## Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling |

Optics Express, Vol. 20, Issue 13, pp. 14321-14337 (2012)

http://dx.doi.org/10.1364/OE.20.014321

Acrobat PDF (1136 KB)

### Abstract

We develop a method for synthesis of a desired intensity profile at the output of a multimode fiber (MMF) with random mode coupling by controlling the input field distribution using a spatial light modulator (SLM) whose complex reflectance is piecewise constant over a set of disjoint blocks. Depending on the application, the desired intensity profile may be known or unknown *a priori*. We pose the problem as optimization of an objective function quantifying, and derive a theoretical lower bound on the achievable objective function. We present an adaptive sequential coordinate ascent (SCA) algorithm for controlling the SLM, which does not require characterizing the full transfer characteristic of the MMF, and which converges to near the lower bound after one pass over the SLM blocks. This algorithm is faster than optimizations based on genetic algorithms or random assignment of SLM phases. We present simulated and experimental results applying the algorithm to forming spots of light at a MMF output, and describe how the algorithm can be applied to imaging.

© 2012 OSA

## 1. Introduction

1. H. J. Shin, M. C. Pierce, D. Lee, H. Ra, O. Solgaard, and R. Richards-Kortum, “Fiber-optic confocal microscope using a MEMS scanner and miniature objective lens,” Opt. Express **15**(15), 9113–9122 (2007). [CrossRef] [PubMed]

2. P. M. Lane, A. L. P. Dlugan, R. Richards-Kortum, and C. E. Macaulay, “Fiber-optic confocal microscopy using a spatial light modulator,” Opt. Lett. **25**(24), 1780–1782 (2000). [CrossRef] [PubMed]

3. K. M. Tan, M. Mazilu, T. H. Chow, W. M. Lee, K. Taguichi, B. K. Ng, W. Sibbett, C. S. Herrington, C. T. A. Brown, and K. Dholakia, “In-fiber common-path optical coherence tomography using a conical-tip fiber,” Opt. Express **17**(4), 2375–2384 (2009). [CrossRef] [PubMed]

4. N. Sim, D. Bessarab, C. M. Jones, and L. Krivitsky, “Method of targeted delivery of laser beam to isolated retinal rods by fiber optics,” Biomed. Opt. Express **2**(11), 2926–2933 (2011). [CrossRef] [PubMed]

6. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics **4**(6), 388–394 (2010). [CrossRef]

7. G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. **99**(19), 191102 (2011). [CrossRef]

*a priori*, one approach is to use an SLM to control the electric field at the MMF input, use a camera to monitor the intensity profile at the MMF output, and use an adaptive algorithm for finding the optimal SLM pattern. Typically, the SLM reflectance is piecewise constant over a disjoint set of blocks, each of which may comprise one or more pixels. Recent work has pursued two classes of methods for adaptively setting the SLM.

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. **104**(10), 100601 (2010). [CrossRef] [PubMed]

10. M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express **19**(2), 933–945 (2011). [CrossRef] [PubMed]

*N*measurements, where

_{SLM}*N*is the number of SLM blocks. Once the transmission matrix has been measured, however, one can synthesize

_{SLM}*any number*of desired field or intensity profiles at the MMF output.

11. R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express **19**(1), 247–254 (2011). [CrossRef] [PubMed]

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

*N*measurements for synthesis of

_{SLM}*each*desired intensity profile. The number of measurements required in our algorithm for the synthesis of a single desired intensity profile is on the same order as the methods measuring the electric field transmission matrix of the system. Those methods could be preferable to our algorithm when a large number of known intensity profiles need to be synthesized. However, our algorithm has the advantage that it can be applied to problems where the desired intensity profile is not known

*a priori*, such as targeted light delivery to fluorophores at unknown locations. It is not clear how algorithms based on measurement of the electric field transmission matrix could be used in such applications. We also show how control of the intensity profile at a MMF output can be used to realize a single-fiber scanning microscope, and evaluate the performance of this imaging scheme.

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

15. O. Katz, E. Small, Y. Bomberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics **5**(6), 372–377 (2011). [CrossRef]

## 2. Optimization problems

### 2.1 Input-output relationship of the system

*a priori*, the system shown in Fig. 1 is employed. Light from a laser is collimated, reflected from an SLM, and focused into the MMF input. At the MMF output, a microscope and camera are used to measure the intensity distribution. The measured intensity profile is fed back to a personal computer that controls the SLM pattern. The feedback data are used to determine an SLM pattern such that the measured intensity distribution optimally approximates the desired distribution. In this section, we show how this can be posed as an optimization problem, and in Section 3 we develop adaptive algorithms for finding the optimal SLM pattern.

