Researching | Long-Period Fiber Gratings

Laser & Optoelectronics Progress, Volume. 60, Issue 9, 0900001(2023)

Long-Period Fiber Gratings

Yunqi Liu^1、*, Chen Jiang¹, Zuyao Liu^1,2, and Xinyi Zhao^1,3

Author Affiliations

¹Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Ministry of Education, School of Communication and Information Engineering, Shanghai University, Shanghai 200444, China

²Key Laboratory of Optoelectronic and Telecommunication of Jiangxi Province, School of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, Jiangxi, China

³Henan Key Laboratory of Optoelectronic Sensing Integrated Application, College of Electronic and Electrical Engineering, Henan Normal University, Xinxiang 453007, Henan, China

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References (183)

Paper Information

Equations(35)

λ = (n_{e f f}^{01} - n_{e f f}^{n m}) Λ / N

，（1）

\{\begin{array}{l} \nabla \times E = - \frac{\partial B}{\partial t} \\ \nabla \times H = \frac{\partial D}{\partial t} \\ \nabla \cdot D = 0 \\ \nabla \cdot B = 0 \end{array}

，（2）

\{\begin{array}{l} \nabla^{2} E + k^{2} E = 0 \\ \nabla^{2} H + k^{2} H = 0 \end{array}

，（3）

\{\begin{array}{l} E (x, y, z) = e (x, y) e x p (- i β z) \\ H (x, y, z) = h (x, y) e x p (- i β z) \end{array}

。（4）

\{\begin{array}{l} [\nabla_{t}^{2} + (k^{2} - β^{2})] e = 0 \\ [\nabla_{t}^{2} + (k^{2} - β^{2})] h = 0 \end{array}

，（5）

[\begin{matrix} E \\ H \end{matrix}] (x, y, z) = \sum_{i} [\begin{matrix} a_{i} e_{i} \\ b_{i} h_{i} \end{matrix}] (x, y) e x p (- i β_{i} z)

，（6）

\{\begin{array}{l} H E_{1 n} \leftrightarrow L P_{0 n} \\ H E_{2 n} + T E_{0 n} \leftrightarrow L P_{1 n} \\ H E_{2 n} + T M_{0 n} \leftrightarrow L P_{1 n} \\ H E_{m + 1, n} + E H_{m - 1, n} \leftrightarrow L P_{m n} (m \geq 2, n \geq 1) \end{array}

。（7）

h_{x} (r, ϕ) = - \frac{ω ε}{β} \{\begin{array}{l} a_{0} J_{m} (\frac{U}{a_{1}} r) e x p (j m ϕ), (0 \leq r \leq a_{1}) \\ b_{0} K_{m} (\frac{W}{a_{1}} r) e x p (j m ϕ), (r > a_{1}) \end{array}

，（8）

e_{z} (r, ϕ) = \{\begin{array}{l} a_{0} \frac{j}{β} e x p (j m ϕ) [\frac{U s i n ϕ}{a_{1}} J_{m}^{'} (\frac{U r}{a_{1}}) + j \frac{m c o s ϕ}{r} J_{m} (\frac{U r}{a_{1}})], (0 \leq r \leq a_{1}) \\ b_{0} \frac{j}{β} e x p (j m ϕ) [\frac{W s i n ϕ}{a_{1}} K_{m}^{'} (\frac{W r}{a_{1}}) + j \frac{m c o s ϕ}{r} K_{m} (\frac{W r}{a_{1}})], (r > a_{1}) \end{array}

，（9）

h_{z} (r, ϕ) = \{\begin{array}{l} a_{0} \frac{- j}{ω μ_{0}} e x p (j m ϕ) [\frac{U c o s ϕ}{a_{1}} J_{m}^{'} (\frac{U r}{a_{1}}) - j \frac{m s i n ϕ}{r} J_{m} (\frac{U r}{a_{1}})], (0 \leq r \leq a_{1}) \\ b_{0} \frac{- j}{ω μ_{0}} e x p (j m ϕ) [\frac{W c o s ϕ}{a_{1}} K_{m}^{'} (\frac{W r}{a_{1}}) - j \frac{m s i n ϕ}{r} K_{m} (\frac{W r}{a_{1}})], (r > a_{1}) \end{array}

，（10）

n_{c o} (z) = n_{1} + δ n (z)

。（11）

δ n (z) = Δ n r e c t (z / Λ)

。（12）

\begin{matrix} r e c t (z / Λ) = \{\begin{matrix} 1, & n Λ \leq z \leq n Λ + p Λ \\ 0, & n Λ + p Λ < z \leq (n + 1) Λ \end{matrix}, \\ (n = 0, 1, 2, \cdot \cdot \cdot, N; 0 < p < 1) \end{matrix}

