## Lensless digital holography with diffuse illumination through a pseudo-random phase mask |

Optics Express, Vol. 19, Issue 25, pp. 25113-25124 (2011)

http://dx.doi.org/10.1364/OE.19.025113

Acrobat PDF (5443 KB)

### Abstract

Microscopic imaging with a setup consisting of a pseudo-random phase mask, and an open CMOS
camera, but without an imaging objective, is demonstrated. The pseudo random
phase mask acts as a diffuser for an incoming laser beam, scattering a speckle
pattern to a CMOS chip, which is recorded once, as a reference. A sample which
is afterwards inserted *somewhere* in the optical beam path
changes the speckle pattern. A single (non-iterative) image processing step,
comparing the modified speckle pattern with the previously recorded one,
generates a sharp image of the sample. After a first calibration the method
works in real-time and allows quantitative imaging of complex (amplitude and
phase) samples in an extended three-dimensional volume. Since no lenses are
used, the method is free from lens aberrations. Compared to standard inline
holography the diffuse sample illumination improves the axial sectioning
capability by increasing the effective numerical aperture in the illumination
path, and it suppresses the undesired twin images. For demonstration, a high
resolution spatial light modulator (SLM) is programmed to act as the
pseudo-random phase mask. We show experimental results, imaging microscopic
biological samples, such as insects, within an extended volume at a distance of
15 cm with a transverse and longitudinal resolution of about 60
*μ*m and 400 *μ*m,
respectively.

© 2011 OSA

## 1. Introduction

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

2. D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London Ser. A **197**, 454–487 (1949). [CrossRef]

3. I. Moon, M. Daneshpanah, A. Anand, and B. Javidi, “Cell identification with computational 3-D holographic microscopy,” Opt. Photon. News **22**, 18–23 (2011). [CrossRef]

^{2}, this effect is particularly significant. We demonstrate this by simultaneously imaging two millimeter-sized samples with a longitudinal separation of 5 cm in a single speckle pattern, demonstrating that both of the samples can be reconstructed independently. Furthermore the images can be reconstructed without the appearance of disturbing twin images. Due to the diffuse reference beam, only the desired first diffraction order is sharply reconstructed, whereas all other orders (including the minus first order which is responsible for the twin image) are dispersed in a uniform background [4

4. T. Nomura and M. Imbe, “Single-exposure phase-shifting digital holography using a random-phase reference wave,” Opt. Lett. **35**, 2281–2283 (2010). [CrossRef] [PubMed]

5. C. Maurer, A. Schwaighofer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Suppression of undesired diffraction orders of binary phase holograms,” Appl. Opt. **47**, 3994–3998 (2008). [CrossRef] [PubMed]

*π*with a resolution of 8 micron (pixelsize). From the known phase distribution in the SLM plane, the complex amplitude of the diffracted field in the camera plane is calculated, which corresponds to the desired reference wave. In order to map position and size of the numerically calculated speckle pattern in the camera plane with the actually recorded images, a preliminary image registration step is required, which, however, can also straightforwardly be performed by using the programmability of the SLM, as will be shown later. These calibration steps have to be done only

*once*for a given setup, and from then on imaging can be done in real time, i.e. from each recorded camera frame the light field in the entire volume between SLM and camera plane can be reconstructed.

## 2. Experimental approach

*T*(

_{R}*x*,

_{S}*y*) = exp(

_{S}*i*Φ

*(*

_{SLM}*x*,

_{S}*y*)) is displayed in the SLM Plane (

_{S}*x*,

_{s}*y*), consisting of uniformly distributed phase levels in an interval between 0 and 2

_{s}*π*. The correspondingly diffracted speckle image

*R*(

*x,y*) in the camera plane (

*x,y*) is recorded as a reference. Thus

*R*= |

*F*{exp(

*i*Φ

*)}|*

_{SLM}^{2}, where the operator

*F*{...} denotes Fresnel (or Fourier) propagation of the wave field from the SLM to the camera plane.

