## High-resolution, wide-field object reconstruction with synthetic aperture Fourier holographic optical microscopy

Optics Express, Vol. 17, Issue 10, pp. 7873-7892 (2009)

http://dx.doi.org/10.1364/OE.17.007873

Acrobat PDF (1642 KB)

### Abstract

We utilize synthetic-aperture Fourier holographic microscopy to resolve micrometer-scale microstructure over millimeter-scale fields of view. Multiple holograms are recorded, each registering a different, limited region of the sample object’s Fourier spectrum. They are “stitched together” to generate the synthetic aperture. A low-numerical-aperture (NA) objective lens provides the wide field of view, and the additional advantages of a long working distance, no immersion fluids, and an inexpensive, simple optical system. Following the first theoretical treatment of the technique, we present images of a microchip target derived from an annular synthetic aperture (NA = 0.61) whose area is 15 times that due to a single hologram (NA = 0.13); they exhibit a corresponding qualitative improvement. We demonstrate that a high-quality reconstruction may be obtained from a limited sub-region of Fourier space, if the object’s structural information is concentrated there.

© 2009 Optical Society of America

## 1. Introduction

1. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett . **97**, 168102 (2006). [CrossRef] [PubMed]

2. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Digital Fourier holography enables wide-field, superresolved, microscopic characterization,” in ‘Optics in 2007’, Opt. Photonics News **18**, 29 (Dec. 2007). [CrossRef]

*entire*sample, specific to a particular, limited region of its Fourier spectrum. The Fourier spectra are then superposed to generate a large synthetic aperture, from which the high-quality image reconstruction can be obtained.

*a priori*sample structural information, specific regions of its Fourier spectrum with high information density may be targeted, allowing high-quality, wide-field reconstructions to be rapidly generated, despite the fact that other Fourier-spectral regions may be excluded.

7. D. Gabor, “A new microscopic principle,” Nature (London) **161**, 777–778 (1948). [CrossRef]

8. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett . **11**, 77–79 (1967). [CrossRef]

10. B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express **13**, 9361–9373 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-23-9361. [CrossRef] [PubMed]

13. J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett . **27**, 2179–2181 (2002). [CrossRef]

16. F. Le Clerc, M. Gross, and L. Collot, “Synthetic-aperture experiment in the visible with on-axis digital heterodyne holography,” Opt. Lett . **26**, 1550–1552 (2001). [CrossRef]

17. R. Binet, J. Colineau, and J. C. Lehureau, “Short-range synthetic aperture imaging at 633 nm by digital holography,” Appl. Opt . **41**, 4775–4782 (2002). [CrossRef] [PubMed]

18. J. R. Price, P. R. Bingham, and C. E. Thomas Jr., “Improving resolution in microscopic holography by computationally fusing multiple, obliquely illuminated object waves in the Fourier domain,” Appl. Opt . **46**, 827–833 (2007). [CrossRef] [PubMed]

26. Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy - approaching the linear systems limits of optical resolution,” Opt. Express **15**, 6651–6663 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-11-6651. [CrossRef] [PubMed]

27. Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy,” J. Opt. Soc. Am. A **25**, 811–822 (2008). [CrossRef]

28. T. Turpin, L. Gesell, J. Lapides, and C. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE **2566**, 230–240 (1995). [CrossRef]

*three-dimensional*spatial frequencies accessed by each recording. We quantify this effect, highlighting a recording scheme for which it is minimally problematic.

## 2. Theory

### 2.1. Fundamentals of the technique

1. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett . **97**, 168102 (2006). [CrossRef] [PubMed]

*j*2

*πvt*), where

*v*represents optical frequency, and

*t*time. We further prescribe that the Fourier transform operation,

*ℱ*, be applied to the complex amplitude distributions in the input plane (and its respective conjugates), but the

*inverse Fourier*transform operation,

*ℱ*, be applied to those in the output plane (and its conjugates). The spatial frequency variables are defined accordingly:

_{-1}*ξ, η*) to domain (

*v*), is defined:

_{ξ}, v_{η}*M*=

*λf*

_{1}

*f*

_{3}/

*f*

_{2}is a scaling constant, with units of squared length. (The quantity

*λ*is optical wavelength.)

