Backscatter analysis based algorithms for increasing transmission through highly scattering random media using phase-only-modulated wavefronts

Curtis Jin; Raj Rao Nadakuditi; Eric Michielssen; Stephen C. Rand

doi:10.1364/JOSAA.31.001788

Journal of the Optical Society of America A
Vol. 31,
Issue 8,
pp. 1788-1800
(2014)
•https://doi.org/10.1364/JOSAA.31.001788

Backscatter analysis based algorithms for increasing transmission through highly scattering random media using phase-only-modulated wavefronts

Curtis Jin, Raj Rao Nadakuditi, Eric Michielssen, and Stephen C. Rand

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Curtis Jin, Raj Rao Nadakuditi, Eric Michielssen, and Stephen C. Rand, "Backscatter analysis based algorithms for increasing transmission through highly scattering random media using phase-only-modulated wavefronts," J. Opt. Soc. Am. A 31, 1788-1800 (2014)

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Abstract

Recent theoretical and experimental advances have shed light on the existence of so-called “perfectly transmitting” wavefronts with transmission coefficients close to 1 in strongly backscattering random media. These perfectly transmitting eigen-wavefronts can be synthesized by spatial amplitude and phase modulation. Here, we consider the problem of transmission enhancement using phase-only-modulated wavefronts. Motivated by biomedical applications, in which it is not possible to measure the transmitted fields, we develop physically realizable iterative and non-iterative algorithms for increasing the transmission through such random media using backscatter analysis. We theoretically show that, despite the phase-only modulation constraint, the non-iterative algorithms will achieve at least about $25 π % \approx 78.5 %$ transmission with very high probability, assuming that there is at least one perfectly transmitting eigen-wavefront and that the singular vectors of the transmission matrix obey the maximum entropy principle such that they are isotropically random. We numerically analyze the limits of phase-only-modulated transmission in 2D with fully spectrally accurate simulators and provide rigorous numerical evidence confirming our theoretical prediction in random media, with periodic boundary conditions, that is composed of hundreds of thousands of non-absorbing scatterers. We show via numerical simulations that the iterative algorithms we have developed converge rapidly, yielding highly transmitting wavefronts while using relatively few measurements of the backscatter field. Specifically, the best performing iterative algorithm yields $\approx 70 %$ transmission using just 15–20 measurements in the regime, where the non-iterative algorithms yield $\approx 78.5 %$ transmission, but require measuring the entire modal reflection matrix. Our theoretical analysis and rigorous numerical results validate our prediction that phase-only modulation with a given number of spatial modes will yield higher transmission than amplitude and phase modulation with half as many modes.

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Vector Operation	Physical Operation
1: ${\underset{̲}{a}}_{1}^{-} = S_{11} \cdot {\underset{̲}{a}}_{1, (k)}^{+}$	1: ${\underset{̲}{a}}_{1, (k)}^{+} \overset{Backscatter}{\to} {\underset{̲}{a}}_{1}^{-}$
2: $\bar{a} = (\sum_{n = - N}^{N} \| a_{1, n}^{-} \| / \sqrt{M})$	2: $\bar{a} = (\sum_{n = - N}^{N} \| a_{1, n}^{-} \| / \sqrt{M})$
3: ${\underset{̲}{a}}_{1}^{-} \leftarrow \underset{̲}{p} ({\underset{̲}{a}}_{1}^{-})$	3: ${\underset{̲}{a}}_{1}^{-} \leftarrow \underset{̲}{p} ({\underset{̲}{a}}_{1}^{-})$
4: ${\underset{̲}{a}}_{1}^{+} = F \cdot {({\underset{̲}{a}}_{1}^{-})}^{*}$	4: ${\underset{̲}{a}}_{1}^{-} \overset{PCM}{\to} {\underset{̲}{a}}_{1}^{+}$
5: ${\underset{̲}{a}}_{1}^{-} = S_{11} \cdot {\underset{̲}{a}}_{1}^{+}$	5: ${\underset{̲}{a}}_{1}^{+} \overset{Backscatter}{\to} {\underset{̲}{a}}_{1}^{-}$
6: ${\underset{̲}{a}}_{1}^{+} = F \cdot {({\underset{̲}{a}}_{1}^{-})}^{*}$	6: ${\underset{̲}{a}}_{1}^{-} \overset{PCM}{\to} {\underset{̲}{a}}_{1}^{+}$
7: ${\underset{̲}{\tilde{a}}}_{1}^{+} = {\underset{̲}{a}}_{1, (k)}^{+} - 2 μ \bar{a} {\underset{̲}{a}}_{1}^{+}$	7: ${\underset{̲}{\tilde{a}}}_{1}^{+} = {\underset{̲}{a}}_{1, (k)}^{+} - 2 μ \bar{a} {\underset{̲}{a}}_{1}^{+}$
8: ${\underset{̲}{a}}_{1, (k + 1)}^{+} = \underset{̲}{p} ({\underset{̲}{\tilde{a}}}_{1}^{+})$	8: ${\underset{̲}{a}}_{1, (k + 1)}^{+} = \underset{̲}{p} ({\underset{̲}{\tilde{a}}}_{1}^{+})$

