Review on Methods for Laser Linewidth Measurement

Mingbin Cui; Jungang Huang; Xiulun Yang

doi:10.3788/LOP202158.0900005

Journals >Laser & Optoelectronics Progress >Volume 58 >Issue 9 >Page 0900005 > Article

Laser & Optoelectronics Progress
Vol. 58, Issue 9, 0900005 (2021)

Review on Methods for Laser Linewidth Measurement

Mingbin Cui¹, Jungang Huang², and Xiulun Yang^1、*

Author Affiliations

¹School of Information Science and Engineering, Shandong University, Qingdao , Shandong 266237, China

²Guangdong Raying Laser Technology Co. Ltd., Dongguan , Guangdong 523808, China

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DOI: 10.3788/LOP202158.0900005 Cite this Article Set citation alerts

Mingbin Cui, Jungang Huang, Xiulun Yang. Review on Methods for Laser Linewidth Measurement[J]. Laser & Optoelectronics Progress, 2021, 58(9): 0900005 Copy Citation Text

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Abstract

This paper reviews and summarizes various methods for linewidth measurement, introduces their development process, principles, and the required experimental equipment. According to the difference in principle, the methods for linewidth measurement are divided into two categories based on signal power spectra or phase noises. The limiting factors of linewidth measurement accuracy for these methods are analyzed and the precautions for the operation of these methods are summarized. In addition, the advantages and disadvantages of these methods as well as the corresponding detection ranges are comprehensively listed. According to the performance parameters and experimental conditions for different lasers, suitable linewidth detection methods are recommended. Finally, as for their practical applications, the methods for linewidth measurement of narrow-linewidth tunable lasers (widely used for coherent communications) are analyzed and discussed in detail.

Keywords

coherent communications linewidth detection narrow-linewidth tunable lasers phase noise signal power spectra spectroscopy

I (t) = R [P_{1} (t) + P_{2} (t) + 2 \sqrt[]{P_{1} (t) P_{2} (t)} c o s (2 π v_{b} + Δ φ)]

，（1）

I (t) = α E_{1} E_{2} c o s (2 π v_{b} + Δ φ)

，（2）

S_{b} (v) = \frac{δ v_{b}}{2 π [{(v - v_{b})}^{2} + {(\frac{δ v_{b}}{2})}^{2}]}

，（3）

Φ (n_{1}) = \sqrt[]{2 π Δ f Δ t} \sum_{0}^{n} X (n_{1})

，（4）

f (k_{1}) = \frac{1}{2 π Δ t} a r g [x_{i n} (k_{1}) x_{i n}^{*} (k_{1} - 1)]

，（5）

Δ v = \frac{2 π Δ t}{N_{w i n} - 1} \sum_{1}^{N_{w i n}} \{{[f (k_{1}) - u]}^{2}\}

，（6）

E (t) = E_{0} e x p [i (ω_{0} t + φ_{0})]

，（7）

E_{T} (t) = E (t) + E (t + τ_{d})

，（8）

R_{l} (τ) = 〈E (t) E^{*} (t) E (t + τ_{d}) E^{*} (t + τ_{d})〉

，（9）

S_{T} (v) = I_{0} [\frac{\frac{2}{τ}}{{(\frac{2}{τ})}^{2}} + v^{2}]

，（10）

E (t) = E_{0} c o s [ω (t) + φ (t)]

，（11）

E_{1} (t) = E_{A 1} c o s [ω (t - τ_{d}) + φ (t - τ_{d})]

，（12）

E_{2} (t) = α_{r} E_{A 2} c o s [(ω - f_{0}) t + φ (t)]

，（13）

E_{L} (t) = E_{1} (t) + E_{2} (t)

。（14）

I (t) = E_{L} (t) E_{L}^{*} (t) = E_{L} c o s [- f_{0} t + (ω - f_{0}) τ_{d} + φ (t) - φ (t - τ_{d})]

，（15）

S_{L} (ω, τ_{d}) = \frac{\frac{1}{2} E_{L}^{2}}{1 + {(ω \pm f_{0})}^{2} τ_{c}}

，（16）

J_{F} = [\begin{matrix} 0 & - 1 \\ - 1 & 0 \end{matrix}]

