## Point spread function and two-point resolution in Fresnel incoherent correlation holography |

Optics Express, Vol. 19, Issue 16, pp. 15603-15620 (2011)

http://dx.doi.org/10.1364/OE.19.015603

Acrobat PDF (1963 KB)

### Abstract

Fresnel Incoherent Correlation Holography (FINCH) allows digital reconstruction of incoherently illuminated objects from intensity records acquired by a Spatial Light Modulator (SLM). The article presents wave optics model of FINCH, which allows analytical calculation of the Point Spread Function (PSF) for both the optical and digital part of imaging and takes into account Gaussian aperture for a spatial bounding of light waves. The 3D PSF is used to determine diffraction limits of the lateral and longitudinal size of a point image created in the FINCH set-up. Lateral and longitudinal resolution is investigated both theoretically and experimentally using quantitative measures introduced for two-point imaging. Dependence of the resolving power on the system parameters is studied and optimal geometry of the set-up is designed with regard to the best lateral and longitudinal resolution. Theoretical results are confirmed by experiments in which the light emitting diode (LED) is used as a spatially incoherent source to create object holograms using the SLM.

© 2011 OSA

## 1. Introduction

1. D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science **300**(5616), 82–86 (2003). [CrossRef] [PubMed]

*z*-scanning and complicated data processing. 3D imaging of incoherently illuminated objects can also be realized by the scanning holography [2

2. 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]

3. Y. Li, D. Abookasis, and J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. **40**, 2864–2870 (2001). [CrossRef]

4. Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. **28**, 2518–2520 (2003). [CrossRef] [PubMed]

5. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. **32**, 912–914 (2007). [CrossRef] [PubMed]

5. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. **32**, 912–914 (2007). [CrossRef] [PubMed]

5. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. **32**, 912–914 (2007). [CrossRef] [PubMed]

6. J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express **15**, 2244–2250 (2007). [CrossRef] [PubMed]

7. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics **2**, 190–195 (2008). [CrossRef]

8. B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express **18**, 962–972 (2010). [CrossRef] [PubMed]

9. B. Katz and J. Rosen, “Could SAVE concept be applied for designating a new synthetic aperture telescope?,” Opt. Express **19**, 4924–4936 (2011). [CrossRef] [PubMed]

**32**, 912–914 (2007). [CrossRef] [PubMed]

6. J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express **15**, 2244–2250 (2007). [CrossRef] [PubMed]

10. B. Katz, D. Wulich, and J. Rosen, “Optimal noise suppression in Fresnel incoherent correlation holography (FINCH) configured for maximum imaging resolution,” Appl. Opt. **49**, 5757–5763 (2010). [CrossRef] [PubMed]

11. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express **19**, 5047–5062 (2011). [CrossRef] [PubMed]

10. B. Katz, D. Wulich, and J. Rosen, “Optimal noise suppression in Fresnel incoherent correlation holography (FINCH) configured for maximum imaging resolution,” Appl. Opt. **49**, 5757–5763 (2010). [CrossRef] [PubMed]

11. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express **19**, 5047–5062 (2011). [CrossRef] [PubMed]

**32**, 912–914 (2007). [CrossRef] [PubMed]

11. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express **19**, 5047–5062 (2011). [CrossRef] [PubMed]

**19**, 5047–5062 (2011). [CrossRef] [PubMed]

## 2. Computational model of FINCH

*f*

_{0}and then illuminated by incoherent source. Light scattered by the object is collimated, transformed by the SLM and captured by the CCD camera. The SLM acts as a diffractive beam splitter, so the waves impacting it are doubled. The light from each point of the object is split into signal and reference waves that interfere and create a Fresnel zone structure on the CCD. The resulting hologram of the observed object is created as an incoherent superposition of interference patterns of individual points. In experimental part of FINCH, three holograms of the object

*I*,

_{j}*j*= 1,2,3, are recorded with different phase shift of signal and reference wave. The recorded holograms are then processed and observed object is digitally reconstructed using the Fresnel transform.