*N*is the number of blocks,

*k*

^{th}block:and

*k*

^{th}block. For a phase-only SLM

*N*

_{m}modes (including spatial and polarization degrees of freedom), we can expand the field at the MMF input in the basis of these propagating modes. Let

*i*

^{th}propagating mode in the MMF. Thenwhere the summation is carried over the 2

*N*

_{m}modes supported by the fiber and the expansion coefficients

**U**, which describes phase shifts and mode coupling [16]. Mode-independent loss, while not considered here, can be described by scaling

**U**by a constant. The field at the MMF output can be represented by a state vector

### 2.2 Synthesis of a known intensity profile as an optimization problem

*a priori*. In order to pose this as an optimization problem, we define an objective function

_{1}is the region over the MMF core over which

_{2}is the rest of the fiber core area (See inset of Fig. 1). We found that among various possible objective functions, Eq. (18) performs best in terms of convergence speed and closeness of the final output intensity profile to the desired intensity profile.

### 2.3 Synthesis of an unknown intensity profile as an optimization problem

18. B. D. Mangum, C. Mu, and J. M. Gerton, “Resolving single fluorophores within dense ensembles: contrast limits of tip-enhanced fluorescence microscopy,” Opt. Express **16**(9), 6183–6193 (2008). [CrossRef] [PubMed]

*a priori*. In the setup of Fig. 1, a test object containing fluorophores may be placed at the MMF output, and the fluorescence power back-scattered from one or more fluorophore(s) may be measured by a power meter. In order to pose this as an optimization problem, we define an objective function that is the negative of the total measured fluorescence intensity. When this objective function is minimized, the intensity profile at the MMF output is expected to closely match the unknown distribution of the fluorophores. We define the objective function aswhere

*n*is set to 1 for single-photon fluorescence and 2 for two-photon fluorescence. The integral is carried over the whole core area of the MMF. The optimization problem for SLM reflectances is identical to Eq. (19). Like the problem described in Section 2.2, the present optimization problem is non-convex, but can be solved using adaptive algorithms in practice. In the next section, we describe adaptive optimization algorithms that find the optimal SLM settings based on measurements of the objective function.

## 3. Adaptive optimization algorithms

### 3.1 Adaptive sequential coordinate ascent

19. R. A. Panicker and J. M. Kahn, “Algorithms for compensation of multimode fiber dispersion using adaptive optics,” J. Lightwave Technol. **27**(24), 5790–5799 (2009). [CrossRef]

*i*th SLM block:where

*i*

^{th}block isWe can then set

*i*

^{th}block is

*i*

^{th}SLM blockwhere

*i*

^{th}block is given by Eq. (24), and the optimal amplitude is given by

### 3.2 Lower bound on the optimal value of the objective function

**V**, which is a matrix that depends on the SLM reflectance vector

**V**. Rewriting the objective function Eq. (18) as a function of

**V**we obtainand rewriting the objective function Eq. (20) for

*n*= 1 as a function of

**V**we obtain

**V**, and the optimization problem of Eq. (19) can be rewritten as

20. A. d'Aspremont and S. P. Boyd, “Relaxations and randomized methods for nonconvex QCQPs,” http://www.stanford.edu/class/ee364b/lectures/relaxations.pdf.

## 4. Results

### 4.1 Simulation results for spot formation in known locations

**U**, which is a worst-case model for a real fiber, assuming negligible mode-dependent loss. We assume that a Fourier transform lens is used to focus the SLM output field to the MMF input plane, with only the zeroth diffraction order incident on the MMF core.

*N*

_{m}modes in two polarizations, we can ignore polarization, and all the summations from Section 2 can be performed over the

*N*

_{m}modes in one polarization.

