Transferring of laser frequency from a reference laser to desired wavelengths is a very important technique in metrology and fundamental physics, e.g., for generating ultra-stable clock lasers to improve the stability of the optical clocks from low-thermal-noise Si resonators, which are not transparent at the wavelength of smaller than 1.2 µm^[1–3], and for remote optical clock comparison^[4–6]. In the remote optical clock comparison, the frequency information of the clock laser is sent to a remote site over a telecom fiber, which is demonstrated to be at the level of ${10}^{-19}$ or better^[7–10] and is sufficient for the comparison of state-of-the-art optical clocks. In order to send the frequency information of optical clocks to remote sites over an optical fiber, the frequency of the clock laser has to be converted to communication wavelength, i.e., around 1.5 µm. Usually optical frequency combs (OFCs) are used to coherently bridge the phase between lasers with different wavelengths^[11].

A direct method to transfer the frequency stability from a source laser to a target laser is realized by transferring the frequency stability of the source laser to the OFC teeth firstly by phase locking their beat signal to a radio frequency (RF) reference and then from the OFC teeth to the target laser by a similar scheme. This method requires that the frequency of the OFC be tuned with high bandwidth and limits the OFC only to be used to transfer frequency from one source laser. Telle et al. introduced a so-called transfer oscillator scheme^[12] to bridge the lasers at different wavelengths without phase locking of the OFC to lasers. The scheme generated a ‘virtual beat’ signal between the source and the target laser, which could be stabilized to the RF reference to realize the frequency stability transfer^[13]. This scheme reduced the requirement on the OFC. Because the external RF reference was involved in the scheme, in the remote comparison, the RF references in the two laboratories had to be referred to the International System of Units (SI) second, or the RF reference should also be sent from one lab to the other. In 2016, Yao et al. introduced an improved method by generating the RF reference from the source laser itself instead of using an external reference^[14]. With Yao et al.’s method, the generated target laser frequency was free from external reference. In previous works, researchers demonstrated that by using an OFC it was possible to transfer laser frequency to other wavelengths with precision at the level of ${10}^{-21}$^[14,15].

In this work, we converted the frequency of a 729 nm ultra-stable laser for a ${\mathrm{Ca}}^{+}$ clock to 1.5 µm by using a self-reference transfer oscillator method^[12,16]. By mathematically deriving the relation between the lasers and the OFC, we constructed the RF processing system for the frequency conversion. Two individual 1560 nm lasers were phase locked to the 729 nm laser with two close preset ratios. The fractional instability of the frequency difference between the two target lasers was measured to be $1.6\times {10}^{-17}$ at 1 s and $2\times {10}^{-20}$ at 10,000 s averaging time, and the relative offset between the measured value and the theoretical value was $4.2\times {10}^{-21}\pm 4.5\times {10}^{-20}$. The ratio between the two lasers was measured to be a level of $1.3\times {10}^{-20}$ in 1 day’s data accumulation.

The frequency transferring system was built for the comparison of the optical clocks at the Innovation Academy for Precision Measurement Science and Technology, CAS^[17–19] and the optical clock at the Huazhong University of Science and Technology^[20]. The system could also be used to synthesize ultra-stable lasers for other purposes.

The RF processing scheme of the experiment is derived as follows. When two lasers with frequencies of ${\nu }_1$ and ${\nu }_2$ beat with an OFC, the beating frequencies $f_{\mathrm{b}1}$ and $f_{\mathrm{b}2}$ between the lasers and the OFC are $$\begin{gathered} f_{\mathrm{b}1}={\nu }_1-m_1{f}_r-f_{\mathrm{CEO}}, \\ {f}_{\mathrm{b}2}={\nu }_2-m_2{f}_r-f_{\mathrm{CEO}}, \end{gathered}$$(1)where $f_{\mathrm{CEO}}$ is the carrier-envelope offset (CEO) frequency, $f_r$ is the repetition rate of the OFC, and $m_1$ and $m_2$ are the corresponding mode numbers of the OFC teeth beating with the lasers. The frequencies $f_{\mathrm{CEO}}$ and $f_r$ are normally stabilized to the RF reference to eliminate the long-term drift of the comb frequency. Therefore, the frequency ratio $R$ between the two lasers can be written in the form of the OFC and the beat frequencies, i.e., $$R=\frac{{\nu }_2}{{\nu }_1}=\frac{m_2{f}_r+f_{\mathrm{CEO}}+f_{\mathrm{b}2}}{m_1{f}_r+f_{\mathrm{CEO}}+f_{\mathrm{b}1}}=\frac{m_2+x}{m_1},$$(2)where $$x=\frac{m_1{\nu }_2-m_2{\nu }_1}{{\nu }_1}=\frac{m_1(f_{\mathrm{CEO}}+f_{\mathrm{b}2})-m_2(f_{\mathrm{CEO}}+f_{\mathrm{b}1})}{m_1{f}_r+f_{\mathrm{CEO}}+f_{\mathrm{b}1}}.$$(3)

