## Quantitative control over the intensity and phase of light transmitted through highly scattering media |

Optics Express, Vol. 21, Issue 22, pp. 25890-25900 (2013)

http://dx.doi.org/10.1364/OE.21.025890

Acrobat PDF (845 KB)

### Abstract

We experimentally demonstrate the use of the transmission matrix (TM) to quantitatively control the amplitude and phase of the light transmitted through highly scattering media. This is achieved by measuring the absolute value of the TM elements. We also use the fact that the cross-correlations between the contributions of different input channels at the observation plane is important in describing the transmitted optical field. In addition, we demonstrate both quantitative control of the intensity at multiple output spatial modes, each with a different intensity, as well as a “dark” area of low intensity. Our experiments are carried out using a low cost (less than US$600) spatial binary amplitude modulator that we modify for phase-only operation, as well as a novel optical setup that enables independent control of a reference and control signal while maintaining interferometric stability. The optical implementation used in this paper will make such experiments widely accessible to many researchers. Furthermore, the results presented could serve as the foundation for many useful potential applications ranging from the biomedical sciences to optical communications.

© 2013 OSA

## 1. Introduction

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

5. S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint Jr., “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express **20**, 16067–16076 (2012). [CrossRef] [PubMed]

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

2. I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered meta-materials,” Opt. Express **16**, 67–80 (2008). [CrossRef] [PubMed]

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

3. 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). [CrossRef] [PubMed]

3. 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). [CrossRef] [PubMed]

3. 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). [CrossRef] [PubMed]

**104**, 100601 (2010). [CrossRef] [PubMed]

4. D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express **20**, 1733–1740 (2012). [CrossRef] [PubMed]

7. A. Francois, A. Salvadori, A. Bressenot, L. Bezdetnaya, F. Guillemin, and M. A. D’Hallewin, “How to avoid local side effects of bladder photodynamic therapy: Impact of the fluence rate,” J. Urology **190**, 731–736 (2013). [CrossRef]

8. L. Novotny and B. Hecht, *Principles of Nano-Optics* (Cambridge University, 2006). [CrossRef]

10. Q. Zhan, “Cylindrical vector beams: From mathematical concepts to applications,” Adv. Opt. Photon. **1**, 1–57 (2009). [CrossRef]

10. Q. Zhan, “Cylindrical vector beams: From mathematical concepts to applications,” Adv. Opt. Photon. **1**, 1–57 (2009). [CrossRef]

13. S. Tripathi and K. C. Toussaint Jr., “Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection,” Opt. Express **17**, 21396–21407 (2009). [CrossRef] [PubMed]

5. S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint Jr., “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express **20**, 16067–16076 (2012). [CrossRef] [PubMed]

**104**, 100601 (2010). [CrossRef] [PubMed]

5. S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint Jr., “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express **20**, 16067–16076 (2012). [CrossRef] [PubMed]

## 2. Measurement of the transmission matrix

**104**, 100601 (2010). [CrossRef] [PubMed]

**20**, 16067–16076 (2012). [CrossRef] [PubMed]

15. W. H. Lee, “Binary synthetic holograms,” Appl. Optics **13**, 1677–1682 (1974). [CrossRef]

4. D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express **20**, 1733–1740 (2012). [CrossRef] [PubMed]

15. W. H. Lee, “Binary synthetic holograms,” Appl. Optics **13**, 1677–1682 (1974). [CrossRef]

*g*(

*x*,

*y*), we implement Lee’s synthetic binary hologram

*f*(

*x*,

*y*) using the following expression [15

15. W. H. Lee, “Binary synthetic holograms,” Appl. Optics **13**, 1677–1682 (1974). [CrossRef]

*T*is the period of the grating in the hologram. The parameter

*q*defines the duty cycle of the grating [15

15. W. H. Lee, “Binary synthetic holograms,” Appl. Optics **13**, 1677–1682 (1974). [CrossRef]

*f*(

*x*,

*y*) only at the center of each of the available micro-mirrors. The calculated binary hologram is then displayed on the DMD-MMA which results in several diffraction orders. The desired phase modulation is in the first diffraction order, which is selected by using an iris to obstruct the other orders at the Fourier plane of lens L1. The selected order is then collimated by lens L2, wherein the desired phase modulation is observed at its back focal plane. This field is then focused onto the sample S by an infinity corrected, 10X microscope objective OBJ1 (Spencer) with numerical aperture (NA) 0.25. Part of the scattered light is then collected by a second, infinity corrected objective OBJ2 (Reichert) with magnification 45X and NA of 0.66, and passed through an analyzer P (Thorlabs LPNIR100-MP) before finally being recorded by a CMOS camera (Thorlabs DCC1545M). The scattering samples were prepared by depositing a mixture of ZnO and ethanol onto the standard microscope slides which resulted in ZnO films with average thickness of 100