。（17）

\{\begin{array}{l} J_{s} = [\begin{matrix} \frac{\sqrt[]{2}}{2} & 0 \\ 0 & \frac{\sqrt[]{2}}{2} \end{matrix}] \\ J_{a} = [\begin{matrix} \frac{\sqrt[]{2}}{2} i & 0 \\ 0 & \frac{\sqrt[]{2}}{2} i \end{matrix}] \end{array}

。（18）

\{\begin{array}{l} J_{+} = [\begin{matrix} a & - b^{*} \\ b & a^{*} \end{matrix}] \\ J_{-} = [\begin{matrix} a & - b \\ b^{*} & a^{*} \end{matrix}] \end{array}

，（19）

\{\begin{array}{l} T_{1} = J_{a} J_{-} J_{F} J_{+} J_{s} = - \frac{i}{2} (a a^{*} + b b^{*}) [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] \\ T_{2} = J_{s} J_{-} J_{F} J_{+} J_{a} = - \frac{i}{2} (a a^{*} + b b^{*}) [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] \end{array}

。（20）

E_{I} (t) = E_{r} c o s [ω (t) + φ (t)]

，（21）

I_{o u t} (t) = I_{0} + \sum_{n_{2} = 1}^{\infty} I_{n_{2}} (t)

，（22）

I_{0} = E^{2} {(\frac{α_{t}}{2})}^{\frac{n_{2}}{2}} c o s [n_{2} ω_{s} t - n_{2} ω τ + φ (t - n_{2} τ_{d}) φ (t)]

，（23）

S_{n_{2}} (ω) = {(\frac{α_{t}}{2})}^{2} (\frac{Δ v}{π}) c o s [{(ω - n_{2} ω_{s})}^{2}]

。（24）

V (ν) = \int_{- \infty}^{\infty} G (ν) L (ν - ν^{'}) d ν

，（25）

G (ν) = \frac{2 \sqrt[]{l n 2}}{\sqrt[]{π} Δ ν_{G}} e x p [- 4 l n 2 {(ν - ν_{0})}^{2} / {(Δ ν_{G}^{})}^{2}]

，（26）

L (ν) = \frac{Δ ν_{L}}{2 π} \frac{1}{(ν - ν_{0})^{2} + Δ ν_{L}^{2} / 4}

，（27）

Δ ν_{V} = \frac{1}{2} [1.0692 Δ ν_{L} + \sqrt[]{0.866639 {(Δ ν_{L}^{})}^{2} + 4 {(Δ ν_{G}^{})}^{2}}]

，（28）

S (f, Δ f) = S_{1} S_{2} + S_{3}

，（29）

S_{1} = \frac{P_{0}^{2}}{4 π} \times \frac{Δ f}{Δ f^{2} + {(f - f_{0})}^{2}}

，（30）

S_{2} = 1 - e x p (- 2 π Δ f τ_{d}) \{c o s [2 π (f - f_{0}) τ_{d}] + 2 π Δ f τ_{d} s i n c [2 π (f - f_{0}) τ_{d}]\}

，（31）

S_{3} = \frac{π P_{0}^{2}}{2} \times e x p (- 2 π Δ f τ_{d}) δ (f - f_{0})

，（32）

Δ S = \frac{12.82 Δ f_{e s t} + 38470}{Δ f_{e s t} + 1685}

。（33）

\begin{array}{l} Δ S (Δ f) = 10 l g S_{p e a k} - 10 l g S_{t r o u g h} = 10 l g \frac{S (f_{0} - \frac{2 l_{1} - 1}{2}, Δ f)}{S (f_{0} - m \frac{c}{n l}, Δ f)} = \\ 10 l g \frac{[1 + {(\frac{2 c}{n f l})}^{2}] [1 + e x p (- 2 π \frac{n Δ f_{P} l}{c})]}{[1 + {(\frac{3 c}{n f l})}^{2}] [1 - e x p (- 2 π \frac{n Δ f_{P} l}{c})]}, l_{1} = 2, m = 2, \end{array}