*r*(

*ξ,η*), the distribution we wish to extract. If the object is illuminated by a wavefield described by the complex amplitude distribution

*A*(

_{i}*ξ, η*), then the complex amplitude distribution in Plane 1 is given by

*θ*with respect to the optical axis and azimuthal angle

_{i}*ϕ*with respect to the lateral coordinate system. Then

_{i}*A*(

_{i}*ξ,η*) =

*A*

_{0}exp [-

*j*2

*π*(

*γ*+

_{ξ}ξ*γ*)], where

_{η}η*A*

_{0}is, in general, a complex constant,

*γ*+

_{ξ}^{2}*γ*= [sin

_{ξ}^{2}*θ*/

_{i}*λ*] , and

*γ*

_{η}/

*γ*

_{ξ}= tan

*ϕ*. The effect of this wave is to impart a phase ramp to the distribution

_{i}*r*(

*ξ,η*), effectively shifting a “bandpass” range of spatial frequencies to “baseband”. That is, if the output-plane rectangular recording area is defined by the region |

*x*| ≤

*L*/2, |

*y*| ≤

*H*/2, that is, its dimensions are

*L*×

*H*, then the range of object spatial frequencies (

*v*) accessible to the recording are defined by the inequalities:

_{ξ}, v_{η}*λ, θ*, and

_{i}*ϕ*(through the auxiliary variables

_{i}*M, γ*=, and

_{ξ}*γ*). That is, by recording multiple holograms under different illumination-wave conditions, multiple regions of the object’s Fourier spectrum can be acquired and combined to generate high-resolution reconstructions. The illumination-wave directions

_{η}*θ*are illustrated in Fig. 1(a),(b). The equivalent parameters for the plane reference wave,

_{i}, ϕ_{i}*θ*and

_{r}*ϕ*, are also displayed (in parts (a),(c)).

_{r}*V*

_{1}(

*ξ, η*) which can be detected in the “far field” are not infinite, no matter how far the recording plane is “extended”. Instead, they correspond to scattered propagating waves near-orthogonal to the optical axis [36], and have modulus equal to 1/

*λ*. Thus, the detectable

*object*spatial frequencies (

*v*) are those which satisfy the inequality:

_{ξ}, v_{η}*γ*) is also restricted to values having modulus less than 1/

_{ξ},γ_{η}*λ*, then the object spatial frequencies which are accessible to the synthetic aperture approach when the illumination and detection angles are allowed to vary freely in all reflection configurations are those whose modulus is less than 2/

*λ*. Considering scattering within the plane of incidence, as depicted in Fig. 2(a), the detected object spatial frequency corresponding to the plane-of-incidence illumination/detection-angle pair (

*θ*), is equal to:

_{i}, θ_{d}*θ*may range from -

_{d}*π*/2 to

*π*/2.) We make the observation here that the effect of varying the detector position can be simulated by keeping it fixed on-axis and tilting the object instead [17

17. R. Binet, J. Colineau, and J. C. Lehureau, “Short-range synthetic aperture imaging at 633 nm by digital holography,” Appl. Opt . **41**, 4775–4782 (2002). [CrossRef] [PubMed]

26. Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy - approaching the linear systems limits of optical resolution,” Opt. Express **15**, 6651–6663 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-11-6651. [CrossRef] [PubMed]

*coherent transfer function*(CTF) for the microscopic imaging system can be constructed, that is, corresponding to the near-spatially invariant linear system relating the object structure to the output plane complex amplitude distribution. Ideally, aberration and apodisation effects can be ignored or corrected for; the CTF magnitude and phase will be near-constant over the region of support.

4. T. R. Hillman, S. A. Alexandrov, T. Gutzler, and D. D. Sampson, “Microscopic particle discrimination using spatially-resolved Fourier-holographic light scattering angular spectroscopy,” Opt. Express **14**, 11088–11102 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-23-11088. [CrossRef] [PubMed]

### 2.2. Procedure for optimizing individual-hologram reconstruction

*V*

_{obj}(

*ξ,η*) and

*U*

_{rec}(

*x,y*). (The distances

*d, g*, depicted are positive.) Then

_{obj}= ℱ(

*V*

_{obj}). In writing Eq. (9), we have noted that ℱ

^{-1}[

*k*

_{∆z}(

*x,y*)] = 𝓚

_{∆z}(

*ν*), that is, the Fourier transform of

_{x},ν_{y}*k*

_{∆z}is equal to its inverse Fourier transform.

