## Spatiotemporal focusing in opaque scattering media by wave front shaping with nonlinear feedback |

Optics Express, Vol. 20, Issue 28, pp. 29237-29251 (2012)

http://dx.doi.org/10.1364/OE.20.029237

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### Abstract

We experimentally demonstrate spatiotemporal focusing of light on single nanocrystals embedded inside a strongly scattering medium. Our approach is based on spatial wave front shaping of short pulses, using second harmonic generation inside the target nanocrystals as the feedback signal. We successfully develop a model both for the achieved pulse duration as well as the observed enhancement of the feedback signal. The approach enables exciting opportunities for studies of light propagation in the presence of strong scattering as well as for applications in imaging, micro- and nanomanipulation, coherent control and spectroscopy in complex media.

© 2012 OSA

## 1. Introduction

1. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. **32**(16), 2309–2311 (2007). [CrossRef] [PubMed]

2. T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics **4**, 388–394 (2010). [CrossRef]

3. E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. **106**, 193905 (2011). [CrossRef] [PubMed]

4. I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered metamaterials, ” Opt. Express **16**(1), 67–80 (2008). [CrossRef] [PubMed]

10. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. **106**, 103901 (2011). [CrossRef] [PubMed]

11. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics **5**, 372–377 (2011). [CrossRef]

12. J. Aulbach, A. Bretagne, M. Fink, M. Tanter, and A. Tourin, “Optimal spatiotemporal focusing through complex scattering media,” Phys. Rev. E **85**, 016605 (2012). [CrossRef]

13. F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Meth. **2**, 932–940 (2005). [CrossRef]

14. P. C. Ray, “Size and shape dependent second order nonlinear optical properties of nanomaterials and its application in biological and chemical sensing,” Chem. Rev. **110**(9), 5332–5365 (2010). [CrossRef] [PubMed]

15. C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Second harmonic generation from nanocrystals under linearly and circularly polarized excitations,” Opt. Express **18**, 11917–11932 (2010). [CrossRef] [PubMed]

16. L. L. Xuan, S. Brasselet, F. Treussart, J.-F. Roch, F. Marquier, D. Chauvat, S. Perruchas, C. Tard, and T. Gacoin, “Balanced homodyne detection of second-harmonic generation from isolated subwavelength emitters,” Appl. Phys. Lett. **89**(12), 121118 (2006). [CrossRef]

17. R. Grange, T. Lanvin, C. L. Hsieh, Y. Pu, and D. Psaltis, “Imaging with second-harmonic radiation probes in living tissue,” Biomed. Opt. Express **2**(9), 2532–2539 (2011). [CrossRef] [PubMed]

## 2. Experiment

*L*= 25

*μ*m and

*L*= 50

*μ*m. The mean free path of this medium (

*l*= 3.5

*μ*m) has been determined by measuring the enhanced backscattering cone. The second-harmonic active nanocrystals consist of barium titanate (BaTiO

_{3}) with a tetragonal crystal structure and an average diameter of 200nm. Prepared using the method described in [18

18. C. L. Hsieh, R. Grange, Y. Pu, and D. Psaltis, “Three-dimensional harmonic holographic microcopy using nanoparticles as probes for cell imaging,” Opt. Express **17**(4), 2880–2891 (2009). [CrossRef] [PubMed]

19. J. C. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy,” Appl. Opt. **24**(9), 1270–1282 (1985). [CrossRef] [PubMed]

*μ*m or 5

*μ*m), the stage samples one optical cycle (841nm) in eight steps, from which the cycle-averaged intensity auto-correlation is calculated. Then the wave front shaping algorithm is started. We use a sequential algorithm; it addresses segment by segment one by one, scanning the phase from

*φ*= 0 to

*φ*= 2

*π*in

*N*= 8 steps. For each step we record the time-integrated second-harmonic radiation from a single nanocrystal retrieved from the camera image as feedback signal. The phase value which maximizes the feedback signal is determined by fitting the measured behavior of the feedback signal vs. phase with a cosine function. We perform several consecutive sequences over all segments of the SLM, starting with a low segmentation of the SLM (48 segments) which is increased twice after the first sequences (to 192 and 768 respectively), always starting the new sequence with the phase pattern obtained from the previous one. After the algorithm has finished, another auto-correlation measurement is performed.

