## Programming balanced optical beam splitters in white paint |

Optics Express, Vol. 22, Issue 7, pp. 8320-8332 (2014)

http://dx.doi.org/10.1364/OE.22.008320

Acrobat PDF (6603 KB)

### Abstract

Wavefront shaping allows for ultimate control of light propagation in multiple-scattering media by adaptive manipulation of incident waves. We shine two separate wavefront-shaped beams on a layer of dry white paint to create two enhanced output spots of equal intensity. We experimentally confirm by interference measurements that the output spots are almost correlated like the two outputs of an ideal balanced beam splitter. The observed deviations from the phase behavior of an ideal beam splitter are analyzed with a transmission matrix model. Our experiments demonstrate that wavefront shaping in multiple-scattering media can be used to approximate the functionality of linear optical devices with multiple inputs and outputs.

© 2014 Optical Society of America

## 1. Introduction

6. J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics **3**, 687–695 (2009). [CrossRef]

7. I. Freund, “Looking through walls and around corners,” Physica A **168**, 49–65 (1990). [CrossRef]

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

9. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics **6**, 283–292 (2012). [CrossRef]

10. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics **4**, 320–322 (2010). [CrossRef]

15. Y. Guan, O. Katz, E. Small, J. Zhou, and Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. **37**, 4663–4665 (2012). [CrossRef] [PubMed]

^{3}.

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

15. Y. Guan, O. Katz, E. Small, J. Zhou, and Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. **37**, 4663–4665 (2012). [CrossRef] [PubMed]

16. 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]

*m*linear optical circuits with 1 incident mode projected to

*m*output modes. If one is capable to manipulate

*n*incident modes with wavefront shaping, it becomes possible to program

*n*×

*m*optical circuits with a desired transmission matrix

**T**. To our knowledge, no experiment demonstrating this capability has been reported.

## 2. Interference on a beam splitter

17. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A **40**, 1371–1384 (1989). [CrossRef] [PubMed]

*θ*applied by the beam splitter on the incident and outgoing modes respectively. In this article we use the term input mode for an incident wave that describes a single orthogonal input of the normal beam splitter or wavefront-shaped beam splitter. The phase angle Θ in the center matrix determines the splitting ratio, which has to be Θ = (2

*n*+ 1)

*π*/2 for a 50:50 beam splitter, with

*n*an integer value.

*θ*between mode 1 and 2 can be controlled. The power

*P*

_{1′},

*P*

_{2′}as a function of Δ

*θ*is shown in Fig. 1(b).

*P*

_{1′}and

*P*

_{2′}will oscillate as a function of Δ

*θ*with a phase difference of the power oscillation

*δ*=

*π*for the ideal lossless beam splitter, independent of phases Δ

*θ*and Ψ in Eq. (1). For this graph we have set Ψ = 0. Any nonzero value for Ψ will provide a phase offset to both the output modes, essentially shifting both

*P*

_{1′}and

*P*

_{2′}along the horizontal axes by the same amount. If one of the incident modes is blocked, a constant power is detected in both output modes that is 4 times lower than the maximum power in one of the output modes when both inputs are present. Note that energy is conserved:

*P*

_{1′}+

*P*

_{2′}=

*P*

_{1}+

*P*

_{2}, and a fringe visibility of 100% is observed.

18. T. Kiss, U. Herzog, and U. Leonhardt, “Compensation of losses in photodetection and in quantum-state measurements,” Phys. Rev. A **52**, 2433–2435 (1995). [CrossRef] [PubMed]

22. J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Optic. **47**, 1819–1824 (2000). [CrossRef]

**T**as a non-unitary version of Eq. (1). The phase difference

*δ*=

*π*will still hold if there are losses in any of the input or output modes. In such a case, the damping can be modeled in the left or right matrix in Eq. (1) by making them non-unitary with a determinant smaller than 1. This is valid for the typical beam splitter one uses. However, when the scattering in the beam splitter does not conserve energy (

*i.e.*the damping is in the center matrix in Eq. (1) dictating that not all energy goes to the considered output modes),

