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Photonics Research
Vol. 10, Issue 10, 2293 (2022)

Particle manipulation behind a turbid medium based on the intensity transmission matrix

Kaige Liu^1,2,†, Hengkang Zhang^3,†, Shanshan Du^1,2, Zeqi Liu^1,2..., Bin Zhang^4,5,*, Xing Fu^1,2,6,* and Qiang Liu^1,2,7,*|Show fewer author(s)

Author Affiliations

¹Key Laboratory of Photonics Control Technology, Ministry of Education, Tsinghua University, Beijing 100084, China

²Department of Precision Instrument, State Key Laboratory of Precision Measurement Technology and Instruments, Tsinghua University, Beijing 100084, China

³Beijing Institute of Control Engineering, Beijing 100190, China

⁴Beijing Institute of Electronic System Engineering, Beijing 100854, China

⁵e-mail: zhangbin1931@126.com

⁶e-mail: fuxing@mail.tsinghua.edu.cn

⁷e-mail: qiangliu@mail.tsinghua.edu.cn

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DOI: 10.1364/PRJ.461172 Cite this Article Set citation alerts

Kaige Liu, Hengkang Zhang, Shanshan Du, Zeqi Liu, Bin Zhang, Xing Fu, Qiang Liu, "Particle manipulation behind a turbid medium based on the intensity transmission matrix," Photonics Res. 10, 2293 (2022) Copy Citation Text

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Abstract

Although optical tweezers can manipulate tiny particles, the distortion caused by the scattering medium restricts their application. Wavefront-shaping techniques such as the transmission matrix (TM) method are powerful tools to achieve light focusing behind the scattering medium. In this paper, we propose a method to focus light through a scattering medium in a large area based on the intensity transmission matrix (ITM). Only relying on the intensity distribution, we can calculate the ITM with the number of measurements equal to that of the control segments. Free of the diffraction limit, our method guarantees high energy usage of the light field. Based on this method, we have implemented particle manipulation with a high degree of freedom on single and multiple particles. In addition, the manipulation range is enlarged more than 20 times (compared to the memory effect) to 200 μm.

ε^{in} = H_{0 - 1} \cdot α,

(1)

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ε^{tar} = E^{out} \cdot α,

(2)

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TM \cdot H_{0 - 1} = E^{out},

(3)

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TM \cdot H_{0 - 1} \cdot α = E^{out} \cdot α,

(4)

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TM \cdot ε^{in} = ε^{tar},

(5)

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ε^{in} = {TM}^{†} \cdot ε^{tar},

(6)

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TM = E^{out} \cdot H_{0 - 1}^{- 1} = E^{out} \cdot (\frac{2}{N} \cdot H - (\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 0 \end{matrix})) = \frac{2}{N} \cdot E^{out} \cdot H - (\begin{matrix} ε_{11}^{out} & 0 & \dots & 0 \\ ε_{12}^{out} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ \\ ε_{1 M}^{out} & 0 & \dots & 0 \end{matrix}),

(7)

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ITM = \frac{2}{N} \cdot | E^{out} | \cdot H .

(8)

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ε^{in} = \frac{2}{N} \cdot (\begin{matrix} | e_{m 1} | & | e_{m 2} | & \dots & | e_{m N} | \end{matrix}) \cdot H,

(9)

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(\begin{matrix} t_{m 1} & t_{m 2} & \dots & t_{m N} \end{matrix}) = \frac{2}{N} \cdot (\begin{matrix} | e_{m 1} | & | e_{m 2} | & \dots & | e_{m N} | \end{matrix}) \cdot H,

(10)

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t_{m j} = \frac{2}{N} \cdot (\begin{matrix} | e_{m 1} | & | e_{m 2} | & \dots & | e_{m N} | \end{matrix}) \cdot h_{j} = \frac{2}{N} \sum_{i = 1}^{N} h_{i j} | e_{m i} | = \frac{2}{N} \sum_{h_{i j} = 1} | e_{m i} | - \frac{2}{N} \sum_{h_{i j} = - 1} | e_{m i} |,

(11)

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t_{m j} = ⟨ | a + a_{1} + a_{j} + i (b + b_{1} + b_{j}) | ⟩ - ⟨ | a + a_{1} + i (b + b_{1}) | ⟩,

(12)

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PBR = \frac{I^{foc}}{⟨ I^{bg} ⟩},

(13)

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⟨ | a + a_{1} + a_{j} + i (b + b_{1} + b_{j}) | ⟩ > ⟨ | a + a_{1} + i (b + b_{1}) | ⟩,

(14)

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⟨ {| a + a_{1} + a_{j} + i (b + b_{1} + b_{j}) |}^{2} ⟩ > ⟨ {| a + a_{1} + i (b + b_{1}) |}^{2} ⟩ .