Vector Operation	Physical Operation
1: ${\underset{̲}{a}}_{1}^{-} = S_{11} \cdot \underset{̲}{p} ({\underset{̲}{θ}}_{1, (k)}^{+})$	1: $\underset{̲}{p} ({\underset{̲}{θ}}_{1, (k)}^{+}) \overset{Backscatter}{\to} {\underset{̲}{a}}_{1}^{-}$
2: $\bar{a} = (\sum_{n = - N}^{N} \| a_{1, n}^{-} \| / \sqrt{M})$	2: $\bar{a} = (\sum_{n = - N}^{N} \| a_{1, n}^{-} \| / \sqrt{M})$
3: ${\underset{̲}{a}}_{1}^{-} \leftarrow \underset{̲}{p} ({\underset{̲}{a}}_{1}^{-})$	3: ${\underset{̲}{a}}_{1}^{-} \leftarrow \underset{̲}{p} ({\underset{̲}{a}}_{1}^{-})$
4: ${\underset{̲}{a}}_{1}^{+} = F \cdot {({\underset{̲}{a}}_{1}^{-})}^{*}$	4: ${\underset{̲}{a}}_{1}^{-} \overset{PCM}{\to} {\underset{̲}{a}}_{1}^{+}$
5: ${\underset{̲}{a}}_{1}^{-} = S_{11} \cdot {\underset{̲}{a}}_{1}^{+}$	5: ${\underset{̲}{a}}_{1}^{+} \overset{Backscatter}{\to} {\underset{̲}{a}}_{1}^{-}$
6: ${\underset{̲}{a}}_{1}^{+} = F \cdot {({\underset{̲}{a}}_{1}^{-})}^{*}$	6: ${\underset{̲}{a}}_{1}^{-} \overset{PCM}{\to} {\underset{̲}{a}}_{1}^{+}$
7: ${\underset{̲}{θ}}_{1, (k + 1)}^{+} = {\underset{̲}{θ}}_{1, (k)}^{+} - 2 \sqrt{M} μ \bar{a} Im [diag {\underset{̲}{p} (- {\underset{̲}{θ}}_{1, (k)}^{+})} \cdot {\underset{̲}{a}}_{1}^{+}]$