*O*(

*x*,

_{S}*y*) = 1 + Δ

_{S}*O*(

*x*,

_{S}*y*) (with |Δ

_{S}*O*(

*x*,

_{S}*y*)| ≪ 1) is close to unity, i.e. the sample is only a small disturbance for the transmitted speckle field. Then the complete wave field behind the SLM and the object becomes

_{S}*T*=

_{R}O*T*+ Δ

_{R}*O T*= exp(

_{R}*i*Φ

*) + Δ*

_{SLM}*O*exp(

*i*Φ

*). In this case the image intensity*

_{SLM}*S*(

*x,y*) in the camera plane becomes

*S*=

*F*{

*T*+ Δ

_{R}*O T*}

_{R}*F*

^{*}{

*T*+ Δ

_{R}*O T*}, where the “*” symbol means the complex conjugate.

_{R}*R*of the undisturbed speckle image, whereas the last term can be neglected, considering that |Δ

*O*| ≪ 1. Under this assumption the difference between the two speckle images with and without inserted sample object becomes

*F*

^{−1}denotes the inverse Fresnel transform (from the camera plane to the SLM plane). Note that the second term on the right side contains the ratio

*F*{

*T*}/

_{R}*F*

^{*}{

*T*} as a part of the argument of the inverse Fresnel transform, corresponding to a randomly distributed speckle field. Therefore the inverse Fresnel transform of this term (even after multiplication with the additional factor

_{R}*F*

^{*}{Δ

*O T*}) also results in a random speckle field, which is distributed uniformly in the SLM (=object) plane, and can be regarded as “speckle noise” with a total (integrated) intensity which corresponds to the intensity of the reconstructed object. However, since the object is localized and the speckle field is homogeneously distributed across the whole object plane, the object can still be reconstructed with a high signal to noise ratio against a diluted background. Thus one finally obtains that

_{R}*O*but not

*O*is obtained), where both amplitude and complex phase of the object are reconstructed. This is possible since the term

*R*is measured as the intensity distribution of the reference image in the camera plane, and the corresponding phase Φ

*is numerically calculated from the known transmission function of the pseudo-random phase mask*

_{R}*T*. In the experiment it is advantageous to reduce artifacts due to the division by small absolute values of

_{R}*F*

^{*}{

*T*} by approximating

_{R}_{00}mode) is used for illumination. The beam is expanded by a set of lenses (not shown) and illuminates the surface of a reflective SLM (Holoeye HEO 1080P) with a slightly divergent beam through a non-polarizing beamsplitter cube. Directly behind the laser the linear beam polarization direction is optimized for SLM incidence by a half-wave plate (not shown), such that the SLM acts as an almost pure phase modulator, affecting the diffracted polarization only negligibly. The SLM has a resolution of 1920 x 1080 pixels, each with a quadratic shape and an edge length of 8 micron. The SLM is connected with the digital graphics card output of a computer and displays a copy of the actual computer screen image. The gray values of each pixel at the computer monitor are converted into refractive index variations of the liquid crystals at the corresponding SLM pixels, such that 256 (8-bit) phase levels within a range between 0 and 2

*π*can be displayed by each SLM pixel. Only a quadratic region from the center of the SLM surface consisting of 1024 × 1024 pixels (corresponding to an area of approximately 8 x 8 mm

^{2}) is used in the experiment, the remaining area is shielded by a black card quadratic aperture. The light diffracted off the SLM surface passes through the beamsplitter cube and is reflected by a mirror to the chip of a CMOS camera (Canon EOS 1000D) at a distance of approximately 15 cm from the SLM. The camera chip has a size of 22.2 × 14.8 mm

^{2}and a resolution of 3888 × 2592 "colored" pixels. The distance between the SLM and the camera is chosen such that all first order diffracted light from the SLM reaches the CMOS chip surface. The CMOS camera is connected via a USB cable to a computer for remote control. For adjustment, the camera is operated in video mode, showing a real-time image of the light intensity at the CMOS chip on the computer monitor. However, for image recording the camera operates in full resolution mode, recording the speckle images in uncompressed raw-format. Due to the red laser illumination, only the red channel of the RGB image data is used for further data processing.

