## Proposal of three-dimensional phase contrast holographic microscopy

Optics Express, Vol. 15, Issue 20, pp. 12662-12679 (2007)

http://dx.doi.org/10.1364/OE.15.012662

Acrobat PDF (543 KB)

### Abstract

We propose a three-dimensional phase contrast digital holographic microscopy. The object to be observed is a low-contrast transparent refractive index distribution sample, such as biological tissue. Low contrast phase objects are converted to high contrast images through the microscopy we propose. In order to gain high three-dimensional resolution, the direction of pump plane wave is scanned, and separate holographic images produced at each angle are acquired and decoded into complex amplitude in Fourier space. The three-dimensional image is reconstructed in a computer from all information acquired through the system. The resolution in the direction of the optical axis is increased by utilizing a 4π configuration of objective lenzes.

© 2007 Optical Society of America

## 1. Introduction

4. I. Freund and M. Deutsch, “2nd-harmonic microscopy of biological tissue,” Opt. Lett. **11**, 94 (1986). [CrossRef] [PubMed]

7. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. **70**, 922 (1997). [CrossRef]

8. M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. **191**, 266 (1998). [CrossRef] [PubMed]

9. M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-Stokes Raman microscope,” Opt. Lett. **7**, 350 (1982). [CrossRef] [PubMed]

12. F. Zernike, “How I discovered phase contrast,” Science **121**, 345 (1955). [CrossRef] [PubMed]

13. W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt. **31**, 4973 (1992). [CrossRef] [PubMed]

17. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang,, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science , **254**1178 (1991). [CrossRef] [PubMed]

18. T. Dresel, G. Hausler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. **31**919 (1992). [CrossRef] [PubMed]

*OTF*) and an estimate of the error in object reconstruction are also discussed.

## 2. Setup of 3-D phase contrast holographic microscopy

13. W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt. **31**, 4973 (1992). [CrossRef] [PubMed]

*λ*/2 -A). The 0-order light is collimated by the tube lens and reflected by the scanning mirror again, which changes the angle of the 0-order light back into the original angle of the pump beam emerging from the light source. After reflection by the scanning mirror, the 0-order light converges onto a micromirror of about 200

*μ*m in diameter. While a 50/50 beam splitter inevitably leads to light loss in the detection path, it can be reduced by using an 80/20 beam splitter. The majority of the 0-order light is reflected by the micromirror and is used as a reference wave for the hologram. The remainder of the 0-order light is transmitted through the micromirror and advances toward the digital camera. A certain quantity of 0-order light is required to form the image. The transmitted 0-order light is used for compensation of phase error that will be discussed later.

*λ*/ 2 -B), and it propagates toward the digital camera. In the same way as the transmitted component, a few frequency components of the scattered reflected wave in the vicinity of the 0-order light are intercepted by the micromirror. While the well-known high-NA depolarization effect occurs in the primary objectives, two polarizers placed on the transmission (P-polarization) and reflection (S-polarization) sides ensure the linear polarization.

*λ*/4 ) of the same size as the micromirror. A part of the 0-order light transmitted through the PBS2 propagates to the digital camera along with the transmitted component of the scattered wave and the other part of the 0-order light reflected by the PBS2 propagates with the reflected component of the scattered wave. The two divided 0-order light parts is used to compensate the phase error for the transmitted and reflected components, which will be described in detail in Section 4. Although the micro quarter wave plate can be removed for a simple setup, in this case the phase error of the reflected component cannot be minimized. The digital camera can be divided into two parts, which are used for the transmission and reflection waves. Since the transmitted and reflected scattered waves and the reference wave are P-polarized on the observation plane of the digital camera, the transmission and reflection waves interfere with the reference wave, and holographic images are recorded through the digital camera. Since the central point of the scanning mirror is placed at the conjugate position of the middle point of the sample on the optical axis, the incident angle of the transmission and reflection waves is perpendicular to the digital camera while the pump beam scanned. The circular area of the scattered wave at the digital camera is shifted during scanning. The 0-order light is fixed at the central positions of the observation planes in each of part, as shown in Fig. 2. The incident angle of the reference wave onto the digital camera never varies during the scanning, since the 0-order light is always reflected at the center of the micromirror after being reflected twice by the scanning mirror. Holographic images of the transmission and reflection waves are recorded for each incident direction of the pump wave.

