## Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM) |

Optics Express, Vol. 20, Issue 8, pp. 9109-9121 (2012)

http://dx.doi.org/10.1364/OE.20.009109

Acrobat PDF (1680 KB)

### Abstract

Fresnel incoherent correlation holography (FINCH) records holograms under incoherent illumination. FINCH was implemented with two focal length diffractive lenses on a spatial light modulator (SLM). Improved image resolution over previous single lens systems and at wider bandwidths was observed. For a given image magnification and light source bandwidth, FINCH with two lenses of close focal lengths yields a better hologram in comparison to a single diffractive lens FINCH. Three techniques of lens multiplexing on the SLM were tested and the best method was randomly and uniformly distributing the two lenses. The improved quality of the hologram results from a reduced optical path difference of the interfering beams and increased efficiency.

© 2012 OSA

## 1. Introduction

1. J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science **265**(5173), 749–752 (1994). [CrossRef] [PubMed]

2. F. Mok, J. Diep, H.-K. Liu, and D. Psaltis, “Real-time computer-generated hologram by means of liquid-crystal television spatial light modulator,” Opt. Lett. **11**(11), 748–750 (1986). [CrossRef] [PubMed]

3. J. N. Mait and G. S. Himes, “Computer-generated holograms by means of a magnetooptic spatial light modulator,” Appl. Opt. **28**(22), 4879–4887 (1989). [CrossRef] [PubMed]

4. J. Rosen, L. Shiv, J. Stein, and J. Shamir, “Electro-optic hologram generation on spatial light modulators,” J. Opt. Soc. Am. A **9**(7), 1159–1166 (1992). [CrossRef]

5. C.-S. Guo, Z.-Y. Rong, H.-T. Wang, Y. Wang, and L. Z. Cai, “Phase-shifting with computer-generated holograms written on a spatial light modulator,” Appl. Opt. **42**(35), 6975–6979 (2003). [CrossRef] [PubMed]

6. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. **32**(8), 912–914 (2007). [CrossRef] [PubMed]

13. J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express **19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

6. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. **32**(8), 912–914 (2007). [CrossRef] [PubMed]

11. B. Katz, D. Wulich, and J. Rosen, “Optimal noise suppression in Fresnel incoherent correlation holography (FINCH) configured for maximum imaging resolution,” Appl. Opt. **49**(30), 5757–5763 (2010). [CrossRef] [PubMed]

12. P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, “Point spread function and two-point resolution in Fresnel incoherent correlation holography,” Opt. Express **19**(16), 15603–15620 (2011). [CrossRef] [PubMed]

13. J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express **19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

13. J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express **19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

12. P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, “Point spread function and two-point resolution in Fresnel incoherent correlation holography,” Opt. Express **19**(16), 15603–15620 (2011). [CrossRef] [PubMed]

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

## 2. Dual lens FINCH analysis

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

_{r¯s=(xs,ys)}on the front focal plane of the collimating lens L

_{1}. A more general analysis for other transversal planes is given in Ref [9

9. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express **19**(6), 5047–5062 (2011). [CrossRef] [PubMed]

_{Is}, located at (

*x*,

_{s}*y*,-

_{s}*f*), induces just before the lens L

_{o}_{1}, in the paraxial approximation [14], a complex amplitude of

_{Is C(r¯s )L(−r¯s/fo )Q(1/fo )}, where

*f*is the focal length of lens L

_{o}_{1},

*Q*designates the quadratic phase function, such that

_{Q(b)=exp[iπbλ−1(x2+y2)]},

*L*denotes the linear phase function, such that

_{L(s¯)=exp[i2πλ−1(sxx+syy)]},

*λ*is the central wavelength of the light and

_{C(r¯s)}is a complex constant dependent on the location of the point source. This wave is multiplied by the transparency of the lens L

_{1}

_{[multiplied by Q(−1/fo ) ]}, propagates a distance

*d*[convolved with

_{Q(1/d)]}and meets the SLM. The SLM transparency is

_{[C1Q(−1/f1)+C2exp(iθ)Q(−1/f2)]}, where

*C*

_{1},

*C*

_{2}are constants, and the angle

*θ*is one of the three angles used in the phase shifting procedure in order to eliminate the bias term and the twin image from the final hologram [6

6. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. **32**(8), 912–914 (2007). [CrossRef] [PubMed]

9. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express **19**(6), 5047–5062 (2011). [CrossRef] [PubMed]

*f*

_{1}and

*f*

_{2}. In the past, we presented two different methods to display a constant phase mask with a lens mask on the same SLM. However, one of these methods, based on the polarization sensitivity of the SLM [9

9. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express **19**(6), 5047–5062 (2011). [CrossRef] [PubMed]

*z*[convolved with

_{h}_{Q(1/zh)}] till the camera. On the camera detector, only the area of the beam overlap, denoted by the area of the function

*P*(

*R*), is considered as part of the hologram.