*δ*could in principle take on any value as long as the total output power does not exceed the total input power. To model the most general form of a lossy beam splitter we make the following assumptions: 1. The system is described by an effective transmission matrix

**T**consisting of 2×2 elements; 2. Transmission matrix

**T**provides an equal power splitting ratio; 3. The power losses for both input modes are identical.

*r*|

^{2}and transmitted with transmission |

*t*|

^{2}. For a lossy balanced beam splitter this means |

*r*| = |

*t*| and |

*r*|

^{2}+ |

*t*|

^{2}≤ 1. We relate the complex input amplitudes

*A*

_{1}and

*A*

_{2}to the output amplitudes

*A*

_{1′}and

*A*

_{2′}as

*A*

_{1}→ |

*r*|(

*A*

_{1′}+

*e*

^{i}^{δ}*A*

_{2′}) and

*A*

_{2}→ |

*r*|(

*A*

_{1′}+

*A*

_{2′}), with relative phase term

*δ*. Now we set |

*r*|

^{2}= 1/

*N*, with splitting factor

*N*and

*N*≥ 2. This leads to the following transmission matrix: For the ideal lossless 50:50 beam splitter,

**T**is only unitary when

*N*= 2, with for example

*δ*=

*π*, and would always result to interference as shown in Fig. 1(b). The eigenvalues

*λ*

_{1}and

*λ*

_{2}of the matrix in Eq. (2) are: The observed phase difference in the power oscillations between the output channels in an interference experiment is given by

*δ*. Since

**T**is a square matrix, from the singular values of Eq. (2)

*r*|

^{2}= |

*t*|

^{2}. In addition

*τ*

_{1},

*τ*

_{2}≤ 1 to guarantee a transmission not exceeding 1. This restricts the possible

*δ*to the range 2cos

^{−1}(

*N*/2 − 1) ≤

*δ*≤ 2cos

^{−1}(1 −

*N*/2) for 2 ≤

*N*≤ 4, as marked by the gray area in Fig. 2.

*N*≫ 10

^{2}and therefore any

*δ*is allowed. The scattering statistics of the sample, such as the the singular value distribution, and the intensity enhancement defining

*N*in Eq. (4), determine the combination of

*τ*

_{1}and

*τ*

_{2}that satisfy Eq. (4). Therefore one would not expect in general a constant probability distribution for

*δ*in the gray marked area in Fig. 2. We would like to approximate the behaviour of a beam splitter where

*δ*→

*π*since this mimics the beam splitter one normally uses.

## 3. Optimization algorithm

*ϕ*⊂ [0, 2

_{n}*π*) of the

*n*

^{th}segment is randomly chosen and the output powers

*P*

_{1′}and

*P*

_{2′}are monitored.

*ϕ*is accepted and kept on the SLM if the summed output powers of both spots has increased and the difference power has decreased:

_{n}*P*_{1′,new}+*P*_{2′,new}>*P*_{1′,old}+*P*_{2′,old}+*ε*_{1}- |
*P*_{1′,new}−*P*_{2′,new}| < |*P*_{1′,old}−*P*_{2′,old}| +*ε*_{2}

*ε*

_{1},

*ε*

_{2}→ 0 to compensate for noise. Otherwise the previous

*ϕ*was restored. Next the (

_{n}*n*+ 1)

^{th}segment is addressed, etc. After the final segment has been addressed, the entire optimization is repeated until the desired convergence is reached.

*θ*is controlled with the SLM.

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

## 4. Experimental setup

*μ*m. The layer is approximately 30

*μ*m thick and spray painted on a glass microscope slide of 1 mm thickness. The transmitted speckle pattern is collected with a second objective (NA=0.55, Nikon) and imaged on a CCD camera after reflection on a PBS, see for example Fig. 4(b). The intensity values for the CCD pixels that correspond to the target spots are integrated to obtain the output powers for the enhanced spots. The optimized spots can be transmitted through the PBS, towards a different part of the setup for applications, by rotating the HWP.