*h*=

*F*sin

*θ*is satisfied, where

*h*represents the height of the principal ray on the pupil plane,

*F*denotes the focal length, and

*θ*stands for the angle of the principal ray emitted from the central point on the object plane to the optical axis (see Fig. 3). In particular, a water immersion objective lens with numerical aperture (

*NA*) 1.2 can be designed and produced easily for the Fourier lens, whose specification is that the angular field of view of the incident plane wave on the object side is 1.2 in

*NA*in water and the

*NA*on the detector side is 0.0075 in air. In this case, if the focal length of the tube lens is 200mm and the magnification of the primary objective is 60X, the focal length of the primary objective is about 4.4mm in water, and the field of view in the sample is approximately 50

*μ*m. The digital camera is set to be conjugate to the pupil plane of the primary objective. If we assume that the optical system between the pupil plane of the primary objective and the detector plane comprises the image-formation system of magnification

*β*, we obtain the relation

*μ*represents the size of a pixel of the digital camera and

*N*denotes the number of the pixels along a side of the digital camera. The length of the side of detector plane is assumed to be equivalent to

*μN*, which is twice as long as the diameter of a circular area of the scattered wave on the digital camera.

## 3. Diffraction due to low contrast phase object

### 3.1 Electric field of the scattered wave

*E*

^{(i)}(

*x*′,

*t*) with wave vector

*n*

_{0}

*k*

_{0}is scattered by the object, the total electric field including the 0-order light and the scattered wave on an infinite-radius reference sphere satisfies the following equation [20]

*x*′ is the three-dimensional position on the reference sphere,

*c*is the speed of the light in vacuum,

*α*(

*x*) is the polarizability relative to average refractive index

*n*

_{0}, and

*N*(

*x*) denotes the number of molecules in a unit volume. The product

*α*(

*x*)

*N*(

*x*) is given by [20]

*ω*is the angular frequency of the light used and

*k*

_{0}=

*ω*/

*c*is the magnitude of the wave vector

*k*

_{0}. Since the reference sphere is located far from the object, that is ∣

*x*′-

*x*∣ ≫

*λ*/2

*π*, the integrand of the second term in Eq. (2) approximates [20]

*λ*is the wavelength of light in vacuum, and

*θ*is the angle between the direction of the effective electric field

*E*

_{0}′(

*x*) and the propagation direction of the scattered light exp[

*i n*

_{0}

*k*

_{0}∣

*x*′-

*x*∣]/∣

*x*′-

*x*∣, and

*φ*is azimuthal angle and

*r*is radius in polar coordinates. While Eq. (5) implies the well-known high-NA depolarization effect, two polarizers placed on the transmission and reflection sides ensure the linear polarization of the scattered waves, as mentioned above. Hereafter, the polarization effects of scattering are not taken into account, namely, the vector electric field shown by Eq. (5) is considered to be the scalar field whose amplitude is equivalent to

*E*with an approximation of sin

_{θ}*θ*= 1. Note that this is not the paraxial approximation. If only the scalar electric field is considered, the integrand of the second term in Eq. (2) is

*E*′(

*x*′,

*t*)→

*E*

_{0}′(

*x*′)

*e*

^{-iωt}and

*E*

^{(i)}(

*x*′,

*t*) →

*E*

_{0}

^{(i)}(

*x*′)

*e*

^{-iωt}, and substitution of Eqs. (3) and (6) into Eq. (2) yields

*ε*(the first order Born approximation) and substitution of

*E*

_{0}

^{(i)}(

*x*′) = exp[

*i n*

_{0}

*k*

_{0}·

*x*] yields

**in the sample to arbitrary position**

*x***on the reference sphere is infinite, the second term of the right side in Eq. (8) is proportional to the Fourier transform of**

*x*′*g*(

*x*), which is measured at the digital camera.