_{H}*P*(

*R*) stands for the limiting aperture of the system, where it is assumed that the aperture is a clear disk of radius

_{H}*R*determined by the overlap area of the two interfering beams on the camera plane. Finally, the magnitude of the interference is squared to yield the intensity distribution

_{H}*I*(

_{H}*u,v*) of the recorded hologram, where

_{ρ¯=(u,v)}are the coordinates of the camera plane. All this description is formulated into the following equation,where the asterisk denotes a two dimensional convolution.

*θ*are recorded and superimposed, according to Eq. (5) of Ref [6

**32**(8), 912–914 (2007). [CrossRef] [PubMed]

*C’*is a constant and

*z*is the reconstruction distance from the hologram to the image point calculated to be,The ± indicates that there are two possible reconstructed images, although only one of them is chosen to be reconstructed, as desired.

_{r}_{r¯r}is the transverse location of the reconstructed image point calculated to be,The precise way by which the results of Eqs. (2)-(4) are calculated from Eq. (1) is described in the appendix of Ref [10

10. B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express **18**(2), 962–972 (2010). [CrossRef] [PubMed]

*M*=

_{T}*z*/

_{h}*f*. The PSF of FINCH is obtained by digitally reconstructing the Fresnel hologram given in Eq. (2) at a distance

_{o}*z*from the hologram plane. The expression of the hologram in Eq. (2) contains a transparency of a lens with focal distance

_{r}*z*and hence, according to Fourier optics theory [14], the reconstructed image is

_{r}_{ν[a]f(x)=f(ax)},

_{r¯=(x,y)}are the coordinates of the reconstruction plane and

*is defined as*J i n c

*, where*J i n c ( r ) = J 1 ( r ) / r

*Jinc*'s central lobe defined by the circle of the first zeros of the

*Jinc*function of Eq. (5). This diameter is equal to 1.22

*λz*in the image plane, orin the object plane. The goal here is to find the values of

_{r}/R_{H}*f*

_{1}and

*f*

_{2}which yields the minimum value of the PSF diameter. It is easy to prove that the minimum value the PSF diameter is obtained for the case of a perfect overlap between the two cones of the spherical waves, one is focused at

*f*

_{1}, before the camera plane, and the other at

*f*

_{2}, beyond the camera plane, as is shown in Fig. 1. Under the condition of perfect overlap of the two beam cones, and based on the triangular similarity, it is evident that,where

*R*is the radius of the SLM. Therefore, the SLM-camera distance,

_{o}*z*, under the perfect overlap condition is given by,Substituting Eqs. (3), (7) and (8) into Eq. (6) yields that the PSF diameter at the object plane isThis diameter value is half of the PSF diameter of a regular glass-lens imaging system with the same numerical aperture (NA). This result looks similar to the Rayleigh resolution criterion [14] for incoherent imaging systems. However, recall that Rayleigh criterion is defined as the radius of the PSF with the diameter of 1.22

_{h}*λ*/

*NA*, we realize that the resolution of the optimal FINCH is beyond the Rayleigh resolution.

*f*

_{1}=

*z*/2 and

_{h}*f*

_{2}= ∞ [13

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

*f*

_{2}= ∞ is implemented by a constant phase mask, while

*f*

_{1}=

*z*/2 is realized by the single diffractive lens, both displayed on the SLM. Practically, it is more efficient to multiplex constant phase with a single diffractive lens rather than two diffractive lenses. This is because the constant phase is implemented by the entire SLM pixels from the polarization component orthogonal to the polarization of the SLM [9

_{h}**19**(6), 5047–5062 (2011). [CrossRef] [PubMed]

*C*

_{o}is the speed of light in the free space and Δ

*λ*is the source bandwidth in terms of wavelengths. Note that the maximum OPD can be minimized in the single lens FINCH by increasing the SLM-camera distance,

*z*. However, by increasing

_{h}*z*, one also increases the image magnification, decreases the field of view and decreases the recorded intensity which, in turn, decreases the signal-to-noise ratio. Therefore, in order to avoid all these negative effects, it is preferred to minimize the maximum OPD by adjusting the distance between the two focuses of the diffracted lenses displayed on the SLM.