**A**, we have numerically solved for the exact modes of the MMF under the weak-guidance assumption [21

21. B. K. Garside, T. K. Lim, and J. P. Marton, “Propagation characteristics of parabolic-index fiber modes: linearly polarized approximation,” J. Opt. Soc. Am. **70**(4), 395–400 (1980). [CrossRef]

*m*is the order of the super-Gaussian (

*m*= 1 corresponds to an ordinary Gaussian distribution),

*a*is a parameter determining the full-width at half-maximum (FWHM) of the distribution by

### 4.2 Experimental results for spot formation in known locations

*f*= 10.4 mm lens, and directed through a linear polarizer onto a 256 × 256-pixel phase-only SLM. Each pixel is 18 × 18 µm

^{2}, with a phase controllable from 0 to 2π with 8-bit resolution, and with a switching speed of 50 ms. The SLM output passes through a first 45%-45% polarization-independent beam splitter and is coupled by an

*f*= 5.5 mm lens into a parabolic-index MMF having 50-μm core diameter and an NA of 0.19. The MMF output facet is AR-coated, with a reflectivity <1%. The MMF output is magnified 65 × using an

*f*= 4.5 mm aspherical lens and imaged onto a phosphor-coated CCD array. The camera image is monitored by a personal computer (PC) that controls the SLM. The SLM is adapted to form a desired intensity profile using CPSCA, and the resulting SLM pattern can be stored for later use. When a nominally 16 × 16-block SLM is used, the SLM pixels are grouped into blocks, each comprising 16 × 16 pixels. A set of 224 blocks covers a circle enclosing more than 95% of the energy incident on the SLM (See the upper-right corner of Fig. 1), so only these 224 blocks are adapted. When a nominally 8 × 8-block SLM is used, pixels are grouped into blocks of 32 × 32 pixels, and 60 blocks are adapted. Adaptation of a single SLM block requires about 1.2 s, of which about 0.2 s is allocated for the four phase changes, and about 1 s is allocated for the three objective function measurements. Adaptation of 224 blocks requires a total of about 270 s. Some elements of Fig. 4, including a second beam splitter, a power meter, and a test object (shown in the inset), are used for imaging, as described in Section 5 below.

*x*= 25 μm, and it is difficult to form high-quality spots beyond

*x*

_{0,des}= 20 µm. Two minor discrepancies are observable as the spot centroid approaches this value. In Fig. 2(b), the experimental transverse spot sizes are smaller than the simulated values, and in Fig. 2(c), the experimental spot centroids lie closer to the desired values than the simulated spot centroids. We comment on these discrepancies in Section 6.

### 4.3 Simulation results for spot formation in unknown locations

*n*= 1 and

*a priori*, based only on measurement of the total fluorescence intensity.

## 5. Application to imaging

**U**does not change significantly.

**U**.

## 6. Discussion

**U**undergoes a sufficiently small change between the SLM adaptation phase and the image recording phase. This would be facilitated by using a short fiber (or other multimode waveguide) and by enclosing it in a rigid, slender tube to prevent bending. Methods to enhance the stability of the system are a subject of ongoing research.

**U**, and is a subject of ongoing research.

## 7. Conclusion

*a priori*, and developed adaptation algorithms for both phase-only and amplitude-and-phase SLMs. The proposed algorithm brings the objective function close to the global minimum after a single pass over the SLM. We also described how this method can be used to realize a scanning microscope, and simulated its imaging capabilities.

## Appendix A

*i*

^{th}element of the output field state vector

*i*

^{th}propagating mode of the fiberThereforewhere the

*k*

^{th}SLM block. For an amplitude-and-phase SLM, we take

*k*

^{th}SLM block, and we normalize all the SLM reflectances to ensure that

## Acknowledgments

## References and links

1. | H. J. Shin, M. C. Pierce, D. Lee, H. Ra, O. Solgaard, and R. Richards-Kortum, “Fiber-optic confocal microscope using a MEMS scanner and miniature objective lens,” Opt. Express |

2. | P. M. Lane, A. L. P. Dlugan, R. Richards-Kortum, and C. E. Macaulay, “Fiber-optic confocal microscopy using a spatial light modulator,” Opt. Lett. |

3. | K. M. Tan, M. Mazilu, T. H. Chow, W. M. Lee, K. Taguichi, B. K. Ng, W. Sibbett, C. S. Herrington, C. T. A. Brown, and K. Dholakia, “In-fiber common-path optical coherence tomography using a conical-tip fiber,” Opt. Express |

4. | N. Sim, D. Bessarab, C. M. Jones, and L. Krivitsky, “Method of targeted delivery of laser beam to isolated retinal rods by fiber optics,” Biomed. Opt. Express |

5. | M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics |

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

7. | G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. |

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. | T. Čižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express |

10. | M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express |

11. | R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express |

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

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

14. | I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. |

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

16. | G. P. Agrawal, |

17. | S. P. Boyd and L. Vandenberghe, |

18. | B. D. Mangum, C. Mu, and J. M. Gerton, “Resolving single fluorophores within dense ensembles: contrast limits of tip-enhanced fluorescence microscopy,” Opt. Express |