（34）

\begin{array}{l} S (f) = \frac{P_{0}^{2}}{4 π} \times \frac{Δ ν}{Δ ν^{2} + {(f - f_{F S})}^{2}} \times \{1 - e x p (- \frac{2 π n l Δ ν}{c}) \times \\ [c o s \frac{2 π n l (f - f_{F S})}{c} + Δ ν \frac{s i n 2 π (f - f_{F S}) \frac{n l}{c}}{f - f_{F S}}]\} + \\ \frac{π {P_{0}}^{2}}{2} \times e x p (- \frac{2 π n l Δ ν}{c}) δ (f - f_{F S}), \end{array}

（35）

\frac{c}{n l Δ ν} ≫ 1

。（36）

Δ f_{m_{1}} = \frac{(m_{1} + 1) c}{n l}

，（37）

\begin{array}{l} S (f) \approx \frac{P_{0}^{2}}{4 π} \times \frac{Δ ν}{Δ ν^{2} + {(f - f_{F S})}^{2}} \times \{1 - e x p (- \frac{2 π n l Δ ν}{c}) \times \\ \{c o s \frac{2 π n l (f - f_{F S})}{c} + Δ ν \times \frac{s i n [2 π (f - f_{F S}) \frac{n l}{c}]}{f - f_{F S}}\}\} 。 \end{array}

（38）

Δ S \approx 10 l g \frac{S (f_{F S} + \frac{3 c}{2 n l})}{S (f_{F S} + \frac{c}{2 n l})} \approx 10 l g \frac{16 c}{9 π n l Δ ν}

。（39）

Δ ν = \frac{16 Δ f_{0}}{9 π} 10^{- \frac{Δ s - 0.204}{10}}

。（40）

S_{φ_{b}} (f) = \int_{- \infty}^{\infty} R_{φ_{b}} (τ) e x p (- i 2 π f τ) d τ

。（41）

S_{v} (f) = f^{2} S_{φ_{b}} (f)

，（42）

Δ v = {(8 l n 2 A)}^{(1 / 2)}

，（43）

A = \int_{\frac{1}{T_{0}}}^{\infty} H [S_{v} (f) - \frac{8 l n 2}{{(π)}^{2}}] S_{v} (f) d f

，（44）

I = A + B c o s [φ_{1} (t)] = A + B c o s (\frac{2 π n Δ l v_{L}}{c})

，（45）

φ_{1} (t) = Δ φ_{0} + \frac{2 π n Δ l v_{L}}{c} (\frac{δ n}{n} + \frac{δ l}{l} + \frac{δ v_{L}}{v_{L}})

，（46）

δ_{φ} (t) = \frac{2 π n Δ l δ v}{c} = \frac{n Δ l}{c} δ_{φ} v (t)

，（47）

S_{v} (f) = {(\frac{c}{n Δ l})}^{2} S_{φ} (f)

。（48）

i^{2} = \frac{4 k_{B} T Δ f_{m}}{R_{L}}

，（49）

σ_{s}^{2} = 2 q (I_{P} + I_{d}) Δ f_{m}

，（50）

\begin{array}{l} S_{L} (ω_{B}, τ_{d}) = \frac{\frac{1}{2} E_{L}^{2}}{1 + {(ω_{B} \pm f_{0})}^{2} τ_{c}} [1 - e x p (- \frac{τ_{d}}{τ_{c}}) c o s (ω_{B} \pm f_{0}) τ_{d} + \frac{s i n (ω_{B} \pm f_{0}) τ_{d}}{(ω_{B} \pm f_{0}) τ_{c}}] + \\ \frac{1}{2} p_{0}^{2} π e x p (- \frac{τ_{d}}{τ_{c}}) δ (ω_{B} \pm f_{0}), \end{array}

（51）

S_{L} (ω_{B}, τ_{d}) = \frac{\frac{1}{2} E_{L}^{2}}{1 + (ω_{B} \pm ω_{0})^{2} τ_{c}}

。（52）

Mingbin Cui, Jungang Huang, Xiulun Yang. Review on Methods for Laser Linewidth Measurement[J]. Laser & Optoelectronics Progress, 2021, 58(9): 0900005

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