20 years of developments in optical frequency comb technology and applications
20 年的光頻梳技術和應用發展歷程

single mode-locked laser (MLL), enabling vast simplification to precision optical measurement and rapid progress and development into optical atomic standards.
單鎖模雷射器 （MLL），極大地簡化了精密光學測量，並迅速發展為光學原子標準。

OFCs were developed by drawing on single-frequency laser stabilization techniques and applying them to mode-locked (pulsed) laser systems. The result was a system that could synthesize ${10}^5-{10}^6$${10}^5-{10}^6$10^(5)-10^(6)10^{5}-10^{6} harmonically related optical modes from either an electronic or optical reference with a fidelity better than 1 part in ${10}^{18}$${10}^{18}$10^(18)10^{18}. More importantly, OFCs enabled the direct conversion of optical-to-microwave frequencies and vice versa, enabling the extraction of microwave timing signals from optical atomic clocks. Beyond their application to precision optical metrology, OFCs were quickly recognized for their versatility as high-fidelity optical frequency converters and as sources of precisely timed ultra-short pulses. More broadly, by taking advantage of the nonlinearities possible with the ultra-short pulses, OFCs enable synthesis over broad spectral regions including the near-infrared, the visible domain and as far as the extreme ultraviolet (XUV). Generation of difference frequencies within the optical spectrum also allows for high-fidelity frequency transfer to the mid-infrared, terahertz, and microwave domains. OFCs quickly found application to a multitude of diverse optical, atomic, molecular, and solid-state systems, including X-ray and attosecond pulse generation ${}^4$${}^4$^(4){ }^{4}, coherent control in field-dependent processes ${}^{5,6}$${}^{5,6}$^(5,6){ }^{5,6}, molecular fingerprinting ${}^7$${}^7$^(7){ }^{7}, trace gas sensing in the oil and gas industry ${}^8$${}^8$^(8){ }^{8}, tests of fundamental physics with atomic clocks ${}^9$${}^9$^(9){ }^{9}, calibration of atomic spectrographs ${}^{10}$${}^{10}$^(10){ }^{10}, precision time/frequency transfer over fiber and free-space ${}^{11}$${}^{11}$^(11){ }^{11}, arbitrary waveform measurements for optical communication ${}^{12}$${}^{12}$^(12){ }^{12}, and precision ranging ${}^{13}$${}^{13}$^(13){ }^{13}. To support this broad application space, OFCs have seen rapid changes in laser development to enable coverage at different spectral regions, varying frequency resolutions, and to enable the development of systems that offer lower size, weight and power (SWAP) ${}^{14-17}$${}^{14-17}$^(14-17){ }^{14-17}.
OFC 是通過利用單頻激光穩定技術並將其應用於鎖模（脈衝）激光系統而開發的。結果是一個系統，它可以從電子或光學參考合成 ${10}^5-{10}^6$${10}^5-{10}^6$10^(5)-10^(6)10^{5}-10^{6} 諧波相關的光學模式，保真度優於 中的 1 個 ${10}^{18}$${10}^{18}$10^(18)10^{18} 部分。更重要的是，OFC 實現了光頻率到微波頻率的直接轉換，反之亦然，從而能夠從光學原子鐘中提取微波定時信號。除了應用於精密光學計量之外，OFC 還因其作為高保真光學轉換器和精確定時超短脈衝源的多功能性而很快得到認可。更廣泛地說，通過利用超短脈衝可能的非線性，OFC 可以在寬光譜區域進行合成，包括近紅外、可見光域和遠至極紫外 （XUV）。在光譜內產生不同的頻率還允許將高保真頻率傳輸到中紅外、太赫茲和微波域。OFC 很快被應用於多種不同的光學、原子、分子和固態系統，包括 X 射線和阿秒脈衝產生 ${}^4$${}^4$^(4){ }^{4} 、場相關過程中 ${}^{5,6}$${}^{5,6}$^(5,6){ }^{5,6} 的相幹控制、分子指紋、 ${}^7$${}^7$^(7){ }^{7} 石油和天然氣工業 ${}^8$${}^8$^(8){ }^{8} 中的痕量氣體感測、原子鐘的基礎物理測試 ${}^9$${}^9$^(9){ }^{9} 、原子光譜儀的校準 ${}^{10}$${}^{10}$^(10){ }^{10} 、光纖和自由空間 ${}^{11}$${}^{11}$^(11){ }^{11} 的精確時間/頻率傳輸、光通信 ${}^{12}$${}^{12}$^(12){ }^{12} 的任意波形測量 和精確測距 ${}^{13}$${}^{13}$^(13){ }^{13} 。 為了支援這一廣泛的應用領域，OFC 見證了雷射器發展的快速變化，以實現不同光譜區域的覆蓋、不同的頻率解析度，並支持開發具有更小尺寸、重量和功率 （SWAP） ${}^{14-17}$${}^{14-17}$^(14-17){ }^{14-17} 的系統。