Vector Operation	Physical Operation
1: ${\underset{̲}{a}}_{1}^{-} = S_{11} \cdot {\underset{̲}{a}}_{1, (k)}^{+}$	1: ${\underset{̲}{a}}_{1, (k)}^{+} \overset{Backscatter}{\to} {\underset{̲}{a}}_{1}^{-}$
2: $\bar{a} = (\sum_{n = - N}^{N} \| a_{1, n}^{-} \| / \sqrt{M})$	2: $\bar{a} = (\sum_{n = - N}^{N} \| a_{1, n}^{-} \| / \sqrt{M})$
3: ${\underset{̲}{a}}_{1}^{-} \leftarrow \underset{̲}{p} ({\underset{̲}{a}}_{1}^{-})$	3: ${\underset{̲}{a}}_{1}^{-} \leftarrow \underset{̲}{p} ({\underset{̲}{a}}_{1}^{-})$
4: ${\underset{̲}{a}}_{1}^{+} = F \cdot {({\underset{̲}{a}}_{1}^{-})}^{*}$	4: ${\underset{̲}{a}}_{1}^{-} \overset{PCM}{\to} {\underset{̲}{a}}_{1}^{+}$
5: ${\underset{̲}{a}}_{1}^{-} = S_{11} \cdot {\underset{̲}{a}}_{1}^{+}$	5: ${\underset{̲}{a}}_{1}^{+} \overset{Backscatter}{\to} {\underset{̲}{a}}_{1}^{-}$
6: ${\underset{̲}{a}}_{1}^{+} = F \cdot {({\underset{̲}{a}}_{1}^{-})}^{*}$	6: ${\underset{̲}{a}}_{1}^{-} \overset{PCM}{\to} {\underset{̲}{a}}_{1}^{+}$
7: ${\underset{̲}{\tilde{a}}}_{1}^{+} = {\underset{̲}{a}}_{1, (k)}^{+} - 2 μ \bar{a} {\underset{̲}{a}}_{1}^{+}$	7: ${\underset{̲}{\tilde{a}}}_{1}^{+} = {\underset{̲}{a}}_{1, (k)}^{+} - 2 μ \bar{a} {\underset{̲}{a}}_{1}^{+}$
8: ${\underset{̲}{a}}_{1, (k + 1)}^{+} = \underset{̲}{p} ({\underset{̲}{\tilde{a}}}_{1}^{+})$	8: ${\underset{̲}{a}}_{1, (k + 1)}^{+} = \underset{̲}{p} ({\underset{̲}{\tilde{a}}}_{1}^{+})$

Vector Operation	Physical Operation
1: ${\underset{̲}{a}}_{1}^{-} = S_{11} \cdot \underset{̲}{p} ({\underset{̲}{θ}}_{1, (k)}^{+})$	1: $\underset{̲}{p} ({\underset{̲}{θ}}_{1, (k)}^{+}) \overset{Backscatter}{\to} {\underset{̲}{a}}_{1}^{-}$
2: $\bar{a} = (\sum_{n = - N}^{N} \| a_{1, n}^{-} \| / \sqrt{M})$	2: $\bar{a} = (\sum_{n = - N}^{N} \| a_{1, n}^{-} \| / \sqrt{M})$
3: ${\underset{̲}{a}}_{1}^{-} \leftarrow \underset{̲}{p} ({\underset{̲}{a}}_{1}^{-})$	3: ${\underset{̲}{a}}_{1}^{-} \leftarrow \underset{̲}{p} ({\underset{̲}{a}}_{1}^{-})$
4: ${\underset{̲}{a}}_{1}^{+} = F \cdot {({\underset{̲}{a}}_{1}^{-})}^{*}$	4: ${\underset{̲}{a}}_{1}^{-} \overset{PCM}{\to} {\underset{̲}{a}}_{1}^{+}$
5: ${\underset{̲}{a}}_{1}^{-} = S_{11} \cdot {\underset{̲}{a}}_{1}^{+}$	5: ${\underset{̲}{a}}_{1}^{+} \overset{Backscatter}{\to} {\underset{̲}{a}}_{1}^{-}$
6: ${\underset{̲}{a}}_{1}^{+} = F \cdot {({\underset{̲}{a}}_{1}^{-})}^{*}$	6: ${\underset{̲}{a}}_{1}^{-} \overset{PCM}{\to} {\underset{̲}{a}}_{1}^{+}$
7: ${\underset{̲}{θ}}_{1, (k + 1)}^{+} = {\underset{̲}{θ}}_{1, (k)}^{+} - 2 \sqrt{M} μ \bar{a} Im [diag {\underset{̲}{p} (- {\underset{̲}{θ}}_{1, (k)}^{+})} \cdot {\underset{̲}{a}}_{1}^{+}]$

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Equations (64)

Journal of the Optical Society of America A