*π*rad. This algorithm was repeated approximately 15 times for all segments to obtain two enhanced spots of equal power at 1′ and 2′, see Fig. 4(b). The total optimization procedure for both incident modes takes about 3 hours. In our experiments we achieved

*η*∼ 5. At ≈ 10

^{3}speckles this is not yet close to the theoretical maximum [8

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

## 5. Experimental results

*θ*in gray values. All pictures are subsequently taken with the same integration time. The intensity clearly oscillates between the two target spots.

*P*

_{1′}(red squares) and

*P*

_{2′}(blue diamonds) as a function of the applied phase difference Δ

*θ*. Both curves show sinusoidal behavior and are approximately out-of-phase, mimicking the behavior of an ideal beam splitter as is shown in Fig. 1(b). We expect an error in the phase of about Δ(Δ

*θ*) = 0.1 rad due to interferometric stability during data collection and an additional systematic error of 0.1 rad due to phase calibration (both not shown). We have fitted two functions of the form

*A*sin(Δ

*θ*+

*b*) +

*c*to the measured power, which is in good agreement with the data points. From

*b*we determined the phase difference |

*δ*| = 2.30 ± 0.14 rad, close to but significantly different from the value of

*δ*=

*π*of an ideal beam splitter. Note that since the ordering of the two spots is arbitrary, we only consider the reduced |

*δ*| when discussing the experimental results. Since the Both

*P*

_{1′}and

*P*

_{2′}show a fringe visibility of approximately 100%, which indicates a near-perfect mode matching between the output modes for the two seperate incident modes. The maximum measured power in both spots is approximately the same to within 5%. When one of the incident modes is blocked in the interference experiment, the output power is approximately constant (white diamonds and squares). The small spatial separation between mode 1 and 2 on the SLM gives a small crosstalk, causing fluctuations within 10%. The output power is approximately 4 times lower than the maximum power in one output mode when both input modes are incident, in excellent agreement with Fig. 1(b).

*δ*|. The result is shown in Fig. 6. All measurements were performed under comparable circumstances. Although the number of measurements are not sufficient for any statistical relevant conclusion, our measurements suggest a tendency for |

*δ*| to cluster close to

*π*. This is somewhat unexpected and therefore interesting to explore. In the next section we therefore present a model that predicts this behavior.

## 6. Model for the phase difference

*δ*in the interference experiment based on random matrix theory. Light undergoes isotropic multiple scattering in a sample with a thickness much larger than the scattering mean free path

*l*and

*kl*≫ 1. We therefore expect the transmission matrix

**T**to follow the statistics of a random matrix, as was demonstrated experimentally for ZnO [16

16. 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]

**T**is a subset of the scattering matrix

**S**, and therefore

*δ*can take in principle any value between [0, 2

*π*) with equal probability. This would naively result in a constant probability distribution for

*δ*, which is not observed.

**S**has to be unitary, which sets restrictions on the allowed values for each element

*s*. Consider a random

_{a,b}**S**in a basis where one input mode is one element of the input vector and one target output spot is one element of the output vector. If

**S**contains a beam splitter of equal splitting ratio, there have to be 2 rows and 2 columns in

**S**with corner elements of approximately the same amplitude: with {

*i*,

*j*,

*m*,

*n*} ≤ Dim(

**S**) positive integers and

*c*

_{1}≈

*c*

_{2}≈

*c*

_{3}≈ 1. For a random scattering matrix with dimension Dim(

**S**) = 2, the only possibility for a balanced beam splitter is that

*δ*=

*π*. From matrix algebra it follows that for Dim(

**S**) = 3,

*δ*can only lie on the boundary lines of the gray area of Fig. 2:

*δ*= 2cos

^{−1}(1/(2|

*r*|

^{2}) − 1) or, equivalently by reordering the two outputs,

*δ*= 2cos

^{−1}(1 − 1/(2|

*r*|

^{2})), with |

*r*|

^{2}the intensity reflectivity as defined in section 2. The amplitude coefficients of

