## Focusing light through random photonic media by binary amplitude modulation |

Optics Express, Vol. 19, Issue 5, pp. 4017-4029 (2011)

http://dx.doi.org/10.1364/OE.19.004017

Acrobat PDF (905 KB)

### Abstract

We study the focusing of light through random photonic materials using wavefront shaping. We explore a novel approach namely binary amplitude modulation. To this end, the light incident to a random photonic medium is spatially divided into a number of segments. We identify the segments that give rise to fields that are out of phase with the total field at the intended focus and assign these a zero amplitude, whereas the remaining segments maintain their original amplitude. Using 812 independently controlled segments of light, we find the intensity at the target to be 75±6 times enhanced over the average intensity behind the sample. We experimentally demonstrate focusing of light through random photonic media using both an amplitude only mode liquid crystal spatial light modulator and a MEMS-based spatial light modulator. Our use of Micro Electro-Mechanical System (MEMS)-based digital micromirror devices for the control of the incident light field opens an avenue to high speed implementations of wavefront shaping.

© 2011 Optical Society of America

## 1. Introduction

_{2}sample; one using a liquid crystal on silicon (LC) spatial light modulator (SLM) in amplitude-only modulation mode and the other using a digital micromirror device (DMD). DMDs consist of millions of mirrors that can be independently controlled to reflect light either to a desired position or to a beam dump. This effectively switches light coming from a particular pixel of DMD on or off and provides a way to spatially modulate the amplitude of light in a binary fashion. The advantage of DMDs over LC SLMs lie in their switching speed. An important figure of merit for switching speed is the settling time, which is the time required for a pixel to become stable after changing its state. For a standard DMD the settling time is 18

*μ*s [10], which is approximately three orders of magnitude faster than that for typical LC SLMs used in the previous works [1

1. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. **32**, 2309–2311 (2007). [PubMed]

5. M. Cui, E. J. McDowell, and C. Yang, “An in vivo study of turbidity suppression by optical phase conjugation (TSOPC) on rabbit ear,” Opt. Express **18**, 25–30 (2010). [PubMed]

5. M. Cui, E. J. McDowell, and C. Yang, “An in vivo study of turbidity suppression by optical phase conjugation (TSOPC) on rabbit ear,” Opt. Express **18**, 25–30 (2010). [PubMed]

## 2. The binary amplitude modulation algorithm

*E*. This electric field is a vectorial sum of electric fields of all incident channels multiplied by the corresponding transmission matrix element. With the algorithm all segments of the incident field are successively probed. Each segment is turned on and off while the intensity at the target output channel is being monitored. This procedure can be visualized by following the block arrows in Fig. 1 (a–c; d–f). As a result the segments leading to destructive interference with the resultant electric field are turned off and the intensity at the target is increased as compared to the unoptimized case. This increase can be seen by comparing the magnitudes of the red vectors in Fig. 1 (a) and Fig. 1 (c). The evolution of the amplitude pattern on the SLM can be visualized by following Fig. 1 (d–f).

_{m}## 3. Experiments with a liquid crystal spatial light modulator

14. E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commercial twisted nematic LCDs,” Appl. Opt. **47**, 2076–2081 (2008). [PubMed]

*μ*m thick layer of air-brush paint (rutile TiO

_{2}pigment with acrylic medium). The transport mean free path for similar samples are

*l*=0.55±0.1

_{tr}*μ*m at 632.8 nm wavelength [1

1. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. **32**, 2309–2311 (2007). [PubMed]

*η*is defined as: where

*I*

_{opt}is the optimized intensity inside the target area after spatial binary amplitude modulation is performed for a specific sample and

*I*

_{ref}is the reference intensity. To measure a suitable reference intensity, the wavefront that is shaped to give a bright focus at target is sent to different parts of the sample. The intensities measured in target with changing sample configuration are ensemble averaged to give

*I*

_{ref}. The enhancement we obtain with this definition gives a measure of the contrast between the focus and the background of the image since the reference intensity is approximately the same as the average background intensity. Since nearly half of the segments on the SLM are turned off in the optimized wavefront, the reference intensity is approximately half of the ensemble averaged intensity when all segments are on.