*d,g*are known, then Eq. (9) indicates the procedure for obtaining 𝒱

_{obj}, and thus

*V*

_{obj}, from the recording plane distribution

*U*

_{rec}(

*x,y*). That is, it is necessary to sequentially invert the nested sequence of operations on the right-hand side of the equation. As indicated by Eq. (8), defocus-correction in a given plane is achieved by multiplying the Fourier or inverse-Fourier transformed distribution by a circularly symmetric quadratic phase factor.

*L*×

*H*bounding rectangle in Plane 4.) The trade-off is the simultaneous reduction in the object’s field of view. This can be understood through the influence of the quadratic phase factor 𝓚

_{d}(

*ν*), which exhibits large

_{x}, ν_{y}*local*spatial frequencies when its arguments take on high values, ultimately exceeding the Nyquist limit set by the sampling rate of the (

*ν*)-distribution, which in turn is determined by the finite size of the recording-plane detector. We impose the restriction that the local spatial frequencies be much less than this Nyquist limit, over the entire object field of view. If

_{x}, ν_{y}*D*

_{4}is a representative “maximum diameter” of the recording-plane detection region (i.e.,

*D*

_{4}≈

*L,H*), and

*D*

_{1}is a representative diameter of the object-plane field of view, then we obtain the following inequality for

*d*(and similarly, for

*g*):

*f*

_{2}

*D*

_{1}/

*f*

_{1}), then the spot sizes are determined by the diffraction limit, and geometrical optics, respectively. Their effective areas (defined as the ratio of total optical power to peak intensity) are (

*M*/

*D*

_{1})

^{2}and (

*λdD*

_{1}/

*M*)

^{2}, respectively. The defocusing dynamic range advantage is given by the ratio of these quantities, provided the spots corresponding to separate diffraction orders don’t overlap. The quantities may also be used to determine the number of detector pixels that sample each spot.

_{d}(

*ν*) and 𝓚′

_{x}, ν_{y}_{g}(

*ν*), which should be substituted for their non-primed equivalents in Eq. (9).

_{ξ}, ν_{y}_{d}and 𝓚′

_{g}does not fully address the issue of the order in which these correction operations should be applied. This problem is not so severe as one might suppose because, if the effects are minor, the order in which a large-scale multiplicative factor and a small-scale convolution kernel are applied to a distribution is of negligible consequence. Thus, if we assume that defocus remains the dominant distorting effect, then Eq. (9), incorporating the generalized functions, appropriately describes the optical system. By the same assumption, the inequalities of Eq. (10) remain applicable.

_{d}, 𝓚′

_{g}using polynomial phase factors, restricted to the sixth order for the purposes of this paper. That is,

_{d,g}, both with one independent parameter (

*d*or

*g*), with 𝓚′

_{d,g}, both with

*twenty-five*independent parameters (the “

*D*”s or “

_{n,m}*G*”s). Since these parameters (with the possible exception of object plane defocus) are properties of the optical system, not of the sample under investigation, then they may, in general, be estimated using a target, or “control” sample, and the values determined from this approach applied to more general sample choices. For example, in order to estimate 𝓚′

_{n,m}_{d}, one might choose to utilize a strongly diffracting, structured sample. Then

_{d}can be optimized by ensuring that the spots in the Fourier plane corresponding to multiple diffraction orders are as tightly focused as possible. To this end, we invoke a sharpness metric maximization algorithm [38

38. S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A **25**, 983–994 (2008). [CrossRef]

_{d}has modulus 1, then by Parseval’s theorem, the integral ∬

_{-∞}

^{∞}|

*Û*

_{4}(

*x,y*)|

^{2}d

*x*d

*y*does not depend on it (conservation of total power). However, the integral

*M*= ∬

_{d}_{-∞}

^{∞}|

*Û*

_{4}(

*x,y*)|

^{4}dxdy will be greatest when the optical power is concentrated into few sharp, bright points. Thus, maximizing this quantity will ensure that the focused peaks are maximally distinct from the background.

38. S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A **25**, 983–994 (2008). [CrossRef]

39. M. King, “Matlab m-files for multidimensional nonlinear conjugate gradient method” (2005), http://users.ictp.it/?mpking/cg.html, accessed 12 December 2008.

*M*. The optimization procedure was initially applied to the lowest-order parameters alone, with the estimated results being used to initialize the routine as higher-order parameters were cumulatively incorporated in successive iterations. Indeed, the lower-order parameters (second and third) had the greatest influence in maximizing

_{d}*M*higher-order parameters were fitted with diminishing significance, and robustness. Nonetheless, their increasingly marginal impact was beneficial, which justified their inclusion.