*μm*. Similarly deposited ZnO films have been reported in the literature to have an average mean free path of 6

*μm*[16

16. S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: Transmission matrix approach,” New J. Phys. **13**, 123021 (2011). [CrossRef]

**104**, 100601 (2010). [CrossRef] [PubMed]

**20**, 16067–16076 (2012). [CrossRef] [PubMed]

*n*th observation point corresponding to the

*m*th Hadamard basis element input. Similarly,

*φ*is the phase difference between the control and reference signals. The cross-correlation term [6] |

_{n,m}*g*| defines the correlation between the reference signal and the control signal at the

_{nm}*n*th observation point corresponding to the

*m*th Hadamard basis element input. In the previous studies [3

**104**, 100601 (2010). [CrossRef] [PubMed]

**20**, 16067–16076 (2012). [CrossRef] [PubMed]

*μm*(equivalent to two physical pixels of DCC1545M on each side) and is smaller than the speckle size.

*g*| is calculated as This process of turning on and off the signal in the way described above is not possible using conventional spatial light modulators that are based on nematic liquid crystal materials and is an advantage of the DMD-MMA.

_{nm}*t*through curve fitting consistently provides better results. Here, the intensity at the

_{n,m}*n*th observation point corresponding to the

*m*th Hadamard basis element is phase modulated by

*α*such that the curve fitting is done with |

*g*| and

_{nm}*φ*as free parameters,

_{n,m}*α*and

*I*as the independent and dependent variables, respectively.

_{n,m}*I*is measured for phase shifts corresponding to

_{n,m}*α*= 0,

*π*/2,

*π*, and 3

*π*/2. In the standard approach to measuring the TM elements, 4

*M*phase profiles are required, where

*M*corresponds to the number of input channels [3

**104**, 100601 (2010). [CrossRef] [PubMed]

**20**, 16067–16076 (2012). [CrossRef] [PubMed]

*M*+ 1 phase images are required. However, our approach provides information that is not captured in the standard process, namely, the absolute value of the TM elements as well as the magnitude of the cross-correlation between the reference and the control signals. Note that although the TM elements are measured for a Hadamard basis input, we convert them to the canonical basis input

*T*using a standard Hadamard to canonical conversion [17] before using them in designing of the input phase profiles.

_{n,m}## 3. Input phase profile calculation

*k*=

*k*

_{1},

*k*

_{2}, ··· ,

*k*is calculated by solving the optimization problem of the form where

_{N′}*ϕ*is the phase modulation to be applied to the

_{m}*m*th control segment and

*N′*is the number of points to be optimized. The predicted intensity

*m*th and

*m′*th control segments to the

*k*th observation point, whereas

*M*control segments there are

_{M}C_{2}cross-correlation terms, where the symbol

_{M}C_{2}is used in combinatorics to refer to the combination of

*M*elements taken 2 at a time without repetition. To use Eq. (7) these cross-correlation terms need to be experimentally measured. One approach to measuring these values could be to perform pairwise interference measurements between the contributions of all the available control segments. However, these measurements in canonical basis could lead to a low SNR. An alternative approach would be to measure the transmitted intensities corresponding to at least

_{M}C_{2}distinct input phase profiles and to solve the resulting system of linear equations. However, if one is to limit interest to regions where the measured values of |

*g*| are relatively large, results with sufficient accuracy can be expected with a simpler relation of the form with

_{nm}

_{M}C_{2}cross-correlation terms. Once a phase profile is designed, corresponding holograms are generated using Eq. (1) and then displayed on the DMD-MMA.

**20**, 16067–16076 (2012). [CrossRef] [PubMed]

18. R. H. Byrd, J. Nocedal, and R. A. Waltz, “KNITRO: An integrated package for nonlinear optimization,” in “*Large Scale Nonlinear Optimization*,”, G. D. Pillo and F. Giannessi, eds. (Springer Verlag, 2006), pp. 35–59. [CrossRef]

**20**, 16067–16076 (2012). [CrossRef] [PubMed]

**32**, 2309–2311 (2007). [CrossRef] [PubMed]

## 4. Results and discussion

**104**, 100601 (2010). [CrossRef] [PubMed]

**20**, 16067–16076 (2012). [CrossRef] [PubMed]

19. L. F. Rojas, M. Bina, G. Cerchiari, M. A. Escobedo-Sanchez, F. Ferri, and F. Scheffold, “Photon path length distribution in random media from spectral speckle intensity correlations,” Eur. Phys. J: Spec. Top. **199**, 167–180 (2011). [CrossRef]