The remarkable technical capabilities outlined above gained John “Jan” Hall and Theodor Hänsch recognition by the Nobel Committee in 2005 for their life long contributions to the field of precision optical frequency metrology ${}^{18,19}$${}^{18,19}$^(18,19){ }^{18,19}, as well as for their technical vision and expertise that resulted in the realization of the ${\mathrm{OFC}}^{20}$${\mathrm{OFC}}^{20}$OFC^(20)\mathrm{OFC}^{20}. A quick search on Google Scholar for publications that contain the exact phrase, “OFC,” returns more than 14,000 publications on the topic in the last 20 years. In writing this review we hope to provide a broad historical overview of the origins of OFCs, explain how they work and how they are applied in different contexts. More importantly, we hope to motivate the reader as to why frequency combs are such a powerful tool in the context of precision laboratory experiments, and explain how they are moving beyond precision metrology and toward commerical applications.
上述卓越的技術能力使 John “Jan” Hall 和 Theodor Hänsch 於 2005 年獲得諾貝爾委員會的認可 ${}^{18,19}$${}^{18,19}$^(18,19){ }^{18,19} ，以表彰他們對精密光學頻率計量領域的終身貢獻，以及他們的技術遠見和專業知識，從而實現了 ${\mathrm{OFC}}^{20}$${\mathrm{OFC}}^{20}$OFC^(20)\mathrm{OFC}^{20} .在Google學術搜索中快速搜索包含確切短語「OFC」的出版物，會返回過去20年中有關該主題的14,000多篇出版物。在撰寫這篇評論時，我們希望對 OFC 的起源提供廣泛的歷史概述，解釋它們的工作原理以及它們如何在不同的環境中應用。更重要的是，我們希望激勵讀者瞭解為什麼頻率梳在精密實驗室實驗中是如此強大的工具，並解釋它們如何超越精密計量並走向商業應用。

The traditional answer is that an OFC is a phase-stabilized MLL. While different generation methods have been developed over the past 20 years, MLLs were the original OFC sources. Because of their historical relevance and operational simplicity, we use them here as a starting point to explain the basics of OFC generation.
傳統的答案是 OFC 是相位穩定的 MLL。雖然在過去 20 年中開發了不同的生成方法，但 MLL 是最初的 OFC 來源。由於它們的歷史相關性和操作簡單性，我們在這裡使用它們作為解釋 OFC 生成基礎知識的起點。

