wavelength of the light source. The constant

*γ*indicates that at a plane

*z*=0 there is effectively interference between two spherical waves, one emerging from a point at

*z*=-γ and the other converging to a point at

*z*=γ. This does not necessarily imply that these particular spherical waves are exclusively needed to create the FZP. For a 3D specimen

*S(x,y,z)*the convolution with the FZP of Eq. (1) is,

*O(x,y;γ)*is a 2D function which is different for different values of the parameter

*γ*. This convolution result is similar to a conventional Fresnel on-axis digital hologram, and therefore, it suffers from the same problems. Specifically,

*O(x,y;γ)*of Eq. (2) contains three terms which represent the information on three images namely the 0

^{th}diffraction order, the virtual and the real images. Trying to reconstruct the image of the specimen directly from a hologram of the form of Eq. (2) would fail because of the disruption originated from two images out of the three. This difficulty is solved here with the same solution applied in an on-axis digital holography. Explicitly, at least three holograms of the same specimen are recoded, where for each one of them a FZP with a different phase value is introduced. A linear combination of the three holograms cancels the two undesired terms and the remaining is a complex valued on-axis Fresnel hologram which contents only the information of the single desired image, either the virtual, or the real one, according to our choice. A possible linear combination of the three holograms to extract a single convolution between the object and one of the quadratic phase function of Eq. (2), is

*Oi(x,y;γ)*is the ith recorded hologram of the form of Eq. (2) and

*θ*is the phase value of the

_{i}*i*th FZP used during the recording. The choice between the signs in the exponents of Eq.v(3) determines which image, virtual or real, is kept in the final hologram. If for instance the virtual image is kept,

*O*is the final complex valued hologram of the form,

_{F}(x,y;γ)*O*is the final hologram which contains the information of only one image - the 3D virtual image of the specimen in this case. Such image

_{F}(x,y;γ)*S’(x,y,z)*can be reconstructed from

*O*by calculating in the computer the inverse operation to Eq. (4), as follows,

_{F}(x,y;γ)*D*and the constant γ characterize the system resolution in a similar way to the effect of an imaging lens [5]. Suppose the image is a single infinitesimal point at

*z*=0, then

*O*gets the shape of a quadratic phase function limited by a finite aperture. The reconstructed point image has a transverse diameter of 1.22

_{F}(x,y;γ)*λγ/D*, which defines the transverse resolution, and an axial length of 8

*λγ*which defines the axial resolution. Note also that the width of the FZP’s last ring along its perimeter is about

^{2}/D^{2}*λγ/D*, and therefore the size of the specimen’s smallest distinguishable detail is approximately equal to the width of this ring.

## 3. Experimental results

*λ*=532nm) was split in two beams with beam expanders consisting each of a microscope objective and a 12-cm focal-length achromat as a collimating lens. One of the beams passed through an electro-optic phase modulator (New Focus 4002) driven by three (or more) constant voltage values which induce three (or more) phase difference values between the interfering beams. Note that unlike previous studies [2

2. B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Threedimensional holographic fluorescence microscopy,” Opt. Lett. **22**, 1506–1508 (1997). [CrossRef]

4. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A **17**, 380–390 (2000). [CrossRef]

*X-Y*piezo stage (Physik Instrument P-527). The data were collected by a GageScope CS1602 acquisition system, and data manipulation was performed by programs written in MATLAB.

_{1,2,3}=0, π/2, and π are shown in Fig. 2, respectively. In this figures it clearly appears that the dominant term is the low frequency term [the first in Eq. (2)], and therefore without mixing the three holograms in the linear combination that eliminates the low frequency along with the twin image term, there is no possibility to recover the desired image with a reasonable quality. These three holograms are substituted into Eq. (3) and yield a complex valued hologram shown in Fig. 3. This time the grating lines are clearly revealed in the phase pattern.

## 4. Conclusions

## Acknowledgments

## References and links

1. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

2. | B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Threedimensional holographic fluorescence microscopy,” Opt. Lett. |

3. | G. Indebetouw, A. El Maghnouji, and R. Foster, “Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,” J. Opt. Soc. Am. A |

4. | G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A |

5. | J. W. Goodman, |

**OCIS Codes**

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(090.2880) Holography : Holographic interferometry

(100.6890) Image processing : Three-dimensional image processing

(110.6880) Imaging systems : Three-dimensional image acquisition

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

**ToC Category:**

Holography

**History**

Original Manuscript: March 13, 2006

Revised Manuscript: May 3, 2006

Manuscript Accepted: May 4, 2006

Published: May 16, 2006

**Virtual Issues**

Vol. 1, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Joseph Rosen, Guy Indebetouw, and Gary Brooker, "Homodyne scanning holography," Opt. Express **14**, 4280-4285 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-10-4280

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### References

- I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1269 (1997). [CrossRef] [PubMed]
- B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, "Three-dimensional holographic fluorescence microscopy," Opt. Lett. 22, 1506-1508 (1997). [CrossRef]
- G. Indebetouw, A. El Maghnouji, and R. Foster, "Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus," J. Opt. Soc. Am. A 22, 892-898 (2005). [CrossRef]
- G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, "Imaging properties of scanning holographic microscopy," J. Opt. Soc. Am. A 17,380-390(2000). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), 126-165.

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