*η*

_{ideal}increases linearly with the number of controlled input channels N as However, deviations from the ideal conditions reduce the intensity enhancement. We have derived that intensity enhancement under intensity noise, 〈

*η*

_{non–ideal}〉 can be written as where SNR represents the signal to noise ratio of the signal at target position, and 〈

*A*〉

^{2}/〈

*A*

^{2}〉 is a factor introduced to account for non-uniform illumination of the SLM, with A representing the amplitude of field reflected from each SLM segment. When the illumination pattern of the SLM is investigated, 〈

*A*〉

^{2}/〈

*A*

^{2}〉 is found to be 0.97±0.01. Derivation of Eq. (3) and Eq. (4) can be found in the Appendix. The experimental data are fitted to Eq. (4) using the signal to noise ratio (SNR) as the only adjustable parameter. The value of the adjusted SNR is found to be 24. From a test performed on the experimental setup with a static binary amplitude pattern on the SLM, the intensity fluctuations of the light incident to the sample was measured and found to have an SNR of 165. The fact that the adjusted SNR has a lower value than measured SNR can be caused by several reasons: in the experiments the state of each segment is updated continuously during an optimization, increasing the rate of wrong decisions as the optimization proceeds. However, Eq. (4) assumes that the probability of making a wrong decision for the state of a segment is constant throughout the optimization process. Moreover, Eq. (4) takes only intensity noise into account, which is an incomplete description of possible sources of noise or instabilities in the experimental setup. Further investigation of effects of noise and instabilities on the performance of the presented algorithm is beyond the scope of this paper.

## 4. Experiments with a micro electro-mechanical system based spatial light modulator

*μ*m. Each mirror can exhibit two angles; it either reflects light to the intended target or into a light dump [10]. Light reflected from the DMD is projected onto the sample by a 10X 0.25 NA microscope objective and light transmitted through the sample is passed through a polarizer and projected on the CCD camera with a 50 mm focal length lens. The sample that is used in the experiments described in this section is 18.5±2.4

*μ*m thick layer of airbrush paint (rutile TiO

_{2}pigment with acrylic medium). The transport mean free path for similar samples are

*l*=0.55±0.1

_{tr}*μ*m at a wavelength of 632.8 nm [1

1. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. **32**, 2309–2311 (2007). [PubMed]

*μ*s [10]. In our experiments, the optimizations using a DMD chip was achieved in a time scale of several minutes, which is similar to time scales of optimizations performed using the LC SLMs. This effect is due to addressing the device via the video card of the PC, which was performed in the same manner for both SLMs, limiting the communication speed to 60 Hz. With faster control of the DMD devices and use of faster cameras for detection, the speed of the method will increase close to three orders of magnitude and the method will be useful for focusing through materials whose configuration change in short time scales, like biological tissue and can be used for medical imaging purposes.

## 5. Conclusion

## A. Analytical expression for ideal intensity enhancement

*ϕ*=0, so that

*E*=

_{n}*A*when no optimization is performed and

*E*is either 0 or

_{n}*A*after optimization is complete.

*t*) have a uniform distribution between −

_{mn}*π*and

*π*[12]. The amplitudes of the transmission matrix elements, |

*t*|, on the other hand are approximated by a Rayleigh probability density function. The electric field at the target output channel is a vectorial sum of random phasors: Reference light intensity at the target position is the ensemble average of intensities recorded in the target for different sample configurations: the same wavefront is assumed to be sent to the sample while both the intensity inside the focus and the reference intensity are calculated, so that

_{mn}*N*′ is the number of segments that remain on after the optimization procedure is finished. It is important to emphasize that the wavefront is optimized for a certain configuration of the sample and is effectively a randomly shaped wavefront for a different configuration of the sample. So, while

*I*

_{ref}is calculated, light coming from different input channels have random phases at the target position. In this case we assume that the phase of each vector constituting

*E*is drawn from a distribution that is uniform between −

_{m}*π*and

*π*. Using this assumption and the fact that the transmission matrix elements and the incident field are statistically independent, the reference intensity can be written as: where the modulus of a transmission matrix element,

*t*is a random variable having a Rayleigh probability density function.

*A*〉

^{2}/〈

*A*

^{2}〉 to the theoretically expected enhancement [16]. In the experiments described in Section 3, this prefactor is found to have a value of 0.97±0.01.

