Wavefront interferometers, which can directly obtain the wavefront aberration of optical elements or systems, are widely used in fields such as astronomy^[1], photolithography^[2], and high-power laser systems^[3]. Due to the increased applications of wavefront measurements in environments with vibration and air turbulence, short-exposure interferometers that can freeze interferograms in a short time are urgently needed.

Laser power is one of the crucial specifications of short-exposure interferometers. We previously studied the frequency stability of FBG-based laser diodes with 15 mW output for a dynamic short-exposure interferometer^[4]. Although high power is obtained, the FBG-based laser diode is not suitable for high-precision interferometers with temporal phase shifting, due to its weak frequency stability and narrow range without mode hopping. The frequency-stabilized HeNe laser, which can provide a coherence length of several hundred meters and a reliable wavelength that serves as a traceability standard, is an ideal light source for high-precision interferometers when exposure time is ignored. The output power of the commercial stabilized laser is about 1 mW, which may reduce the signal-to-noise ratios of the wavefront interferograms^[5], especially in the cases of a long interferometric cavity or measuring objects with low reflectivity^[4]. To get higher power, the interferometer is sometimes equipped with a multiple-longitudinal-mode HeNe laser, but the testing position of the analyte needs to be carefully selected to avoid the loss of the interference fringe contrast^[6]. Thus, there is an urgent requirement to design a high-power and frequency-stabilized HeNe laser for short-exposure interferometry.

Generally, the commercial frequency-stabilized HeNe laser utilizes an internal-cavity resonator running in two longitudinal modes as the light source, and its wavelength locking is realized by balancing the intensities of the two modes^[7]. The operating mode of the laser is selected from the nearly orthogonal-polarized adjacent longitudinal modes by employing a polarization component, resulting in a low monomode-power conversion efficiency, which is the ratio of the frequency-stabilized output and the resonator output. To improve the working power, it is possible to consider the laser running in three longitudinal modes, with the intermediate mode as the operating mode. Methods such as the self-heterodyne mixing method^[8], the coherence-based feedback method^[9], the thermal phase-locking method^[10], and the Pound–Drever–Hall method^[11,12] have been adopted to stabilize the frequency of the operating mode. These methods are usually designed for the light sources of the displacement interferometers, and their control strategies are relatively complex. To achieve a high-quality HeNe laser for the short exposure interferometry, the output power and the frequency stability are balanced by ZYGO Corporation^[5]. High power was obtained by optimizing the coatings and shapes of the end mirror as well as the gas fill of the laser. All the lasers mentioned above are achieved through the polarized mode selection from the internal-cavity resonators with random polarization output. Thus, the monomode power conversion efficiency is relatively low. Furthermore, there exists a potential risk of mode hopping in the randomly polarized laser^[13], which not only increases the difficulty of operating mode selection but also challenges the power stabilization.

In this paper, what we believe is a novel high-power and frequency-stabilized light source based on a semi-external-cavity HeNe laser is proposed for short-exposure interferometry. By optimizing the parameters of the semi-external resonant cavity, a high-fineness valley corresponding to monomode operation is generated on the temperature-tuned power curve of the laser. Frequency stabilization is achieved through locking the power valley. Because linearly polarized light can be output directly, high conversion efficiency of monomode power can be realized for short-exposure interferometry.

Figure 1 shows the schematic of a high-power and frequency-stabilized HeNe laser. The laser features a resonator with a semi-external cavity structure, consisting of a plane mirror, a Brewster window plate, and a concave mirror. The plane mirror and the concave mirror are located at the cathode emission end and the anode emission end, respectively. The Brewster window plate inside the cavity is sealed on the capillary end face near the plane mirror to achieve linear polarization radiation output directly. The laser emitted from the plane mirror is split by a beam splitter (BS). The reflected light is utilized as the reference light, which is received by a silicon photodiode (PD). By adjusting the temperature of the laser tube to lock the intensity of the reference light at the high-finesse valley floor of the power tuning curve, the oscillations of the side modes are completely suppressed, and the resonant frequency of the operating mode can be simultaneously stabilized. Thus, the working power of the frequency-stabilized HeNe laser without polarization mode selection can be enhanced.