_{φ}## 3. Theory

### 3.1. Intensity-intensity auto-correlation of speckle pulses

*AC*(

*τ*) [19

19. J. C. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy,” Appl. Opt. **24**(9), 1270–1282 (1985). [CrossRef] [PubMed]

*τ*introduced in the Michelson interferometer. For the discussion, we consider ensemble averaged intensity auto-correlation (AC) with background, which are where

*I*(

*t*) denotes the intensity at the fundamental frequency and the brackets denote the ensemble average over all realizations of the scattering medium. For short pulses the observed contrast ratio between the maximum and the background is AC(

*τ*= 0) : AC(

*τ*→ ∞) =3:1, regardless of the exact temporal shape of the intensity

*I*(

*t*).

### 3.2. Enhancement of the time-integrated second harmonic intensity

#### 3.2.1. Enhancement vs. number of segments

*P*

_{2ω}at the particle positioned at point

*b*can be calculated by

*t*describe the propagation of the electric field

_{ab}*E*from SLM segment

*a*to the particle position

*b*. Here, we have made a number of simplifications; we neglect the polarization of the light, as well as the tensor character of the second-order susceptibility

*χ*

_{(2)}of the nanocrystals. Furthermore we assume that the volume speckle inside the medium is larger than the particle size, and the particle radiates like a dipole at the second harmonic frequency. The radiated second harmonic power can be calculated by with the speed of light

*c*, the particle volume

*V*and the vacuum permittivity

*ε*

_{0}. In a wavefront shaping experiment, the electric field on each SLM segment

*a*,

*E*=

_{a}*A*

_{a}e^{φa}, is adapted with the aim that all contributions are in phase at the particle position, e.g.

*φ*= −arg(

_{a}*t*) for all segments

_{ab}*a*. In our experiment, the phase-only SLM does not significantly modify the amplitudes

*A*. As described above, we use a sequential algorithm for the WFS experiment; it addresses all segments one by one, scanning the phase term

_{a}*φ*from 0 to 2

*π*in discrete steps. Given that the segment of interest contributes the field

*E*

_{1}

*e*≡

^{iφ}*t*at the point of interest, while the sum of unmodified segments contributes

_{ab}A_{a}e^{iφ}*E*

_{2}| ≫ |

*E*

_{1}| holds, such that the last term on the right hand side dominates the varying terms, and the behavior of

*W*

_{2ω}(

*ϕ*) is well described by a cosine function with a constant offset.

*N*into account, the observed enhancement is approximately given by where we reintroduced the distribution of the contributing field amplitudes A from each of the N segments.

#### 3.2.2. Enhancement for speckle pulses

*τ*

_{c}around the point in time at which the optimized pulse is formed. Within this temporal window, the enhancement is given by Eq. (8). At earlier or later moments, the speckle pulse will be modified in a random fashion, but on average it will be unaffected by the optimization.

*η*

_{pulsed}will be lower compared to the hypothetical continuous-wave case. We define the reduction factor

*c*by In a first approximation, we expect the enhancement to be lowered by the effective number of independent temporal speckle grains within a speckle pulse, which is approximately given by the ratio

_{τ}*c*≈

_{τ}*τ*

_{c}/

*τ*

_{d}. As the diffuse decay time is proportional to the thickness squared (see section 3.1),

*c*approximately scales with

_{τ}*L*

^{−2}. However, the ratio

*τ*

_{c}/

*τ*

_{d}does not exactly reflect the temporal distribution of the second harmonic intensity. For a known average intensity of the fundamental light at the particle position 〈

*I*(

*t*)〉 we can calculate the correction factor

*c*more precisely. The average generated second harmonic intensity is consequently proportional to 〈

_{τ}*I*(

*t*)〉

^{2}. The reduction factor

*c*for the enhancement compared to the monochromatic case is calculated from the ratio of the energy of the generated second harmonic in a time-window