(15)

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p_{r, i} (r, i) = \frac{1}{2 π σ^{2}} \exp (- \frac{r^{2} + i^{2}}{2 σ^{2}}),

(16)

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⟨ {| a + a_{1} + i (b + b_{1}) |}^{2} ⟩ = \iint_{\infty} \frac{{(a + a_{1})}^{2} + {(b + b_{1})}^{2}}{2 π σ^{2}} \exp (- \frac{a^{2} + b^{2}}{2 σ^{2}}) d a d b = \iint_{\infty} \frac{a^{2} + b^{2}}{2 π σ^{2}} \exp (- \frac{a^{2} + b^{2}}{2 σ^{2}}) d a d b + \iint_{\infty} \frac{a_{1}^{2} + b_{1}^{2}}{2 π σ^{2}} \exp (- \frac{a^{2} + b^{2}}{2 σ^{2}}) d a d b + \iint_{\infty} \frac{2 a a_{1} + 2 b b_{1}}{2 π σ^{2}} \exp (- \frac{a^{2} + b^{2}}{2 σ^{2}}) d a d b = 2 σ^{2} + a_{1}^{2} + b_{1}^{2},

(17)

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⟨ {| a + a_{1} + a_{j} + i (b + b_{1} + b_{j}) |}^{2} ⟩ = 2 σ^{2} + {(a_{1} + a_{j})}^{2} + {(b_{1} + b_{j})}^{2} .

(18)

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{(a_{1} + a_{j})}^{2} + {(b_{1} + b_{j})}^{2} > a_{1}^{2} + b_{1}^{2} .

(19)

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| t_{m 1} | = \sqrt{\frac{π}{2}} σ, φ_{m 1} = 0,

(20)

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a_{1} = \sqrt{\frac{π}{2}} σ, b_{1} = 0,

(21)

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{(a_{j} + \sqrt{\frac{π}{2}} σ)}^{2} + b_{j}^{2} > \frac{π}{2} σ^{2} .

(22)

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p_{on} = \iint_{{(x + \frac{\sqrt{π}}{2} σ)}^{2} + y^{2} > \frac{π}{2} σ^{2}} \frac{1}{2 π σ^{2}} \exp (- \frac{x^{2} + y^{2}}{2 σ^{2}}) d x d y \approx 0.5809,

(23)

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⟨ | t_{m j} | ⟩ = \frac{1}{p_{on}} \iint_{{(x + \frac{\sqrt{π}}{2} σ)}^{2} + y^{2} > \frac{π}{2} σ^{2}} \frac{\sqrt{x^{2} + y^{2}}}{2 π σ^{2}} \exp (- \frac{x^{2} + y^{2}}{2 σ^{2}}) d x d y \approx 1.5252 σ,

(24)

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⟨ {| t_{m j} |}^{2} ⟩ = \frac{1}{p_{on}} \iint_{{(x + \frac{\sqrt{π}}{2} σ)}^{2} + y^{2} > \frac{π}{2} σ^{2}} \frac{x^{2} + y^{2}}{2 π σ^{2}} \exp (- \frac{x^{2} + y^{2}}{2 σ^{2}}) d x d y \approx 2.7297 σ^{2},

(25)

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⟨ \exp (i φ_{m j}) ⟩ = \frac{1}{p_{on}} \iint_{{(x + \frac{\sqrt{π}}{2} σ)}^{2} + y^{2} > \frac{π}{2} σ^{2}} \frac{x + i y}{2 π σ^{2} \sqrt{x^{2} + y^{2}}} \exp (- \frac{x^{2} + y^{2}}{2 σ^{2}}) d x d y \approx 0.3828,

(26)

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I^{foc} = {| \sum_{n = 1}^{N_{on}} A t_{m n} |}^{2} = \sum_{n = 1}^{N_{on}} A^{2} {| t_{m n} |}^{2} + \sum_{n \neq h}^{N_{on}} A^{2} t_{m n}^{*} t_{m h},

(27)

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⟨ I^{foc} ⟩ = A^{2} \sum_{j = 1}^{N_{on}} ⟨ {| t_{m j} |}^{2} ⟩ + A^{2} \sum_{j \neq k}^{N_{on}} ⟨ | t_{m j} | ⟩ ⟨ | t_{m k} | ⟩ ⟨ \exp (i φ_{m j}) ⟩ ⟨ \exp (i φ_{m k}) ⟩ = A^{2} \times 0.5809 N [2.7297 σ^{2} + (0.5809 N - 1) \times {1.5252}^{2} σ^{2} \times {0.3828}^{2}] .

(28)

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⟨ I^{bg} ⟩ = A^{2} \sum_{j = 1}^{N_{on}} ⟨ {| t_{m j} |}^{2} ⟩ = A^{2} \times 0.5809 N \times 2.7297 σ^{2},

(29)

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PBR = \frac{⟨ I^{foc} ⟩}{⟨ I^{bg} ⟩} = \frac{A^{2} \times 0.5809 N [2.7297 σ^{2} + (0.5809 N - 1) \times {1.5252}^{2} σ^{2} \times {0.3828}^{2}]}{A^{2} \times 0.5809 N \times 2.7297 σ^{2}} = 1 + 0.1249 (0.5809 N - 1) \approx 0.0726 N .

(30)

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PBR = 1 + \frac{1}{π} (\frac{N}{2} - 1) \approx 0.1592 N .

(31)

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I (θ, L) = k_{0} θ L / \sinh (k_{0} θ L),

(32)

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Kaige Liu, Hengkang Zhang, Shanshan Du, Zeqi Liu, Bin Zhang, Xing Fu, Qiang Liu, "Particle manipulation behind a turbid medium based on the intensity transmission matrix," Photonics Res. 10, 2293 (2022)

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