_{d}_{d}has been determined, then the coefficients of

_{g}can be determined in a similar way, utilizing a metric

*M*. (This assumes, of course, that the target object is sufficiently structured so that maximizing the image contrast is equivalent to bringing it into optimum focus.)

_{g}### 2.3. Correlation between separately recorded holograms

*first Born approximation*[29, Sub-section 13.1.2]. Under this assumption, each pair of illumination and detection wavevectors corresponds to a particular 3D spatial frequency component of the sample. That is, if the sample is illuminated by a monochromatic plane wave with wavevector

**k**

_{0}, then the plane-wave component of the scattered light with wavevector

**k**has complex amplitude proportional to the component of the sample 3D

*angular*spatial frequency

**K**, where

**K**=

**k**-

**k**

_{0}. Each such component will correspond to a point of the far-field complex amplitude distribution, or, alternatively, to a point in the Fourier-plane complex amplitude distribution.

**k**

_{0}, the locus of points corresponding to the tip of the vector

**K**is a sphere, of radius

*k*= |

**k**

_{0}|, known as

*Ewald’s sphere of reflection*[29, p.701]. The union of all such spheres is the surface and interior of another sphere, the

*Ewald limiting sphere*, which has radius 2

*k*and is centered at the origin.

40. S. S. Kou and C. J. R. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express **15**, 13640–13648 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-21-13640. [CrossRef] [PubMed]

*F*(

_{S}*ξ, η, z*) [29, p.696], or its Fourier spectrum ℱ

_{S}(

*ν*) = ℱ(

_{ξ}, ν_{z}*F*).

_{S}*F*, which is:

_{S}_{FS}, we may define the autocorrelation function of ℱ

_{S}but consider only variations in the

*ν*direction, represented by the term ∆

_{z}_{νz}. That is,

*θ*is small.

_{i}_{h}

^{2}. Then the expected axial intensity reflectance function

*f*(

_{A}*z*) will be proportional to the probability density function (pdf) of a zero-mean Gaussian variable with this variance, that is:

*θ*= 45°, and the wavelength

_{i}*λ*= 632.8nm, then the normalized correlation function modulus |

*μ*| will not fall below 1/

_{ℱS}*e*provided that σ

_{h}∆

*ν*> 0.23. If the surface roughness is such that σ

_{z}_{h}/

*λ*= 5, then by Eq. (17), the maximum allowed polar angle deviation between measurements would be about 3.6°.

*ϕ*, an effective method for evading the decorrelation issue is to vary this angle alone between holographic recordings, keeping the polar angle

_{i}*θ*fixed. This limits the synthetic CTF to an annular-shaped region (see Fig. 4(c)), but it is the approach we adopt in the current paper. For such a CTF, the synthesized images will resemble those generated using dark-field coherent microscopy. We note finally that virtually all real objects, as opposed to hypothetical, ideal ones such as perfect phase gratings, scatter sufficiently to produce some spectral intensity over all regions of the measured frequency space. Thus, providing the 3D constraint is satisfied, separately recorded, overlapping spatial-frequency spectra should exhibit measurable correlation. For the same reason, images of highly scattering targets will demonstrate a quantitative resolution improvement with increasing synthetic aperture area, no matter what illumination polar angle is chosen.

_{i}## 3. Experimental setup and Methodology

*λ*= 632.8 nm) was split into sample and reference arms using the beamsplitter B1. A telescope system is used to expand the reference beam. The object is plane-wave illuminated off-axis and its scattered and diffracted light follows the optical path described in Section 2. Lenses L1, L2, and L3 have focal lengths 40 mm, 150 mm, and 400 mm, respectively. The objective (L1) is a Mitutoyo infinity-corrected long-working-distance objective, Mitutoyo Plan Apo 5×, with NA = 0.14 and working distance 34 mm. The object is placed on a rotation stage; multiple holograms are recorded by rotating it clockwise in increments of 4°. That is, the illumination conditions were held fixed over the entire sequence of recordings; however, the azimuthal angle

*ϕ*was effectively rotated

_{i}*anti-clockwise*in increments of 4° relative to the sample. For this reason, the rectangle of Fig. 2(c) corresponding to accessible region of the Fourier plane does not maintain the same orientation as the axes shown; instead, it also rotates about the origin. The polar angle of illumination selected was

*θ*= 62°.