*P*

_{0}with the coordinates (

*x*

_{0},

*y*

_{0},

*z*

_{0}), where

*z*

_{0}< 0 is used if the object lies in front of the collimating lens. Its paraxial image

*P*is created by the collimating lens and has coordinates (

_{r}*x*,

_{r}*y*,

_{r}*z*) measured in the coordinate system with origin at the center of the CCD (see Fig. 1). The image is formed by paraxial rays, which represent normals of a paraboloidal wave converging to the image point

_{r}*P*. This paraboloidal wave is incident on the SLM and splits into reference and signal waves with complex amplitudes

_{r}*U*and

_{r}*U*. The splitting of incoming waves is ensured through the SLM transmission function of the form where

_{sj}*a*and

*b*are coefficients enabling an optimal energy distribution between signal and reference waves,

*ϑ*denotes a constant phase shift,

_{j}*θ*=

*k*(

*x*

^{2}+

*y*

^{2})/(2

*f*) is a quadratic phase of a lens with the focal length

_{d}*f*, and

_{d}*k*denotes the wave number. Complex amplitude of the light field behind the SLM can be written as where

*a*and

_{r}*a*are amplitudes and Φ

_{s}*and Φ*

_{r}*phases of reference and signal waves converging to the points*

_{s}*P*and

_{r}*P*, respectively. Using paraxial approximation, the phases Φ

_{s}*and Φ*

_{r}*can be expressed as where (*

_{s}*x*,

_{c}*y*) are coordinates of the CCD camera plane,

_{c}*z*=

_{r}*z*′

_{0}– Δ

_{1}– Δ

_{2}and

*z*=

_{s}*z*′

*– Δ*

_{m}_{2}. Δ

_{1}and Δ

_{2}denote the separation distances between the collimating lens, SLM and CCD, and

*z*

_{0}and

*z*′

_{0}and

*z*and

_{m}*z*′

*are object and image distances for the collimating lens and the SLM lens, respectively (see Fig. 1). The reference and signal waves originate from the same object point*

_{m}*P*

_{0}and are spatially coherent. If the difference of their optical paths is less than coherence length of the source, the waves interfere at the CCD, and create an interference pattern with the shape of the Fresnel zone plate, Holograms of the object are prepared for three different phase settings

*ϑ*

_{1}= 0,

*ϑ*

_{2}= 2

*π*/3 and

*ϑ*

_{3}= 4

*π*/3. Intensity records

*I*are then processed to remove holographic twin image. In this way, a complex function

_{j}*T*is created which is used for digital image reconstruction [5

**32**, 912–914 (2007). [CrossRef] [PubMed]

*T*can be simplified to the form where

*– Φ*

_{s}*is arranged by using Eq. (3), we can write*

_{r}*T*represents a quadratic phase of a lens. In accordance with [11

**19**, 5047–5062 (2011). [CrossRef] [PubMed]

*f*= 1/

_{l}*z*− 1/

_{s}*z*and its axis is shifted laterally relative to the origin of coordinate system of the object space. The focal length

_{r}*f*and the shift of the axis (

_{l}*X*′,

*Y*′) are uniquely determined by the position of object points and system parameters. Applying a lens equation, we can write where where As will be demonstrated later, a digital image is produced coherently. The phase function Ω

_{0}is then used to determine the conditions of interference in the reconstructed image. Its dependence on the object position can be expressed as

*T*, which is created from intensity records

*I*according to Eq. (5). Complex amplitude of the image

_{j}*U*′ is then obtained as a convolution of

*T*and impulse response function of free propagation of light,

*h*. When calculating the PSF, a quadratic phase of the digital lens Eq. (7) is used in the convolution. In this case we can write, where

**r**′ = (

*x*′,

*y*′,

*z*′) is the position vector of image space, defined in the coordinate system with origin at the center of the CCD camera,

*Q*denotes the aperture function, which simulates a transverse confinement of waves interfering at the CCD, and

*h*is the free space impulse response function. Substituting Eq. (7) into Eq. (13) and applying the Fresnel approximation of

*h*[12

12. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (J. Wiley, 1991). [CrossRef]

*λ*denotes the wavelength. Intensity distribution

*I*′ = |

*U*′|

^{2}defines a diffraction limited PSF, including both an optical and digital part of the FINCH imaging.