## B. Analytical expression for intensity enhancement under intensity noise

*P*

_{wrong}is the probability for the algorithm to make a wrong decision for the state of a single segment,

*i.e.*, keeping it on while it should be turned off and vice versa. This probability is: where

*k*segment;

^{th}*I*is positive. Likewise,

_{k}*I*is negative. Here

_{k}*I*and has the probability density function as given in Eq. (11). Similarly,

_{k}*I*and a standard deviation of

_{k}*σ*

_{noise}is the standard deviation for noise. Thus,

*P*

_{wrong}can be evaluated as: From Eq. (13), it is reasonable to assume that

*σ*

_{noise}can be written as

*σ*

_{noise}= 〈

*I*〉/

_{m}*SNR*=

*N*〈

*A*

^{2}

*t*

^{2}〉/

*SNR*. Substituting

*σ*and

*σ*

_{noise}in Eq. (23), we obtain: The possibility of making wrong decisions for a segment leads to observation of a reduced intensity enhancement as compared to the ideal case. The intensity enhancement at target position including the effects of noise and a Gaussian illumination profile, 〈

*η*

_{non–ideal}〉 is given by:

## Acknowledgments

## References and links

1. | I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. |

2. | I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered metamaterials,” Opt. Express |

3. | I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics |

4. | Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics |

5. | M. Cui, E. J. McDowell, and C. Yang, “An in vivo study of turbidity suppression by optical phase conjugation (TSOPC) on rabbit ear,” Opt. Express |

6. | M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to turbidity suppression by phase conjugation,” Opt. Express |

7. | C.-L. Hsieh, Y. Pu, R. Grange, G. Laporte, and D. Psaltis, “Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle,” Opt. Express |

8. | S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. |

9. | S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Communications |

10. | D. Dudley, W. Duncan, and J. Slaughter, “Emerging Digital Micromirror Device (DMD) Applications,” Proc. SPIE |

11. | C. W. J Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. |

12. | J. W. Goodman, |

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

14. | E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commercial twisted nematic LCDs,” Appl. Opt. |

15. | A. Derode, A. Tourin, and M. Fink, “Ultrasonic pulse compression with one-bit time reversal through multiple scattering,” J. Appl. Phys. |

16. | F. van Beijnum, M. Sc. Thesis, University of Twente (2009). |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(110.7050) Imaging systems : Turbid media

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: January 3, 2011

Manuscript Accepted: February 4, 2011

Published: February 15, 2011

**Virtual Issues**

Vol. 6, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

D. Akbulut, T. J. Huisman, E. G. van Putten, W. L. Vos, and A. P. Mosk, "Focusing light through random photonic media by binary amplitude modulation," Opt. Express **19**, 4017-4029 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4017

Sort: Year | Journal | Reset

### References

- I. M. Vellekoop, and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007). [PubMed]
- I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered metamaterials,” Opt. Express 16, 67–80 (2008). [PubMed]
- I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
- Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008). [PubMed]
- M. Cui, E. J. McDowell, and C. Yang, “An in vivo study of turbidity suppression by optical phase conjugation (TSOPC) on rabbit ear,” Opt. Express 18, 25–30 (2010). [PubMed]
- M. Cui, and C. Yang, “Implementation of a digital optical phase conjugation system and its application to turbidity suppression by phase conjugation,” Opt. Express 18, 3444–3455 (2010). [PubMed]
- C.-L. Hsieh, Y. Pu, R. Grange, G. Laporte, and D. Psaltis, “Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle,” Opt. Express 18, 20723–20731 (2010). [PubMed]
- S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010). [PubMed]
- S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
- D. Dudley, W. Duncan, and J. Slaughter, “Emerging Digital Micromirror Device (DMD) Applications,” Proc. SPIE 4985, 14 (2003).
- C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
- J. W. Goodman, Statistical optics (Wiley, New York, 2000).
- M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 2003).
- E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commercial twisted nematic LCDs,” Appl. Opt. 47, 2076–2081 (2008). [PubMed]
- A. Derode, A. Tourin, and M. Fink, “Ultrasonic pulse compression with one-bit time reversal through multiple scattering,” J. Appl. Phys. 85, 6343–6352 (1999).
- F. van Beijnum, and M. Sc, Thesis, University of Twente (2009).

## 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.