_{τ}*τ*

_{c}around the time

*t*

_{max}of the maximum of 〈

*I*(

*t*)〉

^{2}and the total second harmonic energy,

#### 3.2.3. Susceptibility tensor

15. C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Second harmonic generation from nanocrystals under linearly and circularly polarized excitations,” Opt. Express **18**, 11917–11932 (2010). [CrossRef] [PubMed]

*E*

_{ci}are the orthogonal components of the electric field along the three axis in the crystal frame and the

*d*are the second-order susceptibilities of the bulk BaTiO

_{ij}_{3}crystal. The considered values are

*d*

_{15}= −41 · 10

^{−9}esu,

*d*

_{31}= −43 · 10

^{−9}esu and

*d*

_{33}= −16 · 10

^{−9}esu [23]. Note that the second-harmonic response is independent of a rotation around the z-axis. The position of the latter in the lab frame is sufficient to describe the second-harmonic response of the nanocrystals, assuming that they are spherical. From Eq. (11) we can see that all components of the vector on the right hand side of Eq. (11) with a non-zero second-harmonic response (

*E*

_{cy}

*E*

_{cz}, 2

*E*

_{cx}

*E*

_{cz}) compete for optimization during the wave front shaping optimization. We assume that the transmission coefficients connecting the SLM segments with each of the crystal axis are independent.

*E*

_{cx}component. Assuming that the detection efficiency is equal for the second harmonic radiation from all crystal axis, and that before optimization the ensemble averaged fields on the three crystal axis 〈|

*E*

_{ci}|〉 are equal, the average generated second harmonic power is proportional to (

*E*

_{cx}components, which are generating second harmonic proportional to

*E*

_{cz}after optimization, such that the cross-terms

*E*

_{cy}

*E*

_{cz}and

*E*

_{cx}

*E*

_{cz}can be neglected.

*E*

_{ci}|〉, {

*i*=

*x,y,z*} are equal. To calculate the feedback signal, we first apply Eq. (11) to calculate the second-order polarization in the crystal frame. Secondly, the polarization vector is calculated in the lab frame, depending on the orientation of the nanocrystal. Finally, we calculate the second-harmonic intensity as it is collected by a high-NA (NA = 1.4) objective corresponding to our experimental parameters. We apply the sequential optimization algorithm, such as applied in our experiment. As a result, we observe that the algorithm in generally optimizes both the

*E*

_{cz}and the

*E*

_{cx}or

*E*

_{cy}component, with a ratio which varies slightly with crystal orientation. Averaged over all crystal positions, we find that the enhancement of the feedback is modified by the factor

*c*= 0.28 ± 0.04 compared to the scalar model (Eq. (8)). Due to the large collection angle, the light radiated from all crystal axis is approximately collected with equal efficiency. The dependence of the factor

_{α}*c*on the crystal orientation is superseded by variations caused by random variations of the transmission coefficients.

_{α}#### 3.2.4. Correction for tight focus

24. S. Roke and G. Gonella, “Nonlinear light scattering and spectroscopy of particles and droplets in liquids,” Annu. Rev. Phys. Chem. **63**, 353–378 (2012) [CrossRef] [PubMed]

25. E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Optimal concentration of light in turbid materials,” J. Opt. Soc. Am. B **28**(5), 1200–1203 (2011). [CrossRef]

*V*is the particle volume of the nanocrystal, and the fraction is the position-dependent intensity

*I*

_{2ω}(

*ϕ*,

*θ*,

*r*) of the second harmonic radiation integrated over the three crystal axis, normalized to its peak intensity

*I*

_{2ω,peak}at the center of the focus. To assess the focal volume quantitatively, we assume that the focus on the nanocrystal has the same profile as a focus created with a high NA lens with and acceptance angle of 90°. From [15

15. C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Second harmonic generation from nanocrystals under linearly and circularly polarized excitations,” Opt. Express **18**, 11917–11932 (2010). [CrossRef] [PubMed]

26. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A **253**, 358–379 (1959). [CrossRef]

*I*

_{2ω}(

*ϕ*,

*θ*,

*r*) using the susceptibility tensor given in Eq. (11). We perform this calculation for a sufficient number of polarization states between linearly and circularly polarized light. We calculate

*c*, averaged over all orientations of the crystal c-axis. For each angle we thereby consider only the respective polarization state which maximizes the SHG process. We used the parameters of the particle radius

_{R}*R*= 100nm, and the refractive index n = 2.3. We find

*c*= 0.57 ± 0.05. We observe that

_{R}*c*is approximately proportional to the radius in the considered size regime. For the crystals used in our experiment, we estimate a size polydispersity of 25%, which will result in equivalent variations of

_{R}*c*.