## 3. Geometrical optics approach of FINCH imaging

*Q*= 1. At the distance

*z*′ =

*f*, a quadratic phase term in Eq. (14) is eliminated. This situation corresponds to a sharp image determined by the Dirac delta function, Equation (17) shows that the object point

_{l}*P*

_{0}(

*x*

_{0},

*y*

_{0},

*z*

_{0}) is digitally reconstructed as an image point

*P*′ whose position is given by the coordinates

*x*′ =

*X*′,

*y*′ =

*Y*′, and

*z*′ =

*f*. The symbol

_{l}*m*used in Eq. (10) then represents the lateral magnification defined as the ratio between the image size and its true size in object space,

*m*=

*x*′/

*x*

_{0}=

*y*′/

*y*

_{0}. Ray optics provides a simple interpretation of FINCH imaging in which the longitudinal position of the object point

*z*

_{0}determines the focal length of the diffractive lens and the lateral position (

*x*

_{0},

*y*

_{0}) causes displacement of the lens axis. Paraxial image always lies on the axis of the lens, so it has image space telecentricity. This feature of the diffractive lens was used in [11

**19**, 5047–5062 (2011). [CrossRef] [PubMed]

**19**, 5047–5062 (2011). [CrossRef] [PubMed]

*z*

_{0}. It is interesting to note that the lateral magnification is independent of the focal length of the lens implemented by the SLM. The derivative

*dm/dz*

_{0}shows that the dependence of

*m*on

*z*

_{0}can be eliminated by appropriate choice of Δ

_{1}, When the SLM is positioned at the back focal plane of the collimating lens (Δ

_{1}=

*f*

_{0}),

*m*becomes independent of the object distance

*z*

_{0}. In this special case, the lateral magnification is given as

*m*= –Δ

_{2}/Δ

_{1}and the FINCH imaging shows object-space telecentricity.

*α*=

*df*/

_{l}*dz*

_{0}. Using Eq. (8) it can be written as where

*F*is given by Eq. (9). Derived relations for

*f*,

_{l}*m*and

*α*are valid for a general position of observed object and can be used in the analysis of two special cases.

### Special case I: Planar object located in focal plane of collimating lens

### Special case II: Lensless FINCH imaging

10. B. Katz, D. Wulich, and J. Rosen, “Optimal noise suppression in Fresnel incoherent correlation holography (FINCH) configured for maximum imaging resolution,” Appl. Opt. **49**, 5757–5763 (2010). [CrossRef] [PubMed]

*f*

_{0}→ ∞ and with Δ

_{1}= 0,

## 4. PSF for diffraction limited FINCH imaging

*ρ*is the radius of holograms of a point object. It depends on the experiment geometry and can be determined by a ray tracing method. In some cases, the size of the hologram is limited by detection conditions and must be determined by means of the Nyquist criterion. The size of the recorded hologram used in Eq. (26) is then defined as where 2

_{c}*ρ*is transverse size of the CCD camera and Δ

_{CCD}*ρ*and Δ

_{r}*ρ*denote a hologram radius obtained applying the ray tracing method and the Nyquist criterion, respectively. Calculation of Δ

_{N}*ρ*will be given later for the actual configuration of the experiment.

_{c}*x*

_{0}=

*y*

_{0}= 0. Transverse intensity profile of a perfectly focused digital image in the plane

*z*′ =

*f*is given by Eq. (32) used with Eq. (28) and can be written in the form where

_{l}*r*′ = (

*x*′

^{2}+

*y*′

^{2})

^{1/2}and

*NA*′ may be understood as the numerical aperture of a diffractive lens,

*NA*′ = Δ

*ρ*/

_{c}*f*.

_{l}*r*′ can be defined by a fall of normalized intensity to the value

*I*′

*(Δ*

_{N}*r*′) = 1

*/e*

^{2}, For analysis of the longitudinal resolution, a depth of field of the reconstructed image must be known. Image defocusing can be evaluated using the Strehl ratio

*D*defined as It describes a change of axial intensity in dependence on a focus error Δ

*z*′. By using Eq. (28), the Strehl ratio can be arranged to the form The depth of focus Δ

*z*′ is then defined by a tolerated decrease of the Strehl ratio

*D*. Assuming that

*f*>> Δ

_{l}*r*′, we can write Equations (34) and (37) are of general validity, but the numerical aperture

*NA*′ must be analyzed separately for the system with a collimating lens and the lensless experiment.