_{i}*μ*m × 9

**μ**m resulting in an active imaging area of 36 mm × 24 mm (4008×2672 pixels). For each rotation angle, three exposures were taken, the hologram, and “reference” and “sample” recordings (achieved by blocking the beam in the other arms). By subtracting the two last-mentioned recordings from the hologram, the non-interference terms in the reconstruction could be suppressed [4

4. T. R. Hillman, S. A. Alexandrov, T. Gutzler, and D. D. Sampson, “Microscopic particle discrimination using spatially-resolved Fourier-holographic light scattering angular spectroscopy,” Opt. Express **14**, 11088–11102 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-23-11088. [CrossRef] [PubMed]

*μ*rad over 720° of rotation.

_{d}, 𝓚′

_{g}. Based on the assumption that these functions are not sample-dependent, the median values of the Legendre polynomial coefficients obtained from multiple different holograms (illumination angles), were selected as those to be applied globally to the set of 90.

*d*, which was deliberately set to 7 cm, for the reasons given in Sub-section 2.2. Given that the rectangular field stop in Plane 3 limited the field of view of the object to 2.9 mm × 2.9 mm, and that the measured parameter

*g*was always less than 100

*μ*m, then it is readily confirmed that the inequalities of Eq. (10) are easily satisfied. The choice of

*d*enabled Fourier-plane diffraction-limited spots to be sampled by over 40,000 detector pixels instead of merely 7.

*g*exists between successive reconstructions. This can be corrected by applying the multiplicative factor 𝓚

_{∆g}(

*ν*), from Eq. (8), to one Fourier spectrum, and the factor 𝓚

_{ξ}, ν_{η}_{-∆g/2}(

*ν*) to the other. The optimum “relative defocus” parameter ∆g between successive holograms is chosen so that the phase difference between their overlapping Fourier spectra is best approximated by a linear ramp. (Expressing the phase difference as the imaginary argument of an exponential function, this is equivalent to ensuring the modulus of the peak of its inverse Fourier transform is maximized.) Of course, the magnitude and orientation of the phase ramp corresponds to the displacement between the reconstructions. Once it has been compensated for, the phase difference between the overlapping regions should be near-constant. One of the holograms should be multiplied by a constant phase factor to set this constant to zero.

_{ξ}, ν_{η}*first*and the

*final*in the sequence. They are, of course, linked by the chain consisting of all the intermediate holograms, but they also overlap in their own right. Any errors between this pair must be corrected for, of course, completing the chain 1 → 2 → 3 → … → final → 1. The residual translation/relative phase errors associated with the chain must be distributed evenly about it. A final position-dependent phase-correction factor can be applied to the annular synthetic aperture, corresponding to a slowly varying function with a single argument: polar angle. Its functional form should be describable using only a few parameters, in our case its values at integer multiples of

*π*/4, which can be optimized using the sharpness metric maximization approach.

## 4. Results

_{d}yield the apparent systematic errors. Next, we see that the reconstructions clearly selectively highlight those object features oriented orthogonally to the displacement of the spectral region with respect to the origin. Most of the regular structure of the object is aligned in the vertical and horizontal directions, so indeed, the strongest signal in the Fourier spectrum is located in those directions. However, for all directions, object features and imperfections such as scratches are only visible over a narrow range of angles. The varying “brightness” of the seemingly homogeneous rectangular box on the left-hand side of the object reconstruction indicates the magnitude of the high-resolution structure within this region corresponding to each angle. Finally, we note the presence of significant reconstruction artifacts at the angles corresponding to the strongest signal. These are due, in general, to the residual non-image terms of the holographic reconstruction.