In Eqs. (2) and (3), $m_1$ and $m_2$ are measurable integers, and $x$ is a decimal. The instability of $x$ is $m_1$ times larger than $R$. Therefore, in frequency ratio measurement or frequency conversion, measuring or phase locking of the $x$ instead of $R$ directly will reduce the requirement on the processing system $m_1$ times. The numerator and denominator of Eq. (3) are at the level of ${10}^{14}\mathrm{Hz}$. They are reduced to a level of ${10}^6$ to ${10}^7\mathrm{Hz}$ by multiplying with a small coefficient $k$ and denoted as $f_{\mathrm{diff}}$ and $f_{\mathrm{ref}}$, respectively, i.e., $$\begin{gathered} f_{\mathrm{diff}}=k{m}_1(f_{\mathrm{CEO}}+f_{\mathrm{b}2})-k{m}_2(f_{\mathrm{CEO}}+f_{\mathrm{b}1}), \\ {f}_{\mathrm{ref}}=k{m}_1(f_r+f_{\mathrm{CEO}}+f_{\mathrm{b}1})-k(m_1-1)(f_{\mathrm{CEO}}+f_{\mathrm{b}1}). \end{gathered}$$(4)

Here, the $f_{\mathrm{ref}}$ is treated following Yao’s method^[14]. From Eqs. (3) and (4), it can be derived that transfer of optical frequency from ${\nu }_1$ to ${\nu }_2$ with a preset value $R_s$ ($x_s$) is equivalent to controlling ${\nu }_2$ to let the equation $$x_s=\frac{f_{\mathrm{diff}}}{f_{\mathrm{ref}}},\text{i.e.},x_s·f_{\mathrm{ref}}-f_{\mathrm{diff}}=0,$$(5)hold. Equation (5) can be technically kept holding by feedback modulating the frequency of ${\nu }_2$ to keep the RF signals $f_{\mathrm{diff}}$ being phase-locked onto $x_s·f_{\mathrm{ref}}$. Therefore, with the expression of Eqs. (4) and (5), the RF processing scheme for the frequency conversion can be constructed as Fig. 1. In the scheme, the addition and subtraction of the frequencies are realized by mixing the two input signals with frequency mixers and picking up the corresponding signal by using proper bandpass filters. The scaling of the frequency is realized by using direct digital synthesizers (DDSs). A DDS scales the input RF frequency with a factor of $n/2^N$, where $n$ is the frequency tuning word (FTW) of the DDS, which is smaller than $2^N$, and $N$ is the bits of the FTW. For example, in our experiment, Analog Devices AD9956 is used to scale the frequency, whose $N$ is 48. The coefficient $k$ is set to $k=c/2^N$, and therefore, the FTW written to the DDS is $c\times m$ or $x_s\times c\times m$. The scheme in Fig. 1 is also used to measure $x$ by generating an $f_{\mathrm{ref}}$ without multiplying $x_s$ in the left part of the figure and replacing the phase detector with a frequency counter.

Compared with the previous work on the transfer of laser frequency, in this scheme, the frequency relation between the reference laser and the target laser is decomposed to basic RF processing actions, i.e., adding, subtracting, and scaling of RF frequencies. These functions can be realized by simple RF elements, e.g., mixers combined with filters are used to perform the adding and subtracting of frequency, and DDSs are used to scale the RF frequencies. There is no external reference or synthesizer involved in this scheme, and, therefore, the precision and stability of the generated laser’s frequency do not rely on external reference or synthesizers.