**S**should satisfy |

*s*|

_{a,b}^{2}≥ 1/4. For Dim(

**S**) ≥ 4 any phase becomes accessible within the gray marked area of Fig. 2. However, the corresponding phase distribution is strongly dependent on Dim(

**S**), as illustrated in Fig. 7. There we have generated many random scattering matrices with different dimension that contain a balanced beam splitter [24]. The corresponding intensity enhancement is given by

*η*= |

*s*|

^{2}/〈|

*s*|

_{a,b}^{2}〉, with |

*s*|

^{2}= (1/4)(|

*s*|

_{i,j}^{2}+ |

*s*|

_{n,j}^{2}+ |

*s*|

_{i,m}^{2}+ |

*s*|

_{n,m}^{2}), and 〈|

*s*|

_{a,b}^{2}〉 = 1/Dim(

**S**). An increased probability for

*δ*=

*π*is observed with higher

*η*. The probability distribution becomes flat for small

*η*. How this scales, depends strongly on Dim(

**S**). It becomes extremely difficult to observe

*η*> 4 for 〈|

*s*|

_{a,b}^{2}〉 < 0.01, because the probability to get these realizations out of random unitary matrices becomes astronomically small. The SLM in our experiment, however, is programmed to realize exactly such situations.

**S**. Figure 8 shows the observed distribution for |

*δ*| as a function of intensity enhancement

*η*. The main observation is that P(|

*δ*|) has a global maximum at |

*δ*| =

*π*that increases with

*η*. The simulations in Fig. 8 are performed for Dim(

**S**) = 300. We control the phase of 40 input elements representing incident mode 1, and 40 input elements representing incident mode 2. Each controlled input element has a normalized input power of 1. We have set margins

*ε*

_{1}= 0 and

*ε*

_{2}= 0.001. We apply the optimization algorithm 2500 times per mode to guarantee convergence. We select output elements for which the total power of the optimization for mode 1 is within 10% of the optimization for mode 2, approximating our experiment. The intensity enhancement

*η*is given by the observed power in a target spot, divided by 40 × 〈|

*s*|

_{a,b}^{2}〉 = 40/Dim(

**S**), where the factor 40 comes from the number of channels that are controlled per incident mode.

**S**) and several amounts of controlled input channels, always demonstrating a global maximum at a |

*δ*| =

*π*that increases with

*η*. This demonstrates that the two optimized spots approximate better the behavior of a balanced beam splitter with increasing enhancement, using our optimization algorithm. Based on the model with large unitary matrices, it is likely that this result is independent of the type of optimization algorithm used.

*η*. We are free to define a basis for the complete scattering matrix

**S**, which includes the SLM and the scattering material. We write

**S**as: where the first two elements of the input vector correspond with the field in the input modes and the first two elements of the output vector correspond to the field in the output modes. Therefore the relevant elements in

**S**for the beam splitter are the four top left elements. We have chosen a basis where the phase difference

*δ*of the beam splitter, as observed in our interference experiment, is included in

*s*

_{2,1}. The phase difference between the input modes

*α*

_{in}is included in

*s*

_{1,2}and

*s*

_{2,2}, the phase difference between the enhanced spots

*α*

_{out}is included in

*s*

_{2,1}and

*s*

_{2,2}. Since

**S**has to be unitary, each column of the matrix should be orthogonal to the other columns. Therefore the innerproduct between the first two columns becomes:

*η*≪ Dim(

**S**) we assume that all elements

*s*still follows the statistics of the elements of a randomly generated unitary matrix, except for the elements describing the beam splitter. We are dealing with a system where Dim(

_{i,j}**S**) ≫ 2 and therefore we can approximate Dim(

**S**) − 2Dim(

**S**). We assume that for a random scattering matrix all elements

*s*are complex Gaussian distributed with mean 〈

_{i,j}*s*〉 = 0 and standard deviation

_{i,j}*σ*= 〈|

_{s}*s*〉 = 1. From the rules of multiplication and adding Gaussian distributions it follows that

_{i,j}*B*should be a complex Gaussian distributed value with 〈

*B*〉 = 0 and

*B*can be ignored in Eq. (7) and we expect

*δ*→

*π*to satisfy this equation. For the simulations in Fig. 8 this should be the case for