_{R}#### 3.2.5. Noise

*γ*(|

*At*|,

_{ab}*σ*,

*N*) is the correlation between the optimal phase

_{φ}*ϕ*= −arg(

_{a}*t*) and the phase in the presence of noise

_{ab}*ϕ*for a given noise level

_{σ}*σ*and the magnitude of the contribution given by |

_{a}*At*|.

_{ab}*γ*(|

*At*|,

_{ab}*σ*,

_{a}*N*) is calculated. The phase which is obtained from the fit of the feedback signal with a cosine function (see sections 2 and 3.2.1) is equivalent to the phase of the first non-zero frequency component of a discrete

_{φ}*N*-point Fourier transform of the feedback scan. Component

_{φ}*k*of the Fourier transform is given by If a single oscillation of a cosine signal with amplitude |

*At*| is sampled with

_{ab}*N*steps, the amplitude in the first non-zero frequency component of a discrete Fourier transform is

_{φ}*σ*results in a mean amplitude in same component of

_{a}*n*has a random phase with respect to the signal

*s*, leading to the mentioned deviation from the optimal phase. The probability density function of the phase deviation

*θ*=

*ϕ*−

_{σ}*ϕ*is given by [27] with

_{a}*k*=

*s/n*and Consequently the phase correlation can be calculated by When both contributions are present, the ensemble averaged squared amplitude of the first non-zero frequency Fourier component is written as Similarly, the second component squared, 〈|FT

_{2,a}|

^{2}〉, will be the sum of the contribution of the noise,

*A*|〉

_{a}t_{ab}^{4}. The latter contribution, since it is amplified less by the other segments (see Eq. (5)), is expected to be about a factor

_{2,a}|

^{2}〉 can therefore be used to calculate the noise level, with which 〈|

*A*|

_{a}t_{ab}^{2}〉 can be determined from Eq. (18) using the experimental values 〈|FT

_{1,a}|

^{2}〉.

## 4. Results and discussion

### 4.1. Spatiotemporal focus on a single nanocrystal

*μ*mx1

*μ*m around the position of the selected particle. The size of the feedback area was chosen to balance between the collection of the largest possible amount of the scattered SHG signal and the increasing influence of camera noise with an increasing feedback area. The feedback signal increases largely during the first three WFS sequences and typically converges to its maximum value after the second sequence with the highest chosen segmentation (N=768).

*η*

_{exp}= 0.7 · 10

^{2}to 5.5 · 10

^{2}, see Table 1). The image after optimization for one of the particles is shown in Fig. 2(b). We observed that for different nanocrystals the focus of the detection objective had to be moved towards the scattering medium (estimated adjustment in the range of up to a few

*μ*m by the manual adjuster) to obtain the smallest spot size on the camera after optimization. For different particles we observed spot sizes (FWHM of a two-dimensional Gaussian fit) from 0.2

*μ*m (close to the diffraction limit) to about 0.6

*μ*m, where bigger spots were typically more of an irregular shape rather than a homogeneous spot. These observations reflect the distribution of the nanocrystals in the silica layer and the resulting scattering of the SHG radiation by surrounding silica particles.