*μ*m

_{-2}, 0.47

*μ*m

^{-2}, and 2.1

*μ*m

^{-2}. These are equivalent to objective lens NAs (in the absence of a central aperture stop) of 0.13, 0.24, and 0.52. The maximum object spatial frequencies accessed by our synthetic aperture are equal to those of an objective with NA = 0.61. (The discrepancy between the values 0.52 and 0.61 is due to the fact that our synthetic aperture is an annulus, not a solid circle.) To convert the quantities to resolution values in the object-reconstruction domain, we consider the

*effective areas*of the squared moduli (intensity distributions) of the apertures’ associated complex-amplitude point-spread functions. The effective area, defined in a similar manner to the identically named quantity in Sub-section 2.2, is equal to the ratio of the integral (over all space) of an intensity distribution to its peak value. Conveniently, by Parseval’s theorem, it is equal to the inverse of the aperture area. A one-dimensional resolution parameter can be equated with the diameter of a circle whose area is equal to the effective area. For the three cases above, respectively, the resolutions, thus defined, are 3.0

*μ*m, 2.6

*μ*m × 1.0

*μ*m, and 0.77

*μ*m, respectively. (The non-cylindrical symmetry of the intermediate case is represented through the major and minor diameters of an ellipse.) These resolutions represent ideal values, assuming that aberrations over the entire extent of the aperture have been fully compensated for. Clearly, this may not be the case even when the “final phase-correction factor” described in the penultimate paragraph of Sub-section 2.3 has been applied. This issue is discussed further in the following section.

## 5. Conclusions

*a priori*targeting of particular Fourier-spectral frequency ranges will limit the number of holograms required to generate high-quality reconstructions. This is true for the target utilized in our current experiment; most of its spectral information was concentrated in the vertical or horizontal directions.

## Acknowledgment

## References and links

1. | S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett . |

2. | S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Digital Fourier holography enables wide-field, superresolved, microscopic characterization,” in ‘Optics in 2007’, Opt. Photonics News |

3. | S. A. Alexandrov, T. R. Hillman, and D. D. Sampson, “Spatially resolved Fourier holographic light scattering angular spectroscopy,” Opt. Lett . |

4. | T. R. Hillman, S. A. Alexandrov, T. Gutzler, and D. D. Sampson, “Microscopic particle discrimination using spatially-resolved Fourier-holographic light scattering angular spectroscopy,” Opt. Express |

5. | S. A. Alexandrov and D. D. Sampson, “Spatial information transmission beyond a systems diffraction limit using optical spectral encoding of the spatial frequency,” J. Opt. A - Pure Appl. Opt . |

6. | M. Ryle and A. Hewish, “The synthesis of large radio telescopes,” Mon. Not. R. Astron. Soc. |

7. | D. Gabor, “A new microscopic principle,” Nature (London) |

8. | J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett . |

9. | U. Schnars and W. Jueptner, |

10. | B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express |

11. | C. J. Mann, L. Yu, C.-M. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express |

12. | M. Sebesta and M. Gustafsson, “Object characterization with refractometric digital Fourier holography,” Opt. Lett . |

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

14. | L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express |

15. | J. Di, J. Zhao, H. Jiang, P. Zhang, Q. Fan, and W. Sun, “High resolution digital holographic microscopy with a wide field of view based on a synthetic aperture technique and use of linear CCD scanning,” Appl. Opt . |

16. | F. Le Clerc, M. Gross, and L. Collot, “Synthetic-aperture experiment in the visible with on-axis digital heterodyne holography,” Opt. Lett . |

17. | R. Binet, J. Colineau, and J. C. Lehureau, “Short-range synthetic aperture imaging at 633 nm by digital holography,” Appl. Opt . |

18. | J. R. Price, P. R. Bingham, and C. E. Thomas Jr., “Improving resolution in microscopic holography by computationally fusing multiple, obliquely illuminated object waves in the Fourier domain,” Appl. Opt . |

19. | C. Liu, Z. G. Liu, F. Bo, Y. Wang, and J. Q. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett . |

20. | M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro, “Super-resolution in digital holography by a two-dimensional dynamic phase grating,” Opt. Express |

21. | V. Mico, Z. Zalevsky, P. Garcia-Martinez, and J. Garcia, “Single-step superresolution by interferometric imaging,” Opt. Express |

22. | C. Yuan, H. Zhai, and H. Liu, “Angular multiplexing in pulsed digital holography for aperture synthesis,” Opt. Lett . |

23. | V. Mico, Z. Zalevsky, P. García-Martínez, and J. García, “Synthetic aperture superresolution with multiple off-axis holograms,” J. Opt. Soc. Am. A |

24. | V. Mico, Z. Zalevsky, and J. García, “Common-path phase-shifting digital holographic microscopy: A way to quantitative phase imaging and superresolution,” Opt. Commun . |

25. | V. Mico, O. Limon, A. Gur, Z. Zalevsky, and J. García, “Transverse resolution improvement using rotating-grating time-multiplexing approach,” J. Opt. Soc. Am. A |

26. | Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy - approaching the linear systems limits of optical resolution,” Opt. Express |

27. | Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy,” J. Opt. Soc. Am. A |

28. | T. Turpin, L. Gesell, J. Lapides, and C. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE |

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

30. | V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc . |

31. | F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett . |

32. | B. Simon, M. Debailleul, V. Georges, V. Lauer, and O. Haeberlé, “Tomographic diffractive microscopy of transparent samples,” Eur. Phys. J. Appl. Phys . |

33. | W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods |

34. | W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Extended depth of focus in tomographic phase microscopy using a propagation algorithm,” Opt. Lett . |

35. | S. S. Kou and C. J. R. Sheppard, “Image formation in holographic tomography,” Opt. Lett . |

36. | J. W. Goodman, |

37. | H. H. Arsenault and G. April, “Fourier holography,” in |

38. | S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A |

39. | M. King, “Matlab m-files for multidimensional nonlinear conjugate gradient method” (2005), http://users.ictp.it/?mpking/cg.html, accessed 12 December 2008. |

40. | S. S. Kou and C. J. R. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express |

41. | J. W. Goodman, |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(090.0090) Holography : Holography

(090.1000) Holography : Aberration compensation

(170.0180) Medical optics and biotechnology : Microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Microscopy

**History**

Original Manuscript: February 9, 2009

Revised Manuscript: April 15, 2009

Manuscript Accepted: April 23, 2009

Published: April 28, 2009

**Virtual Issues**

Vol. 4, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Timothy R. Hillman, Thomas Gutzler, Sergey A. Alexandrov, and David D. Sampson, "High-resolution, wide-field object
reconstruction with synthetic aperture
Fourier holographic optical microscopy," Opt. Express **17**, 7873-7892 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-7873