### FINCH with collimating lens

*f*is given by Eq. (21) and the radius of intensity records Δ

_{l}*ρ*can be simply estimated using the ray tracing and the Nyquist criterion. By means of ray optics, the radius Δ

_{c}*ρ*can be determined by a simple relation where 2

_{r}*ρ*denotes a transverse size of the SLM. When the spatial period of recorded interference patterns is required to fulfill the Nyquist criterion, the radius Δ

_{SLM}*ρ*is approximately given by where Δ

_{N}*x*is the pixel size of the CCD camera. For real experimental parameters and assumption Δ

_{CCD}_{2}< 2

*f*, Δ

_{d}*ρ*= Δ

_{c}*ρ*can be used.

_{r}*ρ*given by Eq. (38), the aperture of the diffractive lens is equal to the aperture of the modulator lens,

_{c}*NA*′ =

*ρ*/

_{SLM}*f*. For the Strehl coefficient

_{d}*D*= 1/

*e*

^{2}, the transverse and longitudinal size of the diffraction spot can be written as It is interesting to note that the size of diffraction spot remains the same when the separation distance Δ

_{2}is changed.

*x*′

*,z*′) obtained using Eq. (28) is illustrated in Fig. 2(b) for parameters

*f*

_{0}= 200 mm,

*f*= 750 mm, Δ

_{d}_{1}= 250 mm and Δ

_{2}= 600 mm. For comparison, the PSF reconstructed from the CCD records is shown in Fig. 2(a). The parameters of the experiment were the same as those used in the calculation. As can be observed in Figs. 2(a) and 2(b), the size and shape of the demonstrated PSFs is in very good agreement.

### Lensless FINCH

*z*

_{0}| <

*f*, the radius of recorded holograms is given as Δ

_{d}*ρ*=

_{c}*ρ*. The numerical aperture of the diffractive lens then can be written as

_{CCD}*NA*′ =

*ρ*/

_{CCD}*f*where

_{lL}*f*is given by Eq. (23). In this case, the transverse and longitudinal size of the diffraction spot Eq. (40) depends on the separation distance between the SLM and the CCD.

_{lL}## 5. Optimal geometric configuration of FINCH

### FINCH imaging with collimating optics

*m*and the focal length

_{F}*f*of the diffractive lens. Using assumption that transverse dimensions of the collimating lens and the SLM are the same, the object and image numerical apertures can be written as

_{lF}*NA*=

*ρ*/

_{SLM}*f*

_{0}and

*NA*′ = Δ

*ρ*/

_{c}*f*. Using Eq. (38), the ratio of the numerical apertures can be written as

_{lF}*NA/NA*′ = |

*m*|

_{F}*f*

_{d}*/*Δ

_{2}. Lateral resolution in object space is given as Δ

*r*= Δ

*r*′/|

*m*|. Using the image resolution Eq. (40), Δ

_{F}*r*can be expressed as where Δ

*r*

_{0}=

*λf*

_{0}/(

*πρ*) is the diffraction limited object resolution of the used collimating lens. Dependence of Δ

_{SLM}*r*on the ratio Δ

_{2}/

*f*is shown in Fig. 3 for different focal lengths of collimating lenses. If Δ

_{d}_{2}≥

*f*, the lateral resolution of the FINCH imaging is limited by the resolution of the collimating optics. In FINCH set-up with Δ

_{d}_{2}

*< f*, the lateral resolution of the collimating optics is reduced. In [11

_{d}**19**, 5047–5062 (2011). [CrossRef] [PubMed]

*a*. If its value is chosen owing to the technical conditions as

*a*= 1, the separation distance Δ

_{2}=

*f*/2 is obtained [10

_{d}**49**, 5757–5763 (2010). [CrossRef] [PubMed]

**19**, 5047–5062 (2011). [CrossRef] [PubMed]

*r*

_{0}. When Eq. (15) in [11

**19**, 5047–5062 (2011). [CrossRef] [PubMed]

*a*= 0, the CCD position ensuring the best two-point resolution is obtained.

### Lensless FINCH imaging

*z*= Δ

*z*′/

*α*. For chosen parameters of the experiment, the optimal position of the CCD camera Δ

_{L}_{2}can be determined from the condition Unlike the case of lateral resolution, the best longitudinal resolution depends on the location of the object

*z*

_{0}. Dependence of Δ

*z*on the ratio Δ

_{2}/

*f*is shown in Fig. 4.

