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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 1, Iss. 6 — Jun. 13, 2006
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Homodyne scanning holography

Joseph Rosen, Guy Indebetouw, and Gary Brooker  »View Author Affiliations


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

In this paper we combine the two holographic techniques scanning and digital holography to a single method of recording on-line Fresnel holograms of microscopic fluorescence samples after laser excitation. In this modified system, the hologram is recorded without temporal carrier frequency using homodyne interferometer. By doing this we offer an improved method of 3D imaging which can be applied to fluorescence microscopy. To the best of our knowledge, it is first time that scanning holography is demonstrated by a homodyne rather than heterodyne interferometer.

2. Principles of homodyne scanning holography

In classic heterodyne scanning holography a pattern of Fresnel Zone Plates (FZP) scans the object and at each and every scanning position the light intensity is integrated by a detector. The overall process is a 2D convolution operation between the object and the FZP patterns. Our modified scanning holography is also a method of convolution by scanning the object with a set of frozen-in-time FZP patterns. The FZP is created by interference of two mutually coherent spherical waves. As shown in Fig. 1, the interference pattern is projected on the specimen, scans it in 2D, and the reflected light from the specimen is integrated on an area detector. Due to the line-by-line scanning by the FZP along the specimen, the one dimensional signal detected is composed of the entire lines of the convolution matrix between the object function and the FZP. In the computer, the detected signal is reorganized in the shape of a 2D matrix the values of which actually represent the Fresnel hologram of the specimen. The specimen we consider is 3D, and its 3D structure is stored in the hologram by the effect that during the convolution, the number of cycles of the FZP (its Fresnel number) contributed from a distant object point is slightly smaller than the number of cycles of the FZP contributed from closer object points.

Fig. 1. Optical setup: EOPM, electro-optic phase modulator introducing a phase difference between the two beams; BS, beam splitter; PI, piezo XY stage, OBJ, objective; PM, photomultiplier tube detector; LPF, lowpass filter; PC, personal computer.

As mentioned above the FZP is the intensity pattern of the interference between two spherical waves given by,

F(x,y,z)=Ap(x,y){2+exp[iπλ(γ+z)(x2+y2)+iθ]+exp[iπλ(γ+z)(x2+y2)iθ]},
(1)

where 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)=S(x,y,z)*F(x,y,z)=2AS(x,y,z)*p(x,y)dz
+AS(x,y,z)p(xx,yy)exp{iπ[(xx)2+(yy)2]λ(γ+z)+iθ}dxdydz
+AS(x,y,z)p(xx,yy)exp{iπ[(xx)2+(yy)2]λ(γ+z)iθ}dxdydz,
(2)

where the asterisk denotes a 2D convolution. Note that 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 0th 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

Fig. 2. Three recorded holograms with phase difference between the two interferometers arms of (a) 0 (b) π/2 and (c) π.
OF(x,y;γ)=O1(x,y;γ)[exp(±iθ3)exp(±iθ2)]+O2(x,y;γ)[exp(±iθ1)exp(±iθ3)]
+O3(x,y;γ)[exp(±iθ2)exp(±iθ1)],
(3)

where Oi(x,y;γ) is the ith recorded hologram of the form of Eq. (2) and θi is the phase value of the ith 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, OF(x,y;γ) is the final complex valued hologram of the form,

OF(x,y;γ)=S(x,y,z)*p(x,y)exp[iπλ(γ+z)(x2+y2)]dz.
(4)

The function OF(x,y;γ) is the final hologram which contains the information of only one image - the 3D virtual image of the specimen in this case. Such image S’(x,y,z) can be reconstructed from OF(x,y;γ) by calculating in the computer the inverse operation to Eq. (4), as follows,

S(x,y,z)=OF(x,y;γ)*exp[iπ2λz(x2+y2)].
(5)

The resolution properties of this imaging technique are determined by the properties of the FZP. More specifically the diameter D and the constant γ characterize the system resolution in a similar way to the effect of an imaging lens [5

5. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), 126–165.

]. Suppose the image is a single infinitesimal point at z=0, then OF(x,y;γ) gets the shape of a quadratic phase function limited by a finite aperture. The reconstructed point image has a transverse diameter of 1.22λγ/D, which defines the transverse resolution, and an axial length of 8λγ2/D2 which defines the axial resolution. Note also that the width of the FZP’s last ring along its perimeter is about λγ/D, and therefore the size of the specimen’s smallest distinguishable detail is approximately equal to the width of this ring.

Fig. 3. (a) The magnitude and (b) The phase of the final hologram. [Media 1]

3. Experimental results

Fig. 4. Movie of the image reconstruction along the light propagation axis.

The three recorded holograms of the specimen taken with phase difference values of θ1,2,3=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.

The computer reconstruction of two pollen grains along the z axis is shown in the movie of Fig. 4. As can be seen in this movie different parts of the pollen grains are in focus at different transverse planes.

4. Conclusions

Acknowledgments

The authors thank Maria DeBernardi for valuable comments, Wenwei Zhong for technical help with the programming, and Brian Storrie for his role in envisioning the construction of a holographic microscope. This research is supported by the National Science Foundation grant DBI-0420382. J. Rosen’s research is supported in part by the Israel Science Foundation grant 119/03. G. Indebetouw’s research is supported in part by the National Institute of Health grant 5R21 RR18440.

References and links

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]

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 22, 892–898 (2005). [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]

5.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), 126–165.

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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