The utility of MLLs within the context of optical metrology was recognized as early as the late 1980s. The optical pulses from MLLs result from the coherent addition of 100 s of thousands to millions of resonant longitudinal optical cavity modes, spanning up to 100 nm in the optical domain. While the broad optical bandwidth is immediately attractive for spectroscopic applications, the mode-locked optical spectrum has unique properties that are beneficial for precision optical metrology: (1) all the optical modes are harmonically related (perfectly equidistant in frequency) and (2) all-optical modes are phase coherent with one
早在 1980 年代後期，MLL 在光學計量領域的效用就得到了認可。來自 MLL 的光脈衝是相幹添加 100 秒的數千到數百萬個諧振縱向光腔模式的結果，在光域中跨越高達 100 nm。雖然寬光頻寬對光譜應用具有直接吸引力，但鎖模光譜具有有利於精密光學計量的獨特特性：（1） 所有光學模式都是諧波相關的（頻率完全等距），以及 （2） 所有光學模式都與一個相位相幹
another (share a common phase evolution). The consequence of this is that the evolution of the electric field, and consequently the phase and frequency dynamics of every optical mode in the laser OFC spectrum is deterministic. As a result, knowledge about the absolute frequency of one mode can be used to determine the absolute frequency of any other mode.
另一個（共用共同的相演化）。其結果是，電場的演變，以及鐳射 OFC 光譜中每種光學模式的相位和頻率動力學都是確定性的。因此，有關一種模式的絕對頻率的資訊可用於確定任何其他模式的絕對頻率。

The comb equation. The deterministic behavior of the OFC spectrum described above is most succinctly described by the comb equation. To understand the comb equation, we will begin by exploring the relatively simple mathematics that describe the optical field output from a MLL (see Fig. 1). The optical field of the laser pulse train can be described by a carrier frequency, $v_{\mathrm{c}}=$$v_{\mathrm{c}}=$v_(c)=v_{\mathrm{c}}= ${\omega }_{\mathrm{C}}/(2\pi )$${\omega }_{\mathrm{C}}/(2\pi )$omega_(C)//(2pi)\omega_{\mathrm{C}} /(2 \pi), that is modulated by a periodic pulse envelope, $A(t)$$A(t)$A(t)A(t). Typically, the time between optical pulses range between 1 and 10 ns . Due to the pulse periodicity, the optical field can also be described as a periodic Fourier series of optical modes, $v_{\mathrm{N}}={\omega }_{\mathrm{N}}/$$v_{\mathrm{N}}={\omega }_{\mathrm{N}}/$v_(N)=omega_(N)//v_{\mathrm{N}}=\omega_{\mathrm{N}} / $(2\pi )$$(2\pi )$(2pi)(2 \pi), with Fourier amplitude components, $A_{\mathrm{N}}$$A_{\mathrm{N}}$A_(N)A_{\mathrm{N}}, and mode number, $N$$N$NN, such that
梳狀方程。上述 OFC 頻譜的確定性行為由梳狀方程最簡潔地描述。為了理解梳狀方程，我們將首先探索描述 MLL 光場輸出的相對簡單的數學運算（見圖 1）。激光脈衝序列的光場可以用載波頻率 來描述， $v_{\mathrm{c}}=$$v_{\mathrm{c}}=$v_(c)=v_{\mathrm{c}}= ${\omega }_{\mathrm{C}}/(2\pi )$${\omega }_{\mathrm{C}}/(2\pi )$omega_(C)//(2pi)\omega_{\mathrm{C}} /(2 \pi) 該載波頻率由週期性脈衝包絡 $A(t)$$A(t)$A(t)A(t) 調製。通常，光脈衝之間的時間在 1 到 10 ns 之間。由於脈衝週期性，光場也可以描述為具有傅里葉振幅分量 $A_{\mathrm{N}}$$A_{\mathrm{N}}$A_(N)A_{\mathrm{N}} 和模數 的週期性傅里葉級數 $N$$N$NN $v_{\mathrm{N}}={\omega }_{\mathrm{N}}/$$v_{\mathrm{N}}={\omega }_{\mathrm{N}}/$v_(N)=omega_(N)//v_{\mathrm{N}}=\omega_{\mathrm{N}} / $(2\pi )$$(2\pi )$(2pi)(2 \pi) ，使得