The output power of the HeNe laser, which is affected by the hole burning effect, can be represented by the hole area that characterizes the number of stimulated radiation photons^[14]. For the standing-wave-cavity laser with ${\upsilon }_0$ as the center frequency of its gain curve, two Lorentzian-shaped depressions will be burnt out on the gain curve by one longitudinal mode with resonant frequency ${\upsilon }_r$. The two depressions, respectively, center at the frequency ${\upsilon }_r$ and $2{\upsilon }_0-{\upsilon }_r$, and they are called the original hole and the mirror hole, correspondingly, as shown in Fig. 2(a). Under the case of a weak radiation field and low gas filling pressure, the widths of the holes at different frequencies are almost equal and much smaller than the Doppler bandwidth of the gain curve. Therefore, the output power of the laser is proportional to the depth of the hole of the longitudinal mode. Taking $d$ as the hole depth, the steady-state operation condition for the single longitudinal mode in the standing wave case is^[15]$$G({\upsilon }_r)-L_r=d·[1+F(2{\upsilon }_0-2{\upsilon }_r)],$$(1)where $G(\upsilon )$ is the frequency-dependent small-signal gain, $L_r$ is the loss of the resonant cavity, and $F(\delta {\upsilon }_c)$ is the modulation efficient of the hole depth, which is expressed as^[15]$$F(\delta {\upsilon }_c)=1/[1+{(\delta {\upsilon }_c/\mathrm{\Delta }{\upsilon }_L)}^2],$$(2)where $\delta {\upsilon }_c$ is the frequency difference between the center frequency and the frequency of the original hole or the mirror hole and the frequency of the longitudinal mode, and $\mathrm{\Delta }{\upsilon }_L$ is the homogeneously broadening linewidth of the gain medium. When the center frequency of the hole is equal to the resonant frequency of the longitudinal mode, $\delta {\upsilon }_c=0$, and the modulation efficiency has a maximum value of 1.

When there are three longitudinal modes with resonant frequencies ${\upsilon }_1$, ${\upsilon }_2$, and ${\upsilon }_3$, respectively, in the fluorescence spectrum region of the laser, each mode will burn out an original hole near its resonant frequency on the gain curve, and a mirror hole at the symmetric position of its resonant frequency with respect to the frequency ${\upsilon }_0$, as shown in Fig. 2(b). The space between the adjacent longitudinal modes is $\mathrm{\Delta }{\upsilon }_s=c/2L$, where $c$ is the velocity of the light, $L$ is the cavity length of the laser^[14]. If $d_1$, $d_2$, and $d_3$, respectively, represent the depths of the holes caused by the interaction between the gain particles and the different resonant modes, the steady-state operation condition for the three longitudinal modes in the standing wave case is^[15]$$\left[\begin{array}{c}G({\upsilon }_1)-L_r\\ G({\upsilon }_2)-L_r\\ G({\upsilon }_3)-L_r\end{array}\right]=\left[\begin{array}{c}{\phi }_{11}{\phi }_{12}{\phi }_{13}\\ {\phi }_{21}{\phi }_{22}{\phi }_{23}\\ {\phi }_{31}{\phi }_{32}{\phi }_{33}\end{array}\right]·\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right],$$(3)where $[{\phi }_{ij}]$ is a symmetric matrix with the size of $3\times 3$, and each element ${\phi }_{ij}$ in the matrix can be expressed as^[15]$${\phi }_{ij}=F({\upsilon }_i-{\upsilon }_j)+F(2{\upsilon }_{0k}-{\upsilon }_i-{\upsilon }_j),$$(4)where $i$ and $j\in [1,2,3]$.

In the process of frequency tuning, the original hole moves in the same direction as the resonant mode, and the mirror hole moves in the opposite direction. When the three longitudinal modes approach their symmetric distribution with respect to the frequency ${\upsilon }_0$, the original and mirror holes of the side mode with frequency ${\upsilon }_1$ overlap partially with the mirror and original holes of the side mode with frequency ${\upsilon }_3$, respectively. This phenomenon induces the intensity attenuation of the two side modes competing for the gain particles in the overlapping region of the holes^[16,17]. In addition, the hole area of the intermediate mode with frequency ${\upsilon }_2$ is also decreased due to the overlap of its own original hole and mirror hole, further reducing the output power of the laser^[18,19]. As the intermediate mode gradually approaches the frequency ${\upsilon }_0$, the phenomenon of the oscillation suppressions of the side modes as well as the decrease of the laser power will be more obvious.

The gain medium of the HeNe laser can be composed of a gas mixture of two neon isotopes^[20], ${\mathrm{Ne}}^{20}$ and ${\mathrm{Ne}}^{22}$. Each isotope medium has an independent gain curve, where the ${\mathrm{Ne}}^{20}$ gain curve with center frequency ${\upsilon }_{01}$ is located on the low-frequency side due to the isotope shifting effect^[21], and the ${\mathrm{Ne}}^{22}$ gain curve with center frequency ${\upsilon }_{02}$ is located on the high-frequency side, as shown in Fig. 3. Assuming that the abundance ratio of the two isotopes ${\mathrm{Ne}}^{20}$ and ${\mathrm{Ne}}^{22}$ in the gas mixture is 1:1, the two gain curves with similar shapes can be combined into one synthesis gain curve with a larger bandwidth, as indicated by the dotted curves in red, blue, and yellow, respectively, in Fig. 3. If there