To demonstrate the precision of the frequency transfer scheme, two 1560 nm lasers were phase locked to the same 729 nm ultra-stable laser by using the scheme shown in Fig. 1. The optical setup for the experiment is illustrated in Fig. 2. As shown in the figure, a multi-branch Er-doped fiber-based OFC was used to bridge the three lasers. One branch was used to obtain the signal of $f_{\mathrm{CEO}}$ and $f_r$. The second branch was used to beat with the 729 nm laser (${\nu }_1$). Because the center wavelength of the OFC was at about 1.5 µm, the spectrum for the beating of 729 nm was generated by expanding the spectrum of the OFC to about 1460 nm and then frequency doubling to cover the wavelength of 729 nm. The third branch was used to beat with the two 1560 nm lasers. One of the 1560 nm lasers (${\nu }_2$) was frequency pre-locked to a 10 cm long optical resonator to keep the frequency from drifting away. The second 1560 nm laser (${\nu }_2^{\prime }$) was free running, and the frequency was controlled by monitoring the beat frequency with ${\nu }_2$. The two 1560 nm lasers were combined by using a fiber optic coupler. One output of the coupler was sent to beat with the OFC. The second output was sent to a photo detector for measuring the beat frequency between the two lasers.

The repetition frequency of the OFC was about 200 MHz. The $f_{\mathrm{CEO}}$ and $f_r$ of the OFC were phase locked to RF signals referred to as H-maser. An about 10 MHz RF signal ($f_{\mathrm{ref}1}$) was generated from ${\nu }_1$. This signal was used as the reference for the frequency counters for later measuring the beating frequency ($\mathrm{\Delta }\nu ={\nu }_2-{\nu }_2^{\prime }$) between the two 1560 nm lasers. The $f_{\mathrm{ref}1}$ was generated according to Eq. (4), i.e., the left part of Fig. 1 without multiplying $x_s$. The numbers used to generate the RF signal were the comb mode number $m_1=2,055,168$ and $c=6,847,836$.

${\nu }_2$ was phase locked to ${\nu }_1$ by feeding back to the acousto-optic modulator (AOM1) with $x_s=0.75$. The OFC mode number $m_2$ for ${\nu }_2$ beating was 960,542. Therefore, $R_s=960,542.75/2,055,168$. ${\nu }_2^{\prime }$ was tuned about 1.2 GHz lower than ${\nu }_2$ and phase locked to ${\nu }_1$ by feeding back the fast signal to AOM2 and the slow signal to the laser’s piezoelectric transducer (PZT) with $x_s^{\prime }=0.5$. The corresponding OFC mode number $m_2^{\prime }$ was 960,536. Therefore, $R_s^{\prime }=960,536.5/2,055,168$. From these numbers, the beating frequency ($\mathrm{\Delta }\nu$) related to $f_{\mathrm{ref}1}$ and the ratio ($R$) between ${\nu }_2$ and ${\nu }_2^{\prime }$ were calculated to be $$\begin{gathered} \mathrm{\Delta }\nu =\frac{{\nu }_2-{\nu }_2^{\prime }}{f_{\mathrm{ref}1}}\times 10\mathrm{MHz}=\frac{(R_s-R_s^{\prime }){\nu }_1}{k\times {\nu }_1}\times 10\mathrm{MHz} \\ =1,250,026,239.036366754\mathrm{Hz}, \\ R=\frac{{\nu }_2^{\prime }}{{\nu }_2}=\frac{R_s^{\prime }}{R_s}=0.9999934932620125444703. \end{gathered}$$(6)