Because $v_c$$v_c$v_(c)v_{c} is not necessarily an exact multiple of the mode spacing, $f_r$$f_r$f_(r)f_{r}, the individual Fourier frequencies are shifted from integer multiples of $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} by a common offset, $f_0\le f_{\mathrm{r}}$$f_0\le f_{\mathrm{r}}$f_(0) <= f_(r)f_{0} \leq f_{\mathrm{r}}, such that
由於 不一定是模式間距的精確倍數， $f_r$$f_r$f_(r)f_{r} 因此 $v_c$$v_c$v_(c)v_{c} 各個傅里葉頻率從的 $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} 整數倍偏移一個公共偏移量 $f_0\le f_{\mathrm{r}}$$f_0\le f_{\mathrm{r}}$f_(0) <= f_(r)f_{0} \leq f_{\mathrm{r}} ，使得

where $N$$N$NN is an integer mode number between 100,000 and $1,000,000$$1,000,000$1,000,0001,000,000, that multiplies $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} from the microwave domain to the optical domain.
其中 $N$$N$NN 是介於 100,000 和 $1,000,000$$1,000,000$1,000,0001,000,000 之間的整數模式數， $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} 它從微波域乘以光域。

Equation (2) is referred to as the comb equation. What the comb equation states is that while an OFC consists of up to a million optical modes, spanning hundreds of terahertz in the optical domain, only two degrees of freedom: (1) the repetition rate, $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} and the (2) laser offset frequency, $f_0$$f_0$f_(0)f_{0}, are needed to define the frequency of each individual optical mode, $v_{\mathrm{N}}$$v_{\mathrm{N}}$v_(N)v_{\mathrm{N}}. This ability to completely define optical frequencies in terms of microwave frequencies was the original claim to fame for OFCs in precision optical metrology. To summarize, MLLs can enable near-perfect coherent division of optical frequencies to the microwave domain, and coherent multiplication of microwave frequencies to the optical domain.
方程 （2） 稱為梳狀方程。梳狀方程指出，雖然 OFC 由多達一百萬個光學模式組成，在光域中跨越數百太赫茲，但只需要兩個自由度：（1） 重複率和 $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} （2） 鐳射偏移頻率 $f_0$$f_0$f_(0)f_{0} ，即可定義每個單獨光學模式的頻率。 $v_{\mathrm{N}}$$v_{\mathrm{N}}$v_(N)v_{\mathrm{N}} 這種根據微波頻率完全定義光學頻率的能力是 OFC 在精密光學計量學中聲名鵲起的最初原因。總而言之，MLL 可以實現將光頻率近乎完美的相干劃分到微波域，並將微波頻率相乘到光域。

The repetition rate $\left(f_{\mathrm{r}}\right)$$\left(f_{\mathrm{r}}\right)$(f_(r))\left(f_{\mathrm{r}}\right). The microwave mode that ties the spectrum together harmonically is the laser repetition rate, $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}}, which is the inverse of the pulse-to-pulse timing, $T_{\mathrm{r}}$$T_{\mathrm{r}}$T_(r)T_{\mathrm{r}}. Pulses exit the laser cavity once per round trip such that the pulse repetition period, $T_{\mathrm{r}}=2L/v_{\mathrm{g}}$$T_{\mathrm{r}}=2L/v_{\mathrm{g}}$T_(r)=2L//v_(g)T_{\mathrm{r}}=2 L / v_{\mathrm{g}}, where $v_{\mathrm{g}}$$v_{\mathrm{g}}$v_(g)v_{\mathrm{g}} is the pulse group velocity in the laser cavity, is defined and controlled via actuation of the laser cavity length, $L$$L$LL. Changes in $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} result in an accordion-like expansion and contraction of the frequency modes.
重複率 $\left(f_{\mathrm{r}}\right)$$\left(f_{\mathrm{r}}\right)$(f_(r))\left(f_{\mathrm{r}}\right) .將光譜諧波連接在一起的微波模式是激光重複率 $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} ，它是脈衝到脈衝時序的倒數 $T_{\mathrm{r}}$$T_{\mathrm{r}}$T_(r)T_{\mathrm{r}} 。脈衝每次往返離開鐳射腔一次，因此脈衝重複週期 $T_{\mathrm{r}}=2L/v_{\mathrm{g}}$$T_{\mathrm{r}}=2L/v_{\mathrm{g}}$T_(r)=2L//v_(g)T_{\mathrm{r}}=2 L / v_{\mathrm{g}} ，其中 $v_{\mathrm{g}}$$v_{\mathrm{g}}$v_(g)v_{\mathrm{g}} 是鐳射腔中的脈衝群速度，通過驅動鐳射腔長度 $L$$L$LL 來定義和控制。變化 $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} 導致頻率模式出現類似手風琴的擴展和收縮。