*g*| between 0.7 and 1. For each selected region, we calculate 25 phase profiles, each predicted to generate one of the 25 targeted intensities at that region. The 25 targeted intensities range from grayscale intensity level of digital number (DN) = 10 to 250 in steps of 10. This range is chosen since the camera used in our experiment is limited to an 8-bit output with possible outputs ranging from DN = 0 to 255. Then, we generate input optical fields with the prescribed phase profiles using the binary holograms calculated according to Eq. (1). Finally, we measure the respective transmitted intensities at the region of interest and compare the measured intensities with the targeted ones. From the Fig. 4, we see that the measured values follow the targeted ones with a maximum standard deviation of less than 30 DN and a mean standard deviation of 23.8 DN. A trend in the observed intensity values to be larger than the targeted intensities for smaller targeted intensities and smaller than the targeted intensities for larger intensities is consistent with our using Eq. (8) in the calculation of the input phase profile. Specifically, when the magnitude of the cross-correlations is not one, Eq. (8) underestimates the intensities for lower intensities and overestimates for larger intensities. Figure 4 demonstrates that even in the presence of multiple scattering events, it is possible to transmit light of a desired intensity through the scattering medium. It is expected that using Eq. (7) in the optimization process compared to Eq. (8) that is currently used, the quality of the control can further be improved. We also point out that the ability to modulate intensity is dependent upon the magnitude of the cross-correlation values. The maximum range can be expected when the magnitude of the cross-correlations is unity and the range decreases progressively with decreasing magnitudes. In the limit, when all the cross-correlation values are zero, no intensity modulation can be expected.

_{nm}**104**, 100601 (2010). [CrossRef] [PubMed]

*N′*= 2 in Eq. (6). The ability to independently modulate the intensity at multiple points demonstrated in this paper should be useful in projecting grayscale images through highly scattering media. This is in contrast to transmission of phase images through such media as demonstrated in [20

20. S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. **1**, 81 (2010). [CrossRef] [PubMed]

21. D. B. Conkey and R. Piestun, “Color image projection through a strongly scattering wall,” Opt. Express **20**, 27312–27318 (2012). [CrossRef] [PubMed]

*μm*

^{2}is targeted to have a low intensity whereas no control is exercised on the rest of the observation plane. The maximum value of intensity in all the areas is 111.5 DN whereas the maximum intensity value within the area of interest (AOI) is only 19.5 DN; note that this further decreases to 12.5 DN when only the central portion of the AOI which is half the size of the AOI is considered. This type of control could lead to interesting applications such as encoding of information in highly scattering channels.

*g*| larger than 0.7. For each selected region of interest, we calculate the initial input phase profiles necessary to generate a constant targeted intensity of 190 DN, and a concomitant range of phase values from 0 to 330° at an interval of 30° using the approach outlined in Section 3. From these phase profiles we calculate four new phase profiles shifted from the original by 0°, 90°, 180° and 270°. Next, we determine the reference and control signals by judiciously using the area of the DMD as discussed above, and shown in Fig. 2, and measure the corresponding transmitted intensities.

_{nm}*φ*of the transmitted signal using the curve fitting approach outlined in Section 2. Since different regions have different reference signals, to filter out the influence of the reference signal on the measured values, for each region we take the phase value measured corresponding to a targeted phase of 0° as offset and subtract that value from the rest of the measured phase values for that observation region. From the graph we find that, unlike the case of the intensity control, all the targeted phase values show a small standard deviation of about 10°. Further, the mean values of the measured phases are close to the targeted values. It shows that the phase angles are less strongly affected by the amplitude of the cross-correlations compared to the magnitude. It can be easily seen for two channels by looking at the expression for the relative phase in four point phase shifting interferometry which is

*I*

^{α=x}is the resultant intensity with a phase modulation

*α*of

*x*radians. Although the intensities are dependent upon the amplitude of the cross-correlation, the relative phase is not as the amplitude of the cross-correlations in the numerator and the denominator cancel out.