*iπr*

^{2}/

*λz*), where

*r*is the radius measured from the center of the SLM,

*λ*is the light wavelength (633 nm) and

*z*is the desired distance (15 cm) where the hologram should be sharply reconstructed. Due to the offset divergence of the incoming laser beam, the actual reconstruction distance is slightly larger than the programmed Fresnel distance of 15 cm, and the camera is positioned in the experimentally determined sharp image plane. The advantage of the Fresnel setup is that the zeroth diffraction order of the hologram, i.e. the merely reflected component of the light which amounts - due to the limited diffraction efficiency of the SLM - still to about 5% of the total image intensity, does not focus to a point in the image plane (as in a Fourier hologram), but is instead distributed over the CMOS chip surface [9

9. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Diffractive optical tweezers in the Fresnel regime,” Opt. Express **12**, 2243–2250 (2004). [CrossRef] [PubMed]

*intensity*images, one can calculate also the phase of each image pixel in the camera plane from the numerical reconstruction of the SLM pattern.

*π*(step 2 in Fig. 1). In this case the SLM phase mask acts as an almost ideal scatterer which produces a two-dimensional diffuse speckle pattern at the camera. This speckle pattern is then stored as a reference for all further measurements. Note that due to the knowledge of the phase mask displayed at the SLM, also the phase of each pixel in the camera plane can be calculated.

*somewhere*into the beam path, while the SLM is still displaying the same pseudo-random phase pattern as before (step 3 in Fig. 1). The presence of the sample changes the speckle pattern recorded by the camera.

^{2}directly at the surface of the SLM. The insect acts as a mixed amplitude and phase sample, since it contains both transparent (wings) and absorptive (body) parts.

*S*) and without (

*R*) included samples (normalized by

*i*Φ

*) (which is calculated by numerically propagating the SLM phase pattern into the camera plane - see step 4 in Fig. 1). Afterwards the resulting complex number array is back-propagated into the SLM plane by inverting the Fresnel operation used before for the calculation of the camera image from the SLM phase mask, and divided by the phase term exp(*

_{R}*i*Φ

*), thus removing the offset phase of the illumination light. The squared absolute value of this back-propagated image corresponds to a sharp intensity image of the sample in the SLM plane, in this case an image of the fly placed on top of the SLM (shown in (d)). The corresponding phase of the calculated complex amplitude (in (e)) corresponds to the phase of the sample object, i.e. it is a quantitative measure for the optical thickness of the object.*

_{SLM}*μ*m is obtained, and it may be expected that for a better suited sample object this would be improved considerably.

*μ*m. Figure 6(b) and 6(c) show two reconstructed images of the sample, which are numerically sharply focussed in the planes of the horizontal (b) and vertical (c) hairs. A comparison of the two figures shows that the focal planes can be clearly distinguished, i.e. in (b) the horizontal hair is clearly sharper than the vertical, whereas in (c) the situation is reversed. At a separation of two mm, the peak intensity of the blurred hair is about half of the peak intensity of the one in focus. For thinner objects the difference would be even more distinct, which is why we can estimate the achieved axial resolution to be better than 2 mm.

*d*is limited by the pixel size of the SLM (

*p*=8

*μ*m). The theoretical limit

*d*≈

*λ*/2

*NA*(the factor 2 arises because both imaging and an illumination NA are approximately equal) is determined by the numerical aperture

*NA*of the imaging arrangement, which is given by the distance between camera and SLM (

*z*), and the size of the camera chip (

*L*), i.e.