_{d}## 6. Coherent digital reconstruction of two-point object

*ϕ*associated with the observed sources. When Δ

*ϕ*=

*π*/2, the intensity of the image is identical to that resulting from incoherent sources. When the sources are in phase (Δ

*ϕ*= 0), the resolution is worse in coherent light than in incoherent light. If sources are in phase opposition (Δ

*ϕ*=

*π*), destructive interference occurs and the point sources are better resolved with coherent illumination than with incoherent light. Detailed analysis of the resolution in coherent light can be found in [13

13. S. Van Aert and D. Van Dyck, “Resolution of coherent and incoherent imaging systems reconsidered–classical criteria and a statistical alternative,” Opt. Express **14**, 3830–3839 (2006). [CrossRef] [PubMed]

*P*

_{1}and

*P*

_{2}are placed symmetrically with respect to the optical axis (

*x*

_{0}= 0), their diffraction images interfere constructively at axial point

*P*′(0, 0,

*z*′) and Δ

*ϕ*= 0 for an arbitrary separation distance Δ

*x*. For the two-point object laterally shifted to the position

*x*

_{0}≠ 0, the phase difference Δ

*ϕ*oscillates in interval < 0, 2

*π*> if the distance of points Δ

*x*is slightly changed. In calculated visibility, fast oscillations appear in dependence of

*V*on Δ

_{x}*x*. When visibility is determined from experimental data, the oscillations do not occur. The reason is that the change in the relative phase causes slight changes of interference pattern that are not resolved in the CCD detection.

*P*

_{1}(0, 0,

*z*

_{0}+ Δ

*z*/2) and

*P*

_{2}(0, 0,

*z*

_{0}– Δ

*z*/2). The longitudinal visibility

*V*then can be defined as where Comparison of theoretical and measured visibilities is presented in the experimental section.

_{z}## 7. Experimental determination of resolving power of FINCH imaging

### 7.1. Design and parameters of the experimental set-up

*μ*m) with holders mounted to XYZ travel translation stages. The faces of the fibers are used as point sources, which can be precisely positioned in an object space near the focal plane of collimating lens. The collimated beams pass through polarizer, iris diaphragm and beam splitter and fall on the reflective SLM Hamamatsu X10468 (16 mm x 12 mm, 792 x 600 pixels). The polarizer is used to optimize a phase operation of the SLM. The SLM is addressed by computer generated holograms enabling splitting of the input beam to the signal and reference waves. The reference wave is transmitted with unchanged wavefront shape. The signal wave is created by a quadratic phase modulation equivalent to the action of lens with the focal length

*f*and successively three different phase shifts 0, 2

_{d}*π*/3 and 4

*π*/3 are imposed on it. Light reflected by the SLM is diverted to the CCD camera (QImaging Retiga-4000R, 15 mm x 15mm, 2048 x 2048 pixels) and three records of interference patterns created by the signal and reference waves are subsequently acquired. The intensity records are processed in a PC and digital image of observed object is created. In this way, the 3D PSF was restored and the transverse and longitudinal resolution of two-point object was examined for various parameters of the set-up.

### 7.2. Experimental results

*x*′,

*z*′) and (

*y*′,

*z*′) of the image space from which information about transverse and longitudinal size of the image spot can be obtained. The PSF that was reconstructed from the experimental data is compared with the calculated PSF in Fig. 2 for parameters

*f*

_{0}= 200 mm,

*f*= 750 mm, Δ

_{d}_{1}= 250 mm, and Δ

_{2}= 600 mm.