### 3.2 Diffraction efficiency

*g*(

*x*) = cos(

*Κ*·

*x*) = exp[

*i κ*·

*x*]/2 + exp[-

*i κ*·

*x*]/2, where

**is the three-dimensional grating vector. For simplicity, we will consider the diffraction due to only the positive frequency component exp[i**

*κ**κ*·

*x*]/2. Substitution of

*g*(

*x*) = exp[i

*κ*·

*x*]/2 into Eq. (8) yields

*k*,

_{x}*k*and

_{y}*k*are the

_{z}*x*,

*y*, and

*z*components of

*k*

_{0}, respectively. A central position of the sample is defined as origin of the coordinate and

*z*direction is optical axis of the system. While the integral over

*x*and

*y*is conducted from negative infinity to positive infinity, the integral domain

**, which corresponds to the size of the sample, is assumed to be sufficiently small compared with the distance between the origin and the position**

*x***on the reference sphere,**

*x*′*L*represents a thickness of the sample in the

*z*direction, and

*k*′,

_{x}*k*′, and

_{y}*k*′ are

_{z}*x*,

*y*, and

*z*components of wave vector of the scattered wave

*k*

_{0}′, and

*k*

_{0}

*x*′/

*r*′=

*kx*′

*etc*. are used, and

*x*in Eq. (13),

*a*=

*n*

_{0}

*k*+

_{x}*κ*-

_{x}*n*

_{0}

*k*′ and

_{x}*b*=

*n*

_{0}

*k*

_{0}/(2

*r*′). As

*r*′ approaches infinity,

*b*approaches zero. In order to execute the integral

*J*, the integrand is multiplied by a factor exp[-

_{x}*βx*

^{2}], where

*β*is real and positive and approaches zero β → +0 (

*β*≪

*b*). Then Eq. (14) becomes

*β*→ +0 (

*β*≪

*b*). The change of variables

*C*makes an angle

*φ*to the real axis. If

_{βb}*β*= (1/

*r*′)

^{α}(1 < α < 2), the

*C*and the real axis. Since there are no singularities of the integrand anywhere between

*C*and the real axis, one can again deform the contour to the real axis,

*a*=

*n*

_{0}

*k*+

_{x}*κ*-

_{x}*n*

_{0}

*k*and

_{x}*b*=

*n*

_{0}

*k*

_{0}/(2

*r*′) yields

*y*is

*π*/2 relative to 0-order light. When the Bragg’s condition

*n*

_{0}

*k*

_{0}+

*κ*-

*n*

_{0}

*k*

_{0}′=0 is satisfied, the diffraction efficiency

*η*is

*ε*is sufficiently small. This result is approximately true as long as

*πεL*/

*λ*< 1. For example,

*ε*< 0.006 when

*L*= 45

*μ*m and

*λ*= 850 nm. If

*πεL*/

*λ*≫1,

*η*becomes the square of sine function and oscillates, since there is a tight coupling between the 0-order light and the diffraction wave and the energy transfer between the two modes occurs periodically due to multi-scattering. Even if

*ε*is somewhat large

*ε*> 0.006, an image can be reconstructed through this measurement system. However, the image might be deformed compared with the original object.

## 4. Algorithm of object reconstruction

*N*×

*N*pixels. It is useful to consider the detection position of the scattered wave on the camera to correspond to wave number space, since the entrance pupil of the primary objective is located at infinity and the radius of the reference sphere is also infinity. The complex amplitude of the scattered wave on the detection plane is calculated numerically by utilizing a digital off axis holography technique [14–16

14. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. **33**, 179 (1994). [CrossRef] [PubMed]

*NA*of the scattered wave in the pupil plane of the primary objective (PO2) is confined by the apertures to be three times as large as that of the 0-order light.