After the two lasers were phase locked, the $\mathrm{\Delta }\nu$ was measured referred to $f_{\mathrm{ref}}$. Because $\mathrm{\Delta }\nu$ was at the level of gigahertz (GHz), which exceeded the measuring range of the frequency counter, the frequency of the signal was divided by 64 by using two cascading divide-by-eight static dividers HMC434. Symmetricom 3120A with a measuring bandwidth of 0.5 Hz was used as a high-precision counter to measure $\mathrm{\Delta }\nu /64$. The recorded frequency (noted as $f_m$) is shown in Fig. 3(a). The figure shows the frequency difference between $f_m$ and $\mathrm{\Delta }\nu /64$. From the data, the average of $f_m$ was calculated to be $$f_m=19,531,659.984943243\pm 1.4\times {10}^{-7}\mathrm{Hz}.$$(7)

From this data, the offset of the measured and theoretical beating frequency was calculated to be $0.8\pm 9.0\mathrm{\mu Hz}$, which was $6.6\times {10}^{-16}\pm 7.2\times {10}^{-15}$ related to the beating frequency and $4.2\times {10}^{-21}\pm 4.5\times {10}^{-20}$ related to ${\nu }_2$. The fractional instability of the beat frequency is shown in Fig. 3(b). It shows that fractional Allan deviation was $1.6\times {10}^{-17}$ and $2\times {10}^{-20}$ at the averaging time of 1 s and 10,000 s, respectively.

The frequency ratio between ${\nu }_2^{\prime }$ and ${\nu }_2$ ($R={\nu }_2^{\prime }/{\nu }_2$) was also measured to double check the precision of the transfer scheme. $R$ was measured according to Eqs. (2) and (3). The $f_{\mathrm{ref}}$ and $f_{\mathrm{diff}}$ for ${\nu }_2$ and ${\nu }_2^{\prime }$ (noted as $f_{\mathrm{ref}2}$ and $f_{\mathrm{diff}2}$) were generated according to the left and right parts of the scheme shown in Fig. 1 with $x_s=1$, respectively. The $x$ for the two lasers was obtained by dividing the measured frequency $f_{\mathrm{diff}2}$ referred to $f_{\mathrm{ref}2}$ (noted as $f_{\mathrm{dm}}$) with 10 MHz (noted as $x_m$). The recorded $f_{\mathrm{dm}}$ is shown in Fig. 4(a). From the figure, the statistic of $f_{\mathrm{dm}}$ was calculated to be $2,499,951.199465099\pm 1.2\times {10}^{-7}\mathrm{Hz}$. Therefore, $x_m=-0.2499951199465099\pm 1.2\times {10}^{-14}$, and the ratio was $$R_m=0.9999934932620125444698\pm 1.2\times {10}^{-20},$$(8)which was $-5\times {10}^{-22}\pm 1.3\times {10}^{-20}$ offset from the setting $R$ in Eq. (6). The fractional Allan deviation is shown in Fig. 4(b), which shows that the instability of $R_m$ was $4.2\times {10}^{-18}$ at 1 s and $4.5\times {10}^{-21}$ at 10,000 s averaging time.

Because there was only one OFC available in our laboratory, the measurement of the ratio between the two 1.5 µm lasers was carried out by using the same beating signal as the comb. This result on the ratio $R$ demonstrated only the precision and consistency of the RF processing and frequency locking electronics.

In the transfer experiment, two different branches of the OFC were used to beat with the source laser (${\nu }_1$) and the target lasers (${\nu }_2$ and ${\nu }_2^{\prime }$). The inter-branch frequency noise introduced by the non-common mode fiber of the OFC might reduce the transfer accuracy and stability. The frequency transfer was expected to be optimized by using a single branch of the OFC or introducing the inter-branch noise cancellation scheme^[21].

We transferred the frequency of a ${\mathrm{Ca}}^{+}$ clock laser at 729 nm to 1.5 µm in the telecommunication band at an accurate frequency ratio for remote clock comparison. To demonstrate the precision of the frequency transferring process, two 1560 nm lasers were phase locked to the 729 nm laser with slightly different frequency ratios. The relative uncertainty of the frequency difference and frequency ratio between the two transfer lasers were measured to be $4.5\times {10}^{-20}$ and $1.3\times {10}^{-20}$ in one day’s data acquisition, respectively. This result suggests that the transfer stability and accuracy of our setup are enough for remote frequency comparison of optical clocks at the level of ${10}^{-18}$.