*I*

_{d}(

*t*)〉 at the particle position according to [22

22. I. M. Vellekoop, P. Lodahl, and A. Lagendijk, “Determination of the diffusion constant using phase-sensitive measurements,” Phys. Rev. E **71**, 056604 (2005). [CrossRef]

*l*= 3.5

*μ*m, extrapolation length ratio at silica-air interface

*z*

_{e}_{1}= 1.38 and

*z*

_{e}_{2}=0.71 at the silica-glass interface, effective refractive index

*n*

_{eff}= 1.25, beam waist of illumination

*w*= 150nm and detector size

_{I}*w*= 100nm). Since the thickness of the sample varies between about 25 – 50

_{D}*μ*m, we use the thickness

*L*as fit parameter. After WFS, the AC shows a sharp peak, demonstrating that the light is focused as a short pulse. A contribution of the non-optimized part of the pulse (see section 3.2.2), which could be expected as a small signature next to the correlation peak of the focused part, is not visible due to a present higher noise level. The measured AC can be fitted very well with the AC based on a sech

^{2}pulse shape. We calculate the pulse duration from the fit with the well-known deconvolution factor

*τ*

_{pulse}= 0.65

*τ*,

_{AC}*τ*being the FWHM of the fit. We have summarized the results obtained from all six particles in Table 1. For all pulses, the pulse shape changed from speckle pulse before the optimization to a single short pulse after WFS with a duration

_{AC}*τ*

_{pulse}ranging between 102fs to 111fs.

### 4.2. Comparison of the measured and the modeled enhancement

*A*| and the average time-resolved transmission 〈

_{a}t_{ab}*I*

_{d}(

*t*)〉 from the experimental data.

_{2,a}|

^{2}in Fig. 4(b) show a weak dependence on the segment position, which can be caused by several effects. Firstly, cross-talk between pixels of the SLM and diffraction effects for larger phase shifts between neighboring segments can cause a noise term which depends on the intensity present on the segment. Secondly, non-linear effects which are not considered in the above model might be present, such as two-photon absorption in the disordered medium. For all cases, we assume that the linear transmission will be affected by noise or a noise-like contribution of the same magnitude and therefore treat the contributions |FT

_{2,a}|

^{2}as noise term to analyze Eq. (18).

*τ*

_{c}is calculated as the half width at half maximum of the measured Fourier transformed laser spectrum. We model the average time-resolved transmission 〈

*I*(

*t*)〉 at the particle position according to [22

22. I. M. Vellekoop, P. Lodahl, and A. Lagendijk, “Determination of the diffusion constant using phase-sensitive measurements,” Phys. Rev. E **71**, 056604 (2005). [CrossRef]

*η*

_{cw}, and the factors

*c*,

_{τ}*c*and

_{α}*c*, we calculate the enhancement predicted by our model,

_{R}*η*

_{model}=

*c*

_{τ}c_{α}c_{r}η_{cw}. For all particles, the calculated values are listed in Table 1. For the particles 1, 3, 5 and 6 the modeled enhancement predicts the experiment value within the accuracy of our model. For particles 2 and 4 it overestimates the enhancement by a factor 4.6 and 4.1 respectively. An overestimation of this magnitude occurs, if two or more particles are placed in direct vicinity, a case which is difficult to identify from the camera images. All particles would contribute to the average reference signal, which would decrease the calculated enhancement accordingly. Furthermore, an overestimation of the enhancement is not surprising considering the composition of the signal which is being optimized. For a given experimental realization of the sample and the nanocrystal in our experimental setup, the feedback signal is solely a function of the phase values set on the SLM. However, next to a global maximum, for which we perform our calculations, the feedback function has a vast number of local maxima into which the algorithm will converge for a given set of starting phase values. A specific example is the factor

*c*which is calculated for the ideal case, that the optimized pulse is formed exactly at the moment in time where the average time-resolved transmission has its maximum. However, in the experiment the point in time at which the optimized pulse is formed will most likely not coincide with the maximum, leading to a lower observed enhancement. Applying different algorithms for the wave front shaping optimization, e.g. a genetic algorithm [28

_{τ}28. I. M. Vellekoop and A.P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. **281**(11), 3071–3080 (2008). [CrossRef]

29. D. B. Conkey, A. N. Brown, A. M. Caravaca-Aguirre, and R. Piestun, “Genetic algorithm optimization for focusing through turbid media in noisy environments,” Opt. Express **20**(5), 4840–4849 (2012). [CrossRef] [PubMed]