*x*of the observed point sources. The measurements were carried out for different ratios Δ

_{2}/

*f*and the experimental results were compared with theoretical predictions. In the measurements, the fiber faces were placed in the focal plane of the collimating lens and the lateral distance of fibers was successively changed using micrometer displacements. Holograms of two-point object were recorded on the CCD camera and then used for digital image reconstruction. The planes of sharp images were determined using the numerical procedure implemented in software for digital imaging. In separate transverse planes of the image space, the first and second order moments were calculated from the intensity profile and the standard deviations

_{d}*σ*and

_{x}*σ*evaluated. The longitudinal position where

_{y}*V*defined by Eq. (53) determined. Visibility evaluated for transverse distance of point sources Δ

_{x}*x*, which vary from 0 to 0.14 mm is shown in Fig. 6(a). Experimental data were acquired in the system with parameters

*f*

_{0}= 200 mm,

*f*= 750 mm, Δ

_{d}_{1}= 250 mm and with varying distances Δ

_{2}= 0.25

*f*, 0.5

_{d}*f*, 0.95

_{d}*f*, 1.0

_{d}*f*and 1.5

_{d}*f*. Experimental results confirm the theoretical prediction that a transverse resolution of the FINCH imaging is limited by resolution of the collimating lens if Δ

_{d}_{2}≥

*f*. Figure 6(a) shows that for such positions of the CCD, the experimental visibility is close to the theoretical curve, which corresponds to a diffraction limited resolution of the collimating lens (dashed line in Fig. 6(a)). The visibility curves obtained for Δ

_{d}_{2}= 0.25

*f*and 0.5

_{d}*f*show that the resolution of a collimating lens is substantially reduced when Δ

_{d}_{2}<

*f*. This trend is also evident from Fig. 6(b) that shows the separation distance Δ

_{d}*x*corresponding to the visibility

*V*= 0.8 in dependence on the ratio Δ

_{x}_{2}/

*f*. Slightly lower experimental resolution then theoretical limit can be attributed to imperfections of the optical set-up, but general trends are clearly reproduced. Improvement in lateral resolution achieved by changing the position of the CCD camera is illustrated in Fig. 7 for the separation distances of the point sources Δ

_{d}*x*= 25

*μ*m and 65

*μ*m and camera positions Δ

_{2}= 1.2

*f*and 0.4

_{d}*f*.

_{d}*z*was altered by precise mechanical displacements. Holograms were recorded on a CCD for each position of a two-point source and its images were created numerically using the Fresnel transform. Lateral intensity profiles were evaluated in a sequence of plains separated by a fine longitudinal step adapted to the depth of field of the system. In this way, the 3D intensity of two-point source was reconstructed and used to determine the longitudinal visibility

*V*defined by Eq. (56). Dependence of the visibility

_{z}*V*on the longitudinal separation distance of point sources Δ

_{z}*z*was found for different positions of the CCD given by Δ

_{2}=

*f*± 50 mm (Δ

_{d}_{2}/

*f*= 1.06 and 0.93), Δ

_{d}_{2}=

*f*± 375 mm (Δ

_{d}_{2}/

*f*= 1.5 and 0.5) and Δ

_{d}_{2}=

*f*± 550 mm (Δ

_{d}_{2}/fd = 1.73 and 0.26). The results are shown in Fig. 8(a). For each position of the CCD, the distance Δ

*z*for which the visibility drops to a value of

*V*= 0.8 was determined from the dependence of

_{z}*V*on Δ

_{z}*z*. Separation distances Δ

*z*for which visibility

*V*drops to the specified level 0.8 have different values for different positions of the CCD. Their dependence on the ratio Δ

_{z}_{2}/

*f*is shown in Fig. 8(b). Figure 9 shows the intensity distribution, which was obtained by digital reconstruction of holographic records of two point sources located on the axis of the collimating lens. CCD camera was in the position Δ

_{d}_{2}= 0.8

*f*and the longitudinal distances of observed point sources were Δ

_{d}*z*= 2.5 mm and 3.5 mm, respectively. Experimental results again confirm the prediction that the best longitudinal resolution is reached to set Δ

_{2}≈

*f*. In this case, the longitudinal resolution is limited by the longitudinal resolution of the collimating lens. The larger discrepancy in the longitudinal resolution can be justified by the fact, that longitudinal magnification is much more sensitive to parameter inaccuracies than lateral magnification.

_{d}_{2}≥

*f*, hence optimum for both transversal and longitudal two-point resolution can be achieved for Δ

_{d}_{2}equal or slightly larger than

*f*. At this position of detector, the wave generated by the modulator is in focus, so that detected interference patterns have a small size. In the experiment, the two-point image was successfully reconstructed even from records taken directly at distance Δ

_{d}_{2}= 1.0

*f*where transverse dimension of the interference pattern was close to diffraction limited Airy pattern. In this case, however, special care had to be taken to overcome technical limitation caused by large difference in irradiance from signal and reference wave and so multiple exposures from CCD had to be combined for each record to accumulate enough light from reference wave and not to cause overexposure from focused signal wave, at the same time.