*ϕ*of the reference wave to the normal of the detection plane is adjusted as

*λ*/sin

*ϕ*= 4

*μ*, which means that a unit cycle of the phase of the reference wave corresponds to four pixels of the digital camera. The difference in spatial frequency between a top and a bottom or a right side and a left side of the digital camera corresponds to 4

*NA*/

*λ*. In this case, the size of the object

*D*in real space, which is calculated by Fourier transforming the wave number space, is

*R*and the size of the object

*D*have the relation

*R*= 3

*D*/4.

*N*×

*N*pixels are Fourier transformed in the computer. As a result of that computation, the complex amplitude of the object wave is obtained on one side at the matrix and a conjugated wave appears on another side, as shown in Fig. 4. While the object wave and the conjugated wave overlap, due to the relation of

*R*=3

*D*/4, the central areas of the object and conjugated waves of the size of

*N*/4 do not overlap. Because the irradiance of the reference wave is adjusted by an attenuator to be about a hundred times (ten times in amplitude) as intense as that of the diffraction wave at the most intense position, only a central bright spot produced by the dc term appears. The autocorrelation of the object wave, which is supposed to appear around the central bright spot, is negligible. A section of the object wave of

*N*/4 ×

*N*/4 elements is cut out and the inverse Fourier transform is applied to this 2-D matrix, which implies that the size of the object to be reconstructed is restricted to

*D*/4 . As a result, the complex amplitude of the scattered wave on the pupil plane is obtained.

*N*/8 in Fig. 4, the circle is cut out and projected onto the “partial sphere” that corresponds to the reference sphere and is mapped into wave number space. The partial sphere lies in a three-dimensional matrix of

*N*/4 ×

*N*/4 ×

*N*/4 elements. The complex amplitudes in all elements other than those on the partial sphere are set to zero (see Fig. 4). In this step, the complex amplitude is projected in the z direction onto the digital partial sphere composed of a set of voxels across which the analog partial sphere passes. If a (x, y) element of the digital partial sphere consists of two voxels in the z direction, the amplitude is distributed equally to the two voxels. While this step can lead to some artifacts in real space, the error is reduced by using a partial sphere convoluted with a Gaussian function containing a few elements in the z direction. After the resultant partial sphere is digitalized, the amplitude is projected onto the digital partial sphere with a weight of the Gaussian distribution. In this case, the peripheral intensity of the image in real space in the z direction becomes weaker.

*π*/2 is added only to the 0-order light, and the square of the modulus is calculated. The same calculation is performed for every direction of the pump plane wave. The final reconstructed object with the size of

*D*/4 ×

*D*/4 ×

*D*/4 in real space is obtained by adding all of these calculated matrices. Since the square of the modulus is independent of shifts in the Fourier plane, a positioning of the twin partial sphere is not required. This method of the 3-D object reconstruction corresponds to image formation of conventional microscopy with an incoherent Kohler illumination system.

## 5. Three-dimensional image formation feature

*P*

_{T}(

*f*) for the transmission side and

*P*

_{R}(

*f*) for the reflection side which have a relation

*P*

_{T}(

*f*) =

*P**

_{R}(-

*f*), as shown in Fig. 6. For simplicity, it is assumed that

*P*

_{T}(

*f*) (

*P*

_{R}(

*f*)) is unity on the partial sphere for the transmission (reflection) side and zero outside the partial sphere. That is,

*πf*

_{0}(=

*n*

_{0}

*k*

_{0}) is scanned over the pupil function

*P*

_{T}(

*f*) and the scattered wave emerging from the object is transmitted though the both sides of the pupil

*P*(

*f*) =

*P*

_{T}(

*f*) +

*P*

_{R}(

*f*). The amplitude of the 0-order light, which can be assumed to be a real number after the phase is shifted by

*π*/2, is attenuated by the micromirror before arriving at the digital camera by the factor of 0 < a < 1. If the amplitude of the object is given by

*O*(

*x*), the amplitude on the twin partial sphere for a certain wave number of the pump wave 2