The offset frequency $\left(f_0\right)$$\left(f_0\right)$(f_(0))\left(f_{0}\right). Pulse formation necessarily requires that every longitudinal laser mode is perfectly equidistant in frequency and shares a common phase. This unlikely condition is enforced by nonlinearities in the laser cavity that underlie pulse formation and spectral broadening via self-phase modulation and four-wave mixing. The harmonic and coherent connection between laser modes is manifest as a common and additive frequency offset, $f_0$$f_0$f_(0)f_{0}. In the frequency domain, $f_0$$f_0$f_(0)f_{0} translates all the laser modes simultaneously. Because this offset frequency is a measure of coherence it also relates to time-changes of the optical carrier phase relative to the pulse envelope, ${\varphi }_{\text{CEO}}(t)$${\varphi }_{\text{CEO }}(t)$phi_("CEO ")(t)\phi_{\text {CEO }}(t),
偏移頻率 $\left(f_0\right)$$\left(f_0\right)$(f_(0))\left(f_{0}\right) 。脈衝形成必然要求每個縱向鐳射模式的頻率完全等距，並共用一個公共相位。這種不太可能的情況是由鐳射腔中的非線性增強的，這些非線性是通過自相位調製和四波混頻形成和光譜展寬的基礎。雷射模式之間的諧波和相干連接表現為公共和加性頻率偏移 $f_0$$f_0$f_(0)f_{0} 。在頻域中， $f_0$$f_0$f_(0)f_{0} 同時平移所有鐳射模式。因為這個偏移頻率是相幹性的度量，所以它還與光載波相位相對於脈沖包絡的時間變化有關 ， ${\varphi }_{\text{CEO}}(t)$${\varphi }_{\text{CEO }}(t)$phi_("CEO ")(t)\phi_{\text {CEO }}(t)

that result due to dispersion induced phase- and group-velocity differences.
這是由於色散引起的相速度和群速度差異而產生的。