### 4.3. Pulse duration after WFS

*τ*

_{c}= 52fs calculated from the measured spectrum of the laser. This observation resembles that made by the first study of (far-field) spatiotemporal focusing through a turbid medium [10

10. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. **106**, 103901 (2011). [CrossRef] [PubMed]

10. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. **106**, 103901 (2011). [CrossRef] [PubMed]

**106**, 103901 (2011). [CrossRef] [PubMed]

### 4.4. Peak-to-background ratio

*c*, we calculate a peak-to-background ratio between 6 · 10

_{τ}^{2}and 3.2 · 10

^{3}for the investigated particles. The analog peak-to-background ratios for the fundamental intensity are approximately given by the square root of these values.

## 5. Conclusions

## Acknowledgments

## References and links

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

2. | T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics |

3. | E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. |

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

5. | M. Fink, “Time reversed acoustics,” Phys. Today |

6. | G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science |

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

8. | C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Digital phase conjugation of second harmonic radiation emitted by nanoparticles in turbid media,” Opt. Express |

9. | I. M. Vellekoop, M. Cui, and C. Yang, “Digital optical phase conjugation of fluorescence in turbid tissue,” Appl. Phys. Lett. |

10. | J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. |

11. | O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics |

12. | J. Aulbach, A. Bretagne, M. Fink, M. Tanter, and A. Tourin, “Optimal spatiotemporal focusing through complex scattering media,” Phys. Rev. E |

13. | F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Meth. |

14. | P. C. Ray, “Size and shape dependent second order nonlinear optical properties of nanomaterials and its application in biological and chemical sensing,” Chem. Rev. |

15. | C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Second harmonic generation from nanocrystals under linearly and circularly polarized excitations,” Opt. Express |

16. | L. L. Xuan, S. Brasselet, F. Treussart, J.-F. Roch, F. Marquier, D. Chauvat, S. Perruchas, C. Tard, and T. Gacoin, “Balanced homodyne detection of second-harmonic generation from isolated subwavelength emitters,” Appl. Phys. Lett. |

17. | R. Grange, T. Lanvin, C. L. Hsieh, Y. Pu, and D. Psaltis, “Imaging with second-harmonic radiation probes in living tissue,” Biomed. Opt. Express |

18. | C. L. Hsieh, R. Grange, Y. Pu, and D. Psaltis, “Three-dimensional harmonic holographic microcopy using nanoparticles as probes for cell imaging,” Opt. Express |

19. | J. C. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy,” Appl. Opt. |

20. | D. J. Thouless, “Maximum metallic resistance in thin wires,” Phys. Rev. Lett. |

21. | R. Landauer and M. Buttiker, “Diffusive traversal time: effective area in magnetically induced interference,” Phys. Rev. B |

22. | I. M. Vellekoop, P. Lodahl, and A. Lagendijk, “Determination of the diffusion constant using phase-sensitive measurements,” Phys. Rev. E |

23. | R. W. Boyd, |

24. | S. Roke and G. Gonella, “Nonlinear light scattering and spectroscopy of particles and droplets in liquids,” Annu. Rev. Phys. Chem. |

25. | E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Optimal concentration of light in turbid materials,” J. Opt. Soc. Am. B |

26. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A |

27. | J. W. Goodman, |

28. | I. M. Vellekoop and A.P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. |

29. | D. B. Conkey, A. N. Brown, A. M. Caravaca-Aguirre, and R. Piestun, “Genetic algorithm optimization for focusing through turbid media in noisy environments,” Opt. Express |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(110.7050) Imaging systems : Turbid media

(190.3970) Nonlinear optics : Microparticle nonlinear optics

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: October 12, 2012

Revised Manuscript: December 4, 2012

Manuscript Accepted: December 5, 2012

Published: December 17, 2012

**Virtual Issues**

Vol. 8, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Jochen Aulbach, Bergin Gjonaj, Patrick Johnson, and Ad Lagendijk, "Spatiotemporal focusing in opaque scattering media by wave front shaping with nonlinear feedback," Opt. Express **20**, 29237-29251 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-28-29237

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### References

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