*πf*

_{0}is

*I*

_{I}(

*x*′) is given by

*P*(

*f*) =

*P*

_{T}(

*f*) +

*P*

_{R}(

*f*) is equivalent to the 3-D coherent point spread function

*U*(

*x*) =

*U*

_{T}(

*x*) +

*U*

_{R}(

*x*), which has the relations

*O*(

*x*) = 1 +

*ε*

_{0}

*o*(

*x*) where

*o*(

*x*) is proportional to

*g*(

*x*) and

*ε*

_{0}≪ 1. Inserting

*O*(

*x*) = 1 +

*ε*

_{0}

*o*(

*x*) into Eq. (29) yields

*P*

_{T}(

*f*

_{0})∣

^{2}d

*f*

_{0}= ∫∣

*U*

_{T}(

*x*′-

*x*)∣

^{2}d

*x*and ∣

*P*

_{T}(

*f*

_{0})∣

^{4}= ∣

*P*

_{T}(

*f*

_{0})∣

^{2}are used in the last term and the second order terms of

*ε*

_{0}are ignored. A further simple calculation leads to

*U*

_{R}

^{2}(

*x*′-

*x*)d

*x*= ∫

*U*

_{T}

^{2}(

*x*′-

*x*)d

*x*= 0 and õ(

*f*) is the Fourier transform of

*o*(

*x*). The 3-D optical transfer function is defined as

*P*(

*f*) =

*P*

_{T}(

*f*) +

*P*

_{R}(

*f*) and the pupil on transmission side

*P*

_{T}(

*f*) for the pump wave, that is

*OTF*in the case for

*NA*= 1.2 with the primary objective in water. The

*OTF*has rotational symmetry in the

*f*direction, which is the spatial frequency in the direction of the optical axis. A cross section involving the

_{z}*f*axis is described in the figure. Note that spatial frequencies along the optical axis cannot be resolved for both the transmitted and reflected components, since the scattered wave propagated in the vicinity of the 0-order light is intercepted by the micromirror, which is similar to the conventional phase contrast microscopy. The depth resolution is gained from the reflected component, and a part of the

_{z}*OTF*corresponding to the reflected component lies in the region known as the missing cone. Although the gap between the two portions of the

*OTF*corresponding to the transmitted and reflected components exists and the spatial frequency in the gap cannot be resolved, it can be reduced by using higher NA objectives.

*O*(

*x*) =

*O**(

*x*), which holds for a 3-D phase object with a low contrast refractive index distribution, Eq. (32) becomes

*O*(

*x*)∣

^{2}= 1 + 2ε

_{0}

*o*(

*x*) is effectively converted into ∣

*O*

_{a}(

*x*)∣

^{2}= (1 -

*a*) + 2

*ε*

_{0}

*o*(

*x*), which means that the initial contrast of the object

_{2ε0}is enhanced to

_{2ε0/(1-a)}by attenuation of the 0-order light. Finally, a simple equation for object reconstruction is obtained,

*O*(

*x*) = 1 +

*ε*

_{0}

*o*(

*x*) into Eq. (38) yields

*A*= ∫∣

*U*

_{T}(

*x*′-

*x*)\2 d

*x*and the second order terms of

*ε*

_{0}are ignored. The image irradiance of the reconstructed object through the second method is proportional to that of the first algorithm, as long as the sample is a low contrast object. The second method can reduce the computing time, because it requires only one 3-D Fourier transform in the final stage. The images of the low contrast object reconstructed by the two methods show identical optical features.

*OTF*in the same way as a conventional optical system. In the first method, each shifted partial sphere is divided by the

*OTF*before it is Fourier transformed. In the second method, the deconvolution is achieved by dividing the total amplitude in the frequency domain by the

*OTF*before taking the square of the modulus. Both methods are phase error free in the transmitted component, so low-frequency objects can be resolved with almost no error in reconstruction. While the relative phase shifts between different reflected holograms can be minimized by analyzing the phase in the overlap areas, a slight dc phase difference between the transmitted and reflected components remains because of the looped part.