## 5. Conclusion

## 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 meta-materials,” Opt. Express |

3. | 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. |

4. | D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express |

5. | S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint Jr., “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express |

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

7. | A. Francois, A. Salvadori, A. Bressenot, L. Bezdetnaya, F. Guillemin, and M. A. D’Hallewin, “How to avoid local side effects of bladder photodynamic therapy: Impact of the fluence rate,” J. Urology |

8. | L. Novotny and B. Hecht, |

9. | S. Tripathi and K. C. Toussaint Jr., “Versatile generation of optical vector fields and vector beams using a non-interferometric approach,” Opt. Express |

10. | Q. Zhan, “Cylindrical vector beams: From mathematical concepts to applications,” Adv. Opt. Photon. |

11. | S. Tripathi, B. J. Davis, K. C. Toussaint Jr., and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys. |

12. | S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Optics |

13. | S. Tripathi and K. C. Toussaint Jr., “Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection,” Opt. Express |

14. | Texas Instruments, |

15. | W. H. Lee, “Binary synthetic holograms,” Appl. Optics |

16. | S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: Transmission matrix approach,” New J. Phys. |

17. | R. C. Gonzalez and R. E. Woods, |

18. | R. H. Byrd, J. Nocedal, and R. A. Waltz, “KNITRO: An integrated package for nonlinear optimization,” in “ |

19. | L. F. Rojas, M. Bina, G. Cerchiari, M. A. Escobedo-Sanchez, F. Ferri, and F. Scheffold, “Photon path length distribution in random media from spectral speckle intensity correlations,” Eur. Phys. J: Spec. Top. |

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

21. | D. B. Conkey and R. Piestun, “Color image projection through a strongly scattering wall,” Opt. Express |

**OCIS Codes**

(050.1380) Diffraction and gratings : Binary optics

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(110.0113) Imaging systems : Imaging through turbid media

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: August 20, 2013

Revised Manuscript: October 13, 2013

Manuscript Accepted: October 14, 2013

Published: October 22, 2013

**Virtual Issues**

Vol. 9, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Santosh Tripathi and Kimani C. Toussaint, "Quantitative control over the intensity and phase of light transmitted through highly scattering media," Opt. Express **21**, 25890-25900 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-25890

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### References

- I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett.32, 2309–2311 (2007). [CrossRef] [PubMed]
- I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered meta-materials,” Opt. Express16, 67–80 (2008). [CrossRef] [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). [CrossRef] [PubMed]
- D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express20, 1733–1740 (2012). [CrossRef] [PubMed]
- S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint, “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express20, 16067–16076 (2012). [CrossRef] [PubMed]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, 2007).
- A. Francois, A. Salvadori, A. Bressenot, L. Bezdetnaya, F. Guillemin, and M. A. D’Hallewin, “How to avoid local side effects of bladder photodynamic therapy: Impact of the fluence rate,” J. Urology190, 731–736 (2013). [CrossRef]
- L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006). [CrossRef]
- S. Tripathi and K. C. Toussaint, “Versatile generation of optical vector fields and vector beams using a non-interferometric approach,” Opt. Express20, 10788–10795 (2012). [CrossRef] [PubMed]
- Q. Zhan, “Cylindrical vector beams: From mathematical concepts to applications,” Adv. Opt. Photon.1, 1–57 (2009). [CrossRef]
- S. Tripathi, B. J. Davis, K. C. Toussaint, and P. S. Carney, “Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy,” J. Phys. B: At. Mol. Opt. Phys.44, 015401 (2011). [CrossRef]
- S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Optics29, 2234–2239 (1990). [CrossRef]
- S. Tripathi and K. C. Toussaint, “Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection,” Opt. Express17, 21396–21407 (2009). [CrossRef] [PubMed]
- Texas Instruments, DLP®LightCrafterTMEvaluation Module (EVM): User’s Guide(2013).
- W. H. Lee, “Binary synthetic holograms,” Appl. Optics13, 1677–1682 (1974). [CrossRef]
- S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: Transmission matrix approach,” New J. Phys.13, 123021 (2011). [CrossRef]
- R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).
- R. H. Byrd, J. Nocedal, and R. A. Waltz, “KNITRO: An integrated package for nonlinear optimization,” in “Large Scale Nonlinear Optimization,”, G. D. Pillo and F. Giannessi, eds. (Springer Verlag, 2006), pp. 35–59. [CrossRef]
- L. F. Rojas, M. Bina, G. Cerchiari, M. A. Escobedo-Sanchez, F. Ferri, and F. Scheffold, “Photon path length distribution in random media from spectral speckle intensity correlations,” Eur. Phys. J: Spec. Top.199, 167–180 (2011). [CrossRef]
- S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun.1, 81 (2010). [CrossRef] [PubMed]
- D. B. Conkey and R. Piestun, “Color image projection through a strongly scattering wall,” Opt. Express20, 27312–27318 (2012). [CrossRef] [PubMed]

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