19. | R. A. Panicker and J. M. Kahn, “Algorithms for compensation of multimode fiber dispersion using adaptive optics,” J. Lightwave Technol. |

20. | A. d'Aspremont and S. P. Boyd, “Relaxations and randomized methods for nonconvex QCQPs,” http://www.stanford.edu/class/ee364b/lectures/relaxations.pdf. |

21. | B. K. Garside, T. K. Lim, and J. P. Marton, “Propagation characteristics of parabolic-index fiber modes: linearly polarized approximation,” J. Opt. Soc. Am. |

22. | K. J. Boucher, C. Jan, J. M. Kahn, J. P. Wilde, and O. Solgaard, “Spot formation and scanning microscopy via multimode fibers,” in |

**OCIS Codes**

(110.2350) Imaging systems : Fiber optics imaging

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: March 14, 2012

Revised Manuscript: April 29, 2012

Manuscript Accepted: May 31, 2012

Published: June 12, 2012

**Citation**

Reza Nasiri Mahalati, Daulet Askarov, Jeffrey P. Wilde, and Joseph M. Kahn, "Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling," Opt. Express **20**, 14321-14337 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14321

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### References

- H. J. Shin, M. C. Pierce, D. Lee, H. Ra, O. Solgaard, and R. Richards-Kortum, “Fiber-optic confocal microscope using a MEMS scanner and miniature objective lens,” Opt. Express15(15), 9113–9122 (2007). [CrossRef] [PubMed]
- P. M. Lane, A. L. P. Dlugan, R. Richards-Kortum, and C. E. Macaulay, “Fiber-optic confocal microscopy using a spatial light modulator,” Opt. Lett.25(24), 1780–1782 (2000). [CrossRef] [PubMed]
- K. M. Tan, M. Mazilu, T. H. Chow, W. M. Lee, K. Taguichi, B. K. Ng, W. Sibbett, C. S. Herrington, C. T. A. Brown, and K. Dholakia, “In-fiber common-path optical coherence tomography using a conical-tip fiber,” Opt. Express17(4), 2375–2384 (2009). [CrossRef] [PubMed]
- N. Sim, D. Bessarab, C. M. Jones, and L. Krivitsky, “Method of targeted delivery of laser beam to isolated retinal rods by fiber optics,” Biomed. Opt. Express2(11), 2926–2933 (2011). [CrossRef] [PubMed]
- M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics2, 1–32 (2008).
- T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics4(6), 388–394 (2010). [CrossRef]
- G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett.99(19), 191102 (2011). [CrossRef]
- 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(10), 100601 (2010). [CrossRef] [PubMed]
- T. Čižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express19(20), 18871–18884 (2011). [CrossRef] [PubMed]
- M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express19(2), 933–945 (2011). [CrossRef] [PubMed]
- R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express19(1), 247–254 (2011). [CrossRef] [PubMed]
- I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun.281(11), 3071–3080 (2008). [CrossRef]
- I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett.32(16), 2309–2311 (2007). [CrossRef] [PubMed]
- I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett.101(12), 120601 (2008). [CrossRef] [PubMed]
- O. Katz, E. Small, Y. Bomberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics5(6), 372–377 (2011). [CrossRef]
- G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 2002).
- S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, New York, 2004).
- B. D. Mangum, C. Mu, and J. M. Gerton, “Resolving single fluorophores within dense ensembles: contrast limits of tip-enhanced fluorescence microscopy,” Opt. Express16(9), 6183–6193 (2008). [CrossRef] [PubMed]
- R. A. Panicker and J. M. Kahn, “Algorithms for compensation of multimode fiber dispersion using adaptive optics,” J. Lightwave Technol.27(24), 5790–5799 (2009). [CrossRef]
- A. d'Aspremont and S. P. Boyd, “Relaxations and randomized methods for nonconvex QCQPs,” http://www.stanford.edu/class/ee364b/lectures/relaxations.pdf .
- B. K. Garside, T. K. Lim, and J. P. Marton, “Propagation characteristics of parabolic-index fiber modes: linearly polarized approximation,” J. Opt. Soc. Am.70(4), 395–400 (1980). [CrossRef]
- K. J. Boucher, C. Jan, J. M. Kahn, J. P. Wilde, and O. Solgaard, “Spot formation and scanning microscopy via multimode fibers,” in 2011 IEEE Photonics Conference (PHO) (IEEE, 2011), pp. 713–714.

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