*NA*≈

*L*/2

*z*. In the initial alignment a test hologram was programmed such that it used the full resolution of the SLM to diffract a test pattern to the camera, which just filled the camera chip. Since the maximal diffraction angle

*α*of the SLM is limited by its minimal grating constant, corresponding to two pixel diameters

*p*, namely sin(

*α*) =

*λ*/2

*p*, the maximal image size in the camera plane, which corresponds also to the size of the camera chip

*L*, is given by

*L*= 2

*z*sin(

*α*) =

*zλ*/

*p*. Comparing this with the resolution limit

*d*, we find that

*d*≈

*p*, as expected. Note that this does not change considerably when shifting the sample to another axial position, since a shift which, e.g., increases the imaging NA, simultaneously decreases the illumination NA, such that the total NA, given by the sum of the two, remains approximately equal (if the sample is in the middle between the SLM and the camera).

*λ*/

*NA*

^{2}≈ 0.6 mm), which is close to our experimentally estimated value of <2mm.

10. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. **3**, 27–29 (1978). [CrossRef] [PubMed]

12. J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrival from experimental far-field speckle data,” Opt. Lett. **13**, 619–621 (1988). [CrossRef] [PubMed]

## 3. Conclusion

## Acknowledgments

## References and links

1. | D. Gabor, “A new microscopic principle,” Nature |

2. | D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London Ser. A |

3. | I. Moon, M. Daneshpanah, A. Anand, and B. Javidi, “Cell identification with computational 3-D holographic microscopy,” Opt. Photon. News |

4. | T. Nomura and M. Imbe, “Single-exposure phase-shifting digital holography using a random-phase reference wave,” Opt. Lett. |

5. | C. Maurer, A. Schwaighofer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Suppression of undesired diffraction orders of binary phase holograms,” Appl. Opt. |

6. | F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A |

7. | C. Kohler, F. Zhang, and W. Osten, “Characterization of a spatial light modulator and its application in phase retrieval,” Appl. Opt. |

8. | An explanation of iterative Fourier transform algorithms can be found for example in: B. C. Kress and P. Meyrueis (Eds.) “ |

9. | A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Diffractive optical tweezers in the Fresnel regime,” Opt. Express |

10. | J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. |

11. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

12. | J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrival from experimental far-field speckle data,” Opt. Lett. |

**OCIS Codes**

(110.6150) Imaging systems : Speckle imaging

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: September 27, 2011

Revised Manuscript: November 3, 2011

Manuscript Accepted: November 6, 2011

Published: November 23, 2011

**Virtual Issues**

Vol. 7, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Stefan Bernet, Walter Harm, Alexander Jesacher, and Monika Ritsch-Marte, "Lensless digital holography with diffuse illumination through a pseudo-random phase mask," Opt. Express **19**, 25113-25124 (2011)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-25-25113

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### References

- D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948). [CrossRef] [PubMed]
- D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London Ser. A 197, 454–487 (1949). [CrossRef]
- I. Moon, M. Daneshpanah, A. Anand, and B. Javidi, “Cell identification with computational 3-D holographic microscopy,” Opt. Photon. News 22, 18–23 (2011). [CrossRef]
- T. Nomura and M. Imbe, “Single-exposure phase-shifting digital holography using a random-phase reference wave,” Opt. Lett. 35, 2281–2283 (2010). [CrossRef] [PubMed]
- C. Maurer, A. Schwaighofer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Suppression of undesired diffraction orders of binary phase holograms,” Appl. Opt. 47, 3994–3998 (2008). [CrossRef] [PubMed]
- F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007). [CrossRef]
- C. Kohler, F. Zhang, and W. Osten, “Characterization of a spatial light modulator and its application in phase retrieval,” Appl. Opt. 48, 4003–4008 (2009). [CrossRef] [PubMed]
- An explanation of iterative Fourier transform algorithms can be found for example in: B. C. Kress and P. Meyrueis (Eds.) “Digital Diffractive Optics,” 1st ed. (John Wiley & Sons, 2000) ISBN-13: 978-0-471-98447-4.
- A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Diffractive optical tweezers in the Fresnel regime,” Opt. Express 12, 2243–2250 (2004). [CrossRef] [PubMed]
- J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978). [CrossRef] [PubMed]
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrival from experimental far-field speckle data,” Opt. Lett. 13, 619–621 (1988). [CrossRef] [PubMed]

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