_{d}## 8. Conclusions

- In the ray approximation, the relationships between geometrical parameters of the object and its digitally created image were found and used to determine the lateral and longitudinal magnification of the image.
- Three-dimensional diffraction limited PSF was calculated and compared with the image spot reconstructed from experimental data acquired in the FINCH set-up for a point object.
- Transverse and longitudinal resolution of FINCH imaging was examined theoretically and experimentally using the visibility function defined for a two-point source implemented by the LED coupled to single-mode fibers.
- The transverse and longitudinal resolution of two-point object was investigated experimentally for different parameters of the set-up with very good agreement with theory.
- It was verified that the transverse and longitudinal resolution of FINCH imaging is limited by the collimating optics and can be achieved only if distance Δ
_{2}between the CCD camera and the SLM is equal or larger than focal length of the SLM lens,*f*. We proved both theoretically and experimentally that when Δ_{d}_{2}is shorter than*f*, the best two-point resolution is not reached and the transverse and longitudinal resolution provided by the collimating lens is significantly reduced._{d}

## Acknowledgments

## References and links

1. | D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science |

2. | 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 |

3. | Y. Li, D. Abookasis, and J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. |

4. | Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. |

5. | J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. |

6. | J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express |

7. | J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics |

8. | B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express |

9. | B. Katz and J. Rosen, “Could SAVE concept be applied for designating a new synthetic aperture telescope?,” Opt. Express |

10. | B. Katz, D. Wulich, and J. Rosen, “Optimal noise suppression in Fresnel incoherent correlation holography (FINCH) configured for maximum imaging resolution,” Appl. Opt. |

11. | G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express |

12. | B. E. A. Saleh and M. C. Teich, |

13. | S. Van Aert and D. Van Dyck, “Resolution of coherent and incoherent imaging systems reconsidered–classical criteria and a statistical alternative,” Opt. Express |

**OCIS Codes**

(090.1760) Holography : Computer holography

(090.1970) Holography : Diffractive optics

(110.6880) Imaging systems : Three-dimensional image acquisition

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

(230.3720) Optical devices : Liquid-crystal devices

**ToC Category:**

Holography

**History**

Original Manuscript: May 16, 2011

Revised Manuscript: July 6, 2011

Manuscript Accepted: July 11, 2011

Published: July 29, 2011

**Citation**

Petr Bouchal, Josef Kapitán, Radim Chmelík, and Zdeněk Bouchal, "Point spread function and two-point resolution in Fresnel incoherent correlation holography," Opt. Express **19**, 15603-15620 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15603

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### References

- D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science 300(5616), 82–86 (2003). [CrossRef] [PubMed]
- 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]
- Y. Li, D. Abookasis, and J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. 40, 2864–2870 (2001). [CrossRef]
- Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. 28, 2518–2520 (2003). [CrossRef] [PubMed]
- J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32, 912–914 (2007). [CrossRef] [PubMed]
- J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15, 2244–2250 (2007). [CrossRef] [PubMed]
- J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2, 190–195 (2008). [CrossRef]
- B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18, 962–972 (2010). [CrossRef] [PubMed]
- B. Katz and J. Rosen, “Could SAVE concept be applied for designating a new synthetic aperture telescope?,” Opt. Express 19, 4924–4936 (2011). [CrossRef] [PubMed]
- B. Katz, D. Wulich, and J. Rosen, “Optimal noise suppression in Fresnel incoherent correlation holography (FINCH) configured for maximum imaging resolution,” Appl. Opt. 49, 5757–5763 (2010). [CrossRef] [PubMed]
- G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19, 5047–5062 (2011). [CrossRef] [PubMed]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (J. Wiley, 1991). [CrossRef]
- S. Van Aert and D. Van Dyck, “Resolution of coherent and incoherent imaging systems reconsidered–classical criteria and a statistical alternative,” Opt. Express 14, 3830–3839 (2006). [CrossRef] [PubMed]

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