_{h}_{B H¯}and

_{A H¯}, as the followingSubstituting Eqs. (7) and (8) into Eq. (11) yields,Under the condition that

*f*

_{1},

*f*

_{2}>>

*R*, the maximum OPD can be approximated to,where

_{o}_{Δf=f2−f1}. From Eq. (13) it is evident that reducing the gap between the focuses, Δ

*f*, yields a corresponding reduction of the maximum OPD. Therefore, we expect to see a continuous improvement in the quality of the reconstructed images from holograms recorded as the gap is reduced between the two focuses. This is indeed the case as demonstrated in the experimental section. In the case that one of the diffractive elements on the SLM is a constant phase mask [13

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

_{δmax≅Ro2/f1=2Ro2/zh}and it is evident that increasing the distance between the camera and the SLM,

*z*, is a way to reduce the maximum OPD and to improve the quality of the reconstructed images. However, as mentioned above, increasing

_{h}*z*has several negative effects that should be avoided.

_{h}*The upper limit for the distance between the two focuses*for achieving a high visibility hologram for a given source bandwidth and image magnification, can be derived by substituting

*M*/

_{T}= z_{h}*f*,

_{o}*z*and Eq. (10) into Eq. (13) to obtain;Above this limit, parts of the hologram might have low fringe visibility, which means that part of the information on the image might be lost and cannot appear on the reconstructed image.

_{h}^{2}≈f_{1}f_{2}*The lower limit for the distance between the two focuses*is obtained from the condition that the distance between each image and the camera should be long enough to justify using the paraxial (Fresnel) approximation. If both images are too close to the camera and below the Fresnel range, the recorded hologram cannot be considered a Fresnel hologram and the analysis based on Eq. (1) is not valid any more. Based on the discussion in page 69 of Ref [14], the lower limit

_{Δfmin}, for a given system magnification and maximum object size

*O*, isAll the above mentioned analysis is based on the assumption that FINCH is diffraction limited and that the pixel size of the camera does not limit the system resolution. This assumption is fulfilled if the finest fringe of the hologram can be correctly sampled by the camera. Referring to Fig. 1 and recalling that the finest fringe is created by the interfered beams with the largest angle difference between them, the condition that should be satisfied iswhere

_{max}_{ 4Roδc/λ}. Another limitation on the value of

_{Ro/fmin=2Ro/zh=λ/2δs}, where

*z*has mostly negative effects on the hologram and therefore in general we tend to keep the distance

_{h}*z*as close as possible to the lower limit.

_{h}## 3. Experimental methods and results

**Random pixel method -**the phase values of the two quadratic functions are distributed randomly, uniformly and equally among the entire SLM pixels, where each SLM pixel contains only the value of one of the lenses. 2)

**Phase sum method**- the quadratic phase functions of the two lenses are summed and the phase distribution of the sum is displayed on all the pixels of the SLM. 3)

**Random ring method**- the phase values of the two quadratic functions are distributed randomly and uniformly among the SLM rings where each ring on the SLM contains the value of one of the lenses. An example of the SLM phase distribution created by the random pixel method is shown in Fig. 2(a) and 2(d). An example of the SLM phase distribution created by the phase sum method is shown in Fig. 2(b) and 2(e). An example of the SLM phase distribution created by the random ring method is shown in Fig. 2(c) and 2(f).