Fig. 1 Frequency comb representations and detection of the offset frequency. a Time and frequency domain representation of an optical frequency comb. The optical output of a mode-locked laser is a periodic train of optical pulses with pulse period, $T_r$$T_r$T_(r)T_{r}, and pulse envelope $A(t)$$A(t)$A(t)A(t). In the frequency domain, this pulse train can be expressed as a Fourier series of equidistant optical frequencies, with mode spacing, $f_r=1/T_r$$f_r=1/T_r$f_(r)=1//T_(r)f_{r}=1 / T_{r}. It is the regular frequency spacing of the modes in the optical spectrum that inspired the analogy to a comb, although the analogy of a frequency ruler better describes the OFCs measurement capability. The frequency of any optical mode, ${\mathit{v}}_{\mathrm{N}}$${\mathit{v}}_{\mathrm{N}}$v_(N)\boldsymbol{v}_{\mathrm{N}}, is characterized by only two degrees of freedom, $f_r$$f_r$f_(r)f_{r} and $f_0$$f_0$f_(0)f_{0}, such that ${\nu }_N=N\cdot f_r+f_0$${\nu }_N=N\cdot f_r+f_0$nu_(N)=N*f_(r)+f_(0)\nu_{N}=N \cdot f_{r}+f_{0}. The mode spacing, $f_r$$f_r$f_(r)f_{r}, is accessed by directly detecting the amplitude modulation of the optical pulse train using an optical photodetector. This detection results in an electronic pulse train composed of coherently related microwave Fourier harmonics, $n\cdot f_r$$n\cdot f_r$n*f_(r)n \cdot f_{r}. Note that the optical spectrum contains information about the offset of the harmonic comb from $0\mathrm{Hz},f_0$$0\mathrm{Hz},f_0$0Hz,f_(0)0 \mathrm{~Hz}, f_{0}, whereas the microwave spectrum only yields harmonics of $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} because direct photodetection is not sensitive to the optical carrier. In the yellow shaded inset, we show the relationship between $f_0$$f_0$f_(0)f_{0} and the carrier-envelope offset phase, ${\varphi }_{\text{CEO}}(t)$${\varphi }_{\text{CEO }}(t)$phi_("CEO ")(t)\phi_{\text {CEO }}(t). The evolution in the pulse-to-pulse change in the carrier-envelope phase is given by $\mathrm{\Delta }{\varphi }_{\text{CEO}}=2\pi f_0/f_r$$\mathrm{\Delta }{\varphi }_{\text{CEO }}=2\pi f_0/f_r$Deltaphi_("CEO ")=2pif_(0)//f_(r)\Delta \phi_{\text {CEO }}=2 \pi f_{0} / f_{r}. Notably, when $f_0=0$$f_0=0$f_(0)=0f_{0}=0, every optical pulse has an identical carrier-envelope phase. The pulse envelope, $A$$A$AA $(t)$$(t)$(t)(t), depicted by a blue dashed line is related by the periodic Fourier transform to the spectral envelope. $\mathbf{b}$$\mathbf{b}$b\mathbf{b} Offset frequency detection via self-referencing. Frequency depiction of how nonlinear self-comparison can be used to detect $f_0$$f_0$f_(0)f_{0}.
圖 1 頻率梳表示和偏移頻率的檢測。a 光頻梳的時域和頻域表示。鎖模雷射器的光輸出是具有脈衝週期 $T_r$$T_r$T_(r)T_{r} 和脈衝包絡 $A(t)$$A(t)$A(t)A(t) 的週期性光脈衝序列。在頻域中，該脈衝序列可以表示為等距光學頻率的傅里葉級數，模間距為 $f_r=1/T_r$$f_r=1/T_r$f_(r)=1//T_(r)f_{r}=1 / T_{r} 。正是光譜中模式的規則頻率間隔激發了與梳子的類比，儘管頻率尺規的類比更好地描述了 OFC 的測量能力。任何光學模式 ${\mathit{v}}_{\mathrm{N}}$${\mathit{v}}_{\mathrm{N}}$v_(N)\boldsymbol{v}_{\mathrm{N}} 的頻率， 都由兩個自由度 $f_r$$f_r$f_(r)f_{r} 和 $f_0$$f_0$f_(0)f_{0} 來表徵，使得 ${\nu }_N=N\cdot f_r+f_0$${\nu }_N=N\cdot f_r+f_0$nu_(N)=N*f_(r)+f_(0)\nu_{N}=N \cdot f_{r}+f_{0} 。使用光學光電探測器直接偵測光脈衝序列的幅度數據來獲得模式間隔 $f_r$$f_r$f_(r)f_{r} 。該檢測結果產生由相幹相關的微波傅里葉諧波 組成的電子脈衝序列 $n\cdot f_r$$n\cdot f_r$n*f_(r)n \cdot f_{r} 。請注意，光譜包含有關諧波梳 偏移的資訊， $0\mathrm{Hz},f_0$$0\mathrm{Hz},f_0$0Hz,f_(0)0 \mathrm{~Hz}, f_{0} 而微波光譜僅產生諧波， $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} 因為直接光探測對光載波不敏感。在黃色陰影插圖中，我們顯示了 $f_0$$f_0$f_(0)f_{0} 載波包絡偏移相位 之間的關係。 ${\varphi }_{\text{CEO}}(t)$${\varphi }_{\text{CEO }}(t)$phi_("CEO ")(t)\phi_{\text {CEO }}(t) 載波包絡相位中脈衝到脈衝變化的演變由下 $\mathrm{\Delta }{\varphi }_{\text{CEO}}=2\pi f_0/f_r$$\mathrm{\Delta }{\varphi }_{\text{CEO }}=2\pi f_0/f_r$Deltaphi_("CEO ")=2pif_(0)//f_(r)\Delta \phi_{\text {CEO }}=2 \pi f_{0} / f_{r} 式給出。值得注意的是，當 時 $f_0=0$$f_0=0$f_(0)=0f_{0}=0 ，每個光脈衝都具有相同的載波包絡相位。由藍色虛線表示的脈衝包絡 $A$$A$AA $(t)$$(t)$(t)(t) 由週期性傅里葉變換與頻譜包絡相關聯。 $\mathbf{b}$$\mathbf{b}$b\mathbf{b} 通過自引用進行偏移頻率檢測。如何使用非線性自比較進行檢測 $f_0$$f_0$f_(0)f_{0} 的頻率描述。