## 6. Error estimation of object reconstruction

*σ*). The root mean square (RMS) error is given by

*N*represents the number of the elements in the 3-D matrix of the image,

*I*denotes the value of the

_{i}^{σ}*i*-th element in the image with phase error

*σ*, and

*I*is the value of the

_{i}^{0}*i*-th element in the image with no error. Figure 8 shows an example of the calculation result of the normalized RMS for a cubic object of 3.0

*μ*m in size. RMS is evaluated for

*σ*between zero and 2π×0.65. Each data point in Fig. 8 is an average of five hundred images, where a different Gaussian white noise distribution is used for each image. The calculation is performed with deconvoluted images. The object is well reconstructed as long as the standard deviation of the error is less than 0.2

*λ*, which can be achieved by an ordinary mechanical design. Since the phase error is only in the reflected component and the part of the

*OTF*corresponding to the reflected component is located apart from the origin, objects composed of only low spatial frequencies can be resolved almost perfectly. Different object distributions were also calculated, and it turns out that the configuration of the object does not strongly affect the RMS. All results have the tendency to produce a similar RMS versus

*σ*relationship.

## 7. Discussion

*N*

^{2}of the digital camera is 1024×1024 pixels for each of the transmitted and reflected components. In this case, the size of the 3-D matrix (

*N*/4)

^{3}for the image, which is Fourier transformed three-dimensionally in the computer, is 256

^{3}elements. The size of the reconstructed object in real space

*D*/4 is approximately 45

*μ*m (the resolution X/2NA is 354nm) if a pump beam of 850nm in wavelength is used as the light source and NA = 1.2.

^{3}, 128

^{2}×

*π*/4 scan directions are required for maximum resolution, where the coefficient

*π*/4 implies the circular pupil. However, the number of scanning directions can be reduced by balancing the quality of the image with the scanning time.

*π*/2 to the 0-order light, and an image only of the absorbing part is obtained separately from the phase part by not adding the phase shift. While the absorbing object can be resolved by other microscopy techniques, such as transmission confocal microscopy, one of the advantages of this system is to be able to visualize 3-D phase objects. As mentioned above, the first Born approximation is assumed in the generation of the scattered wave through the interaction between the object and the pump wave, which implies that the specimen must be considered to be a low contrast object. If the specimen has a high contrast refractive index distribution, the object cannot be well-reconstructed, and thus the system has this applicative limitation.

## 8. Conclusion

## References and links

1. | T. Wilson, |

2. | W. B. Amos, J. G. White, and M. Fordham, “Use of confocal imaging in the study of biological structures,” Appl. Opt. |

3. | G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, and N. Nanninga, “3-Dimensional imaging of biological structures by high resolution confocal scanning laser microscopy,” Scanning Microsc. |

4. | I. Freund and M. Deutsch, “2nd-harmonic microscopy of biological tissue,” Opt. Lett. |

5. | P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. |

6. | J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. |

7. | Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. |

8. | M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. |

9. | M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-Stokes Raman microscope,” Opt. Lett. |

10. | A. Zumbusch, G. R. Holtom, and X. S. Xie, “Vibrational microscopy using coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. |

11. | F. Zernike, “Das Phasenkontrastverfahren bei der mikroskopischen Beobachtung,“ Z. Tech. Phys. |

12. | F. Zernike, “How I discovered phase contrast,” Science |

13. | W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt. |

14. | U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

15. | J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. |

16. | S. Kostianovski, S. G. Lipson, and E. N. Ribak, “Interference microscopy and Fourier fringe analysis applied to measuring the spatial refractive-index distribution,” Appl. Opt. |

17. | D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang,, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science , |

18. | T. Dresel, G. Hausler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. |

19. | M. Mansuripur, |

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

21. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,“ Bell Syst. Tech. J , |

**OCIS Codes**

(110.0180) Imaging systems : Microscopy

(110.4850) Imaging systems : Optical transfer functions

(180.5810) Microscopy : Scanning microscopy

(180.6900) Microscopy : Three-dimensional microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: August 10, 2007

Revised Manuscript: September 11, 2007

Manuscript Accepted: September 13, 2007

Published: September 18, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Naoki Fukutake and Tom D. Milster, "Proposal of three-dimensional phase contrast holographic microscopy," Opt. Express **15**, 12662-12679 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12662

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### References

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