*mm*× 1.0

*mm*in size, which is a combination of a binary grating with a spatial frequency of 10 cycles per

*mm*and the digits '10.0'. The object was placed 25

*cm*before the collimation lens L

_{1}with a focal length of

*f*= 25

_{o}*cm.*The distance between the SLM (Holoeye Pluto, 1920 × 1080 pixels, 8

*μm*pixel pitch, phase only) and the CMOS camera (PCO.EDGE, pixel size δ

*= 6.5*

_{c}*μm*, 2560 × 2160 pixels array) was

*z*= 49.5

_{h}*cm*. The distance between the camera and SLM,

*z*, was kept constant for all the experiments. A polarizer and a chromatic bandpass filter (BPF) with a 80

_{h}*nm*bandwidth around 650

*nm*central wavelength, was introduced to the setup between the object and the refractive lens. The polarizer direction was adjusted in order to get maximum modulation of the polarized-dependent SLM. The BPF was introduced in order to increase the temporal coherence and by that to improve the fringe visibility of the hologram. The SLM was placed at an angle of 77.5 degrees with respect to the optical axis and the aspect ratio of the Fresnel patterns displayed on the SLM adjusted in one direction for this angle so that the pattern displayed on the camera retained its spherical geometry as if the SLM was normal to the imaging plane. Experiments were carried out with the three different methods of representation of the dual lenses mentioned above. In all the experiments we used the same object, and diffractive masks of 1080 × 1080 pixels displayed on the SLM. FINCH complex holograms created by image capture at three different phases were created for each of eight different values of the gap between the two focuses, ranging from Δ

*f*= 5.3

*cm*(

*f*

_{1}= 47

*cm*,

*f*

_{2}= 52.3

*cm*) to Δ

*f*= 24.9

*cm*(

*f*

_{1}= 40

*cm*,

*f*

_{2}= 64.9

*cm*). This selection of masks satisfied the condition of a complete overlap formulated by Eq. (8).

*f*values and for each method, are depicted in Fig. 4 . Figures 4(a), 4(b) and 4(c) show the best in-focus reconstructions of the holograms captured with the random pixel method, the phase sum method and the random ring method, respectively.

*f*of the two diffractive lenses. This result verifies our prediction that reducing the optical path differences in the system increases the fringe visibility of the recorded holograms.

_{1}and diffractive lens with

*f = z*49.5

_{h}=*cm*) imaging system with the same magnification. The mean-square error (MSE) between the best in-focus reconstructed planes from each hologram and the reference image of the object shown in Fig. 5 is used as a quantitative comparison measure. The MSE calculation is defined in the following equation:where

*i*,

*j*are the coordinates of each pixel,

*M, K*are the dimensions of the considered area,

_{P(i,j)}is the reference image shown in Fig. 5,

_{P˜(i,j)}is one of the reconstructed images shown in Fig. 4, and

*ξ*is a factor that scales the reconstructed images to minimize the MSE given byAll graphs of MSE versus Δ

*f*for the three multiplexing methods are shown in Fig. 6 . The quality improvement is demonstrated both in the visual comparison of Fig. 4, as well as by the quantitative MSE given in Fig. 6. Higher quality is achieved by using the random pixel based method as well as by reducing the focal length difference of the two diffractive lenses on the SLM.

*f*and the SLM-camera distance,

*z*. Substituting the data of

_{h}*R*= 4.3

_{o}*mm*,

*λ*= 650

*nm*, Δλ = 80

*nm*and

*z*= 49.5

_{h}*cm*into Eq. (14) yields that Δ

*f*≈7.5

_{max}*cm*. Substituting the data of

*M*≈2,

_{T}*O*= 2.0

_{max}*mm*and

*λ*= 650

*nm*into Eq. (15) yields that Δ

*f*≈2

_{min}*cm*. The range of our measurements was between Δ

*f*= 5.3

*cm*and Δ

*f*= 24.9

*cm*, and indeed as can be seen in Figs. 4 and 6, in the random pixel method, below Δ

*f*= 10

*cm*the quality of the reconstruction is relatively good and is further improved as Δ

*f*is decreased. Substituting the data of

*R*= 4.3

_{o}*mm*,

*λ*= 650

*nm*and δ

*= 6.7*

_{c}*μm*and δ

*= 8*

_{s}*μm*into Eq. (17) yields that

*z*= 21.2

_{h,min}*cm*. The value of

*z*= 49.5

_{h}*cm*used in our setup ensures that our holographic system is deep inside the regime of a diffracted limited system.