In the simplest terms, $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} controls the pulse-to-pulse timing, and hence the periodicity of the pulse train, permits coarse frequency control of the OFC spectrum, and connects the optical and microwave domains via $Nf$$Nf$NfN f r. The offset frequency, $f_0$$f_0$f_(0)f_{0}, controls the carrier-phase of the pulse train, and enables fine optical frequency tuning. The detection and control of the laser offset frequency $f_0$$f_0$f_(0)f_{0} is the key for allowing precise frequency determination of the comb modes, as well as for control of the pulse electric field in high-field physics and attosecond science experiments ${}^5$${}^5$^(5){ }^{5}. More specifically, with knowledge of $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} alone, a single optical mode can
用最簡單的話來說， $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} 控制脈衝到脈衝的定時，從而控制脈衝序列的週期性，允許對 OFC 頻譜進行粗調頻率控制，並通過 $Nf$$Nf$NfN f r 連接光域和微波域。偏移頻率 $f_0$$f_0$f_(0)f_{0} 控制脈衝序列的載波相位，並支援精細的光學頻率調諧。鐳射偏置頻率 $f_0$$f_0$f_(0)f_{0} 的檢測和控制是精確確定梳狀模式頻率以及控制高場物理學和阿秒科學實驗 ${}^5$${}^5$^(5){ }^{5} 中脈衝電場的關鍵。更具體地說，僅憑 知識 $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} ，單一光學模式就可以
only be known to $\pm f_{\mathrm{r}}/2$$\pm f_{\mathrm{r}}/2$+-f_(r)//2\pm f_{\mathrm{r}} / 2. On an optical frequency, this represents an error of parts in ${10}^6-{10}^5$${10}^6-{10}^5$10^(6)-10^(5)10^{6}-10^{5} depending on the mode spacing.
只有 . $\pm f_{\mathrm{r}}/2$$\pm f_{\mathrm{r}}/2$+-f_(r)//2\pm f_{\mathrm{r}} / 2 在光學頻率上，這表示各部分的誤差 ${10}^6-{10}^5$${10}^6-{10}^5$10^(6)-10^(5)10^{6}-10^{5} 取決於模式間隔。

Full frequency stabilization of the comb is achieved using negative feedback to the laser cavity length and intra-cavity dispersion to physically control $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} and $f_0$$f_0$f_(0)f_{0}. Ensuring good mechanical stability and engineering of the stabilization loops, the above methods can enable control of the average cavity length at resolutions below a femtometer, the diameter of the proton. Applications with the highest stability requirements, or ones that require long-term accuracy and averaging, generally require
梳子的全頻穩定是通過對鐳射腔長度和腔內色散的負反饋來實現的，以物理控制和 $f_{\mathrm{r}}$$f_{\mathrm{r}}$f_(r)f_{\mathrm{r}} $f_0$$f_0$f_(0)f_{0} 。確保穩定的迴路具有良好的機械穩定性和工程設計，上述方法可以在低於飛米（質子直徑）的解析度下控制平均腔體長度。具有最高穩定性要求的應用程式或需要長期精度和平均值的應用程式通常需要