，（13）

\begin{array}{l} \frac{d A^{c o}}{d z} = i κ_{11 - 11}^{c o - c o} A^{c o} + \frac{i}{2} \sum_{v} κ_{1 v - 11}^{c l - c o} A_{v}^{c l} \times \\ \sum_{m = 1}^{+ \infty} [A_{m} e x p (- i 2 δ_{1 v - 11}^{c l - c o} z)] \\ \sum_{v} \{\frac{d A_{v}^{c l}}{d z} + i κ_{1 v - 11}^{c l - c o} A^{c o} \times \sum_{m = 1}^{+ \infty} [A_{m} e x p (- i 2 δ_{1 v - 11}^{c l - c o} z)]\}, \end{array}

（14）

k_{11 - 11}^{c o - c o} (z) = \frac{ω ε_{0} n_{1}^{2} σ (z)}{2} \int_{0}^{2 π} d ϕ \int_{0}^{a 1} r d r ({|E_{r}^{c o}|}^{2} + {|E_{ϕ}^{c o}|}^{2})

，（15）

k_{1 v - 11}^{c l - c o} (z) = \frac{ω ε_{0} n_{1}^{2} σ (z)}{2} \int_{0}^{2 π} d ϕ \int_{0}^{a 1} r d r (E_{r}^{c l} E_{r}^{c o^{*}} + E_{ϕ}^{c l} E_{ϕ}^{c o^{*}})

，（16）

δ_{1 v - 11}^{c l - c o} = \frac{1}{2} (β_{11}^{c o} - β_{1 v}^{c l} - \frac{2 π}{Λ})

。（17）

T (λ) = |A {(L)|}^{2} = c o s^{2} (\sqrt[]{{\hat{σ}}^{2} + κ^{2}} L) + \frac{1}{1 + κ^{2} / {\hat{σ}}^{2}} s i n^{2} (\sqrt[]{{\hat{σ}}^{2} + κ^{2}} L)

，（18）

λ = (n_{e f f}^{01} - n_{e f f, c l}^{n m}) Λ

，（19）

d λ / d T = λ ∙ γ ∙ (α + Γ_{t e m p})

，（20）

d λ / d ε = λ ∙ γ ∙ (1 + Γ_{s t r a i n})

，（21）

d λ / d n_{s u r} = λ ∙ γ ∙ Γ_{s u r}

，（22）

γ = (d λ / d Λ) / (n_{e f f}^{01} - n_{e f f, c l}^{n m})

，（23）

Γ_{t e m p} = \frac{ξ_{c o} n_{e f f}^{01} - ξ_{c l} n_{e f f, c l}^{n m}}{n_{c o}^{e f f} - n_{e f f, c l}^{n m}}

，（24）

Γ_{s t r a i n} = \frac{η_{c o} n_{e f f}^{01} - η_{c l} n_{e f f, c l}^{n m}}{n_{e f f}^{01} - n_{e f f, c l}^{n m}}

，（25）

Γ_{s u r} = - \frac{u_{m}^{2} λ_{r e s}^{3} n_{s u r}}{8 π r_{c l}^{3} n_{c l} (n_{e f f}^{01} - n_{e f f, c l}^{n m}) (n_{c l}^{2} - n_{s u r}^{3})^{3 / 2}}

，（26）

Δ λ^{(m)} = (n_{e f f}^{01} - n_{e f f, c l}^{n m}) Δ Λ + Δ n_{Δ} Λ

，（27）

Δ n = c σ

，（28）

Δ λ^{(m)} \approx - δ n_{e f f, c l}^{n m} \cdot Λ

。（29）

\dot{E} = G {\dot{R}}_{T}

，（30）

Δ λ^{(m)} = (n_{e f f}^{01} - n_{e f f, c l}^{n m}) Δ Λ - δ n_{e f f, c l}^{n m} \cdot Λ

，（31）

T_{o} = T_{c l}^{2} - 2 T_{c o} T_{c l} c o s (φ + δ)

，（32）

T_{c l} = 1 - T_{c o}

，（33）

φ = \frac{2 π Δ n_{e f f, c l}^{n m} L}{λ^{(m)}}

，（34）

R_{F S R} = \frac{λ^{2}}{Δ n_{e f f, c l}^{n m} L}

。（35）

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Yunqi Liu, Chen Jiang, Zuyao Liu, Xinyi Zhao. Long-Period Fiber Gratings[J]. Laser & Optoelectronics Progress, 2023, 60(9): 0900001

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Paper Information

Category: Reviews

Received: Feb. 14, 2022

Accepted: Jun. 30, 2022

Published Online: Apr. 24, 2023

The Author Email: Liu Yunqi (yqliu@shu.edu.cn)

DOI:10.3788/LOP220735

Topics

laser devices and laser physics

Lasers and Laser Optics

laser manufacturing

Instrumentation, Measurement and Metrology