*f*

_{1}= 40cm and

*f*

_{2}= 64.9cm. This Δ

*f*= 24.9 value is the highest point on the graph of Fig. 6, in which the reconstruction quality is poorest of all, in the entire three multiplexing methods. Nonetheless, as can be seen from Figs. 7(a) and 7(b), all the images are relatively clear and the entire details in the images are revealed. How much these images are good can be learned by comparing them to the images at the value of Δ

*f*= 10

*cm*, shown in Figs. 7(c) and 7(d), taken at the distance of 45

*cm*and 55

*cm*from the SLM, respectively, where the focal length of the diffractive lenses multiplexed on the SLM were

*f*

_{1}= 45

*cm*and

*f*

_{2}= 55

*cm*. There is not any significant difference in the quality between the images of Figs. 7(a) and 7(b) and the images of Figs. 7(c) and 7(d), although the quality of the hologram reconstruction for Δ

*f*= 10

*cm*, taken under the random pixel and random ring methods, are much better than that of Δ

*f*= 24.9

*cm.*

## 4. Discussion and conclusions

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

*cm*herein instead of 138

*cm*in Ref [13

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

*nm*herein instead of 10

*nm*in Ref [13

**19**(27), 26249–26268 (2011). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science |

2. | F. Mok, J. Diep, H.-K. Liu, and D. Psaltis, “Real-time computer-generated hologram by means of liquid-crystal television spatial light modulator,” Opt. Lett. |

3. | J. N. Mait and G. S. Himes, “Computer-generated holograms by means of a magnetooptic spatial light modulator,” Appl. Opt. |

4. | J. Rosen, L. Shiv, J. Stein, and J. Shamir, “Electro-optic hologram generation on spatial light modulators,” J. Opt. Soc. Am. A |

5. | C.-S. Guo, Z.-Y. Rong, H.-T. Wang, Y. Wang, and L. Z. Cai, “Phase-shifting with computer-generated holograms written on a spatial light modulator,” Appl. Opt. |

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

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

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

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

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

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

12. | P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, “Point spread function and two-point resolution in Fresnel incoherent correlation holography,” Opt. Express |

13. | J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express |

14. | J. W. Goodman, |

15. | J. W. Goodman, |

16. | E. Wolf, |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.1670) Coherence and statistical optics : Coherent optical effects

(050.1950) Diffraction and gratings : Diffraction gratings

(090.1760) Holography : Computer holography

(090.1970) Holography : Diffractive optics

(090.2880) Holography : Holographic interferometry

(100.6890) Image processing : Three-dimensional image processing

(110.4980) Imaging systems : Partial coherence in imaging

(110.6880) Imaging systems : Three-dimensional image acquisition

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

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: January 20, 2012

Revised Manuscript: March 18, 2012

Manuscript Accepted: March 24, 2012

Published: April 4, 2012

**Citation**

Barak Katz, Joseph Rosen, Roy Kelner, and Gary Brooker, "Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM)," Opt. Express **20**, 9109-9121 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-9109

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### References

- J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science265(5173), 749–752 (1994). [CrossRef] [PubMed]
- F. Mok, J. Diep, H.-K. Liu, and D. Psaltis, “Real-time computer-generated hologram by means of liquid-crystal television spatial light modulator,” Opt. Lett.11(11), 748–750 (1986). [CrossRef] [PubMed]
- J. N. Mait and G. S. Himes, “Computer-generated holograms by means of a magnetooptic spatial light modulator,” Appl. Opt.28(22), 4879–4887 (1989). [CrossRef] [PubMed]
- J. Rosen, L. Shiv, J. Stein, and J. Shamir, “Electro-optic hologram generation on spatial light modulators,” J. Opt. Soc. Am. A9(7), 1159–1166 (1992). [CrossRef]
- C.-S. Guo, Z.-Y. Rong, H.-T. Wang, Y. Wang, and L. Z. Cai, “Phase-shifting with computer-generated holograms written on a spatial light modulator,” Appl. Opt.42(35), 6975–6979 (2003). [CrossRef] [PubMed]
- J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett.32(8), 912–914 (2007). [CrossRef] [PubMed]
- J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express15(5), 2244–2250 (2007). [CrossRef] [PubMed]
- J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics2(3), 190–195 (2008). [CrossRef]
- G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express19(6), 5047–5062 (2011). [CrossRef] [PubMed]
- B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express18(2), 962–972 (2010). [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(30), 5757–5763 (2010). [CrossRef] [PubMed]
- P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, “Point spread function and two-point resolution in Fresnel incoherent correlation holography,” Opt. Express19(16), 15603–15620 (2011). [CrossRef] [PubMed]
- J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express19(27), 26249–26268 (2011). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).
- J. W. Goodman, Statistical Optics (Wiley, 1985).
- E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge, 2007).

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