*η*= 12.2, which we indeed observe in the simulated distributions. In our experiments we have

*η*∼ 5 and Dim(

**S**) ∼ 10

^{3}, and therefore according to this model

*δ*→

*π*for

*η*∼ 10

^{1}. Therefore our experimental observations of Fig. 6 are not convincingly explained by this model. On the other hand, Eq. (7) sets restrictions on the allowed combinations of

*B*,

*α*

_{in},

*η*, and

*δ*, which we have ignored up till now. Therefore more advanced modeling is required that is outside the scope of the present paper.

## 7. Discussion

*η*on the observed

*δ*. In addition this would also allow to address speckle patterns with more complicated correlations.

26. S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. **34**, 2739–2741 (2009). [CrossRef] [PubMed]

27. T. J. Huisman, S. R. Huisman, A. P. Mosk, and P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B. (2013). [CrossRef]

28. P. Lodahl, A. P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. **95**, 173901 (2005). [CrossRef] [PubMed]

35. D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, and J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. **108**, 053601 (2012). [CrossRef] [PubMed]

## 8. Conclusions and outlook

## Acknowledgments

## References and links

1. | E. Hecht, |

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

3. | M. A. Nielsen and I. L. Chuang, |

4. | E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature |

5. | P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. |

6. | J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics |

7. | I. Freund, “Looking through walls and around corners,” Physica A |

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

9. | A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics |

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

11. | 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. |

12. | 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. |

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

14. | D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” Nat. Comm. |

15. | Y. Guan, O. Katz, E. Small, J. Zhou, and Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett. |

16. | 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. |

17. | R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanic lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A |

18. | T. Kiss, U. Herzog, and U. Leonhardt, “Compensation of losses in photodetection and in quantum-state measurements,” Phys. Rev. A |

19. | U. Leonhardt, |

20. | S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A |

21. | L. Knöll, S. Scheel, E. Schmidt, D. -G. Welsch, and A.V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A |

22. | J. Jeffers, “Interference and the lossless lossy beam splitter,” J. Mod. Optic. |

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

24. | We use Matlab 2013 for creating random unitary matrices |

25. | N. P. Puente, E. I. Chaikina, S. Herath, and A. Yamilov, “Fabrication, characterization and theoretical analysis of controlled disorder in the core of the optical fibers,” Appl. Optics |

26. | S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. |

27. | T. J. Huisman, S. R. Huisman, A. P. Mosk, and P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering materials,” Appl. Phys. B. (2013). [CrossRef] |

28. | P. Lodahl, A. P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. |

29. | S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. |

30. | Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. |

31. | A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science |

32. | Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in Anderson localization of indistinguishable particles,” Phys. Rev. Lett. |

33. | J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. |

34. | W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. |

35. | D. Bonneau, M. Lobino, P. Jiang, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, M. G. Thompson, and J. L. O’Brien, “Fast path and polarization manipulation of telecom wavelength single photons in lithium niobate waveguide devices,” Phys. Rev. Lett. |

36. | S. R. Huisman, Light Control with Ordered and Disordered Nanophotonic Media (PhD. thesis, University of Twente, 2013). |

**OCIS Codes**

(290.0290) Scattering : Scattering

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: January 23, 2014

Revised Manuscript: March 20, 2014

Manuscript Accepted: March 21, 2014

Published: April 1, 2014

**Virtual Issues**

Vol. 9, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Simon R. Huisman, Thomas J. Huisman, Sebastianus A. Goorden, Allard P. Mosk, and Pepijn W. H. Pinkse, "Programming balanced optical beam splitters in white paint," Opt. Express **22**, 8320-8332 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8320

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### References

- E. Hecht, Optics (Addison Wesley, 4, 2002).
- B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2, 2007).
- M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 1, 2000).
- E. Knill, R. Laflamme, G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001). [CrossRef] [PubMed]
- P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Millburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]
- J. L. O’Brien, A. Furusawa, J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]
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