Sort: Year | Journal | Reset

### References

- S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Synthetic aperture Fourier holographic optical microscopy," Phys. Rev. Lett. 97,168102 (2006). [CrossRef] [PubMed]
- S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Digital Fourier holography enables wide-field, superresolved, microscopic characterization," in ‘Optics in 2007’, Opt. Photonics News 18, 29 (Dec. 2007). [CrossRef]
- S. A. Alexandrov, T. R. Hillman, and D. D. Sampson, "Spatially resolved Fourier holographic light scattering angular spectroscopy," Opt. Lett. 30, 3305-3307 (2005). [CrossRef]
- T. R. Hillman, S. A. Alexandrov, T. Gutzler, and D. D. Sampson, "Microscopic particle discrimination using spatially-resolved Fourier-holographic light scattering angular spectroscopy," Opt. Express 14, 11088-11102 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-23-11088. [CrossRef] [PubMed]
- S. A. Alexandrov and D. D. Sampson, "Spatial information transmission beyond a systems diffraction limit using optical spectral encoding of the spatial frequency," J. Opt. A - Pure Appl. Opt. 10, 025304 (2008). [CrossRef]
- M. Ryle and A. Hewish, "The synthesis of large radio telescopes," Mon. Not. R. Astron. Soc. 120, 220-230 (1960).
- D. Gabor, "A new microscopic principle," Nature (London) 161, 777-778 (1948). [CrossRef]
- J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967). [CrossRef]
- U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, Berlin, 2005).
- B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, "Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy," Opt. Express 13,9361-9373 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-23-9361. [CrossRef] [PubMed]
- C. J. Mann, L. Yu, C.-M. Lo, and M. K. Kim, "High-resolution quantitative phasecontrast microscopy by digital holography," Opt. Express 13, 8693-8698 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-8693. [CrossRef] [PubMed]
- M. Sebesta and M. Gustafsson, "Object characterization with refractometric digital Fourier holography," Opt. Lett. 30, 471-473 (2005). [CrossRef] [PubMed]
- J. H. Massig, "Digital off-axis holography with a synthetic aperture," Opt. Lett. 27, 2179-2181 (2002). [CrossRef]
- L. Martınez-Leon and B. Javidi, "Synthetic aperture single-exposure on-axis digital holography," Opt. Express 16,161-169 (2008), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-16-1-161. [CrossRef] [PubMed]
- J. Di, J. Zhao, H. Jiang, P. Zhang, Q. Fan, and W. Sun, "High resolution digital holographic microscopy with a wide field of view based on a synthetic aperture technique and use of linear CCD scanning," Appl. Opt. 47, 5654-5659 (2008). [CrossRef] [PubMed]
- F. Le Clerc, M. Gross, and L. Collot, "Synthetic-aperture experiment in the visible with on-axis digital heterodyne holography," Opt. Lett. 26, 1550-1552 (2001). [CrossRef]
- R. Binet, J. Colineau, and J. C. Lehureau, "Short-range synthetic aperture imaging at 633 nm by digital holography," Appl. Opt. 41, 4775-4782 (2002). [CrossRef] [PubMed]
- J. R. Price, P. R. Bingham, and C. E. Thomas, Jr., "Improving resolution in microscopic holography by computationally fusing multiple, obliquely illuminated object waves in the Fourier domain," Appl. Opt. 46, 827-833 (2007). [CrossRef] [PubMed]
- C. Liu, Z. G. Liu, F. Bo, Y. Wang, and J. Q. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002). [CrossRef]
- M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro, "Super-resolution in digital holography by a two-dimensional dynamic phase grating," Opt. Express 16,17107-17118 (2008), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-16-21-17107. [CrossRef] [PubMed]
- V. Mico, Z. Zalevsky, P. Garcia-Martinez, and J. Garcia, "Single-step superresolution by interferometric imaging," Opt. Express 12, 2589-2596 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2589. [CrossRef] [PubMed]
- C. Yuan, H. Zhai, and H. Liu, "Angular multiplexing in pulsed digital holography for aperture synthesis," Opt. Lett. 33, 2356-2358 (2008). [CrossRef] [PubMed]
- V. Mico, Z. Zalevsky, P. Garcıa-Martınez, and J. Garcıa, "Synthetic aperture superresolution with multiple offaxis holograms," J. Opt. Soc. Am. A 23, 3162-3170 (2006). [CrossRef]
- V. Mico, Z. Zalevsky, and J. Garcıa, "Common-path phase-shifting digital holographic microscopy: A way to quantitative phase imaging and superresolution," Opt. Commun. 281, 4273-4281 (2008). [CrossRef]
- V. Mico, O. Limon, A. Gur, Z. Zalevsky, and J. Garcıa, "Transverse resolution improvement using rotatinggrating time-multiplexing approach," J. Opt. Soc. Am. A 25, 1115-1129 (2008). [CrossRef]
- Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, "Imaging interferometric microscopy - approaching the linear systems limits of optical resolution," Opt. Express 15, 6651-6663 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-11-6651. [CrossRef] [PubMed]
- Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, "Imaging interferometric microscopy," J. Opt. Soc. Am. A 25, 811-822 (2008). [CrossRef]
- T. Turpin, L. Gesell, J. Lapides, and C. Price, "Theory of the synthetic aperture microscope," Proc. SPIE 2566, 230-240 (1995). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, UK, 1999, 7th ed.).
- V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002). [CrossRef] [PubMed]
- F. Charriere, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy, " Opt. Lett. 31, 178-180 (2006). [CrossRef] [PubMed]
- B. Simon, M. Debailleul, V. Georges, V. Lauer, and O. Haeberl’e, "Tomographic diffractive microscopy of transparent samples," Eur. Phys. J. Appl. Phys. 44, 29-35 (2008). [CrossRef]
- W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nat. Methods 4, 717-719 (2007). [CrossRef] [PubMed]
- W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Extended depth of focus in tomographic phase microscopy using a propagation algorithm," Opt. Lett. 33, 171-173 (2008). [CrossRef] [PubMed]
- S. S. Kou and C. J. R. Sheppard, "Image formation in holographic tomography," Opt. Lett. 33, 2362-2364 (2008). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, Englewood, Colorado, 2005, 3rd ed.).
- H. H. Arsenault and G. April, "Fourier holography," in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic Press, New York, 1979), pp. 165-180.
- S. T. Thurman and J. R. Fienup, "Phase-error correction in digital holography," J. Opt. Soc. Am. A 25, 983-994 (2008). [CrossRef]
- M. King, "Matlab m-files for multidimensional nonlinear conjugate gradient method" (2005), http://users.ictp.it/∼mpking/cg.html, accessed 12 December 2008.
- S. S. Kou and C. J. R. Sheppard, "Imaging in digital holographic microscopy," Opt. Express 15,13640-13648 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-21-13640. [CrossRef] [PubMed]
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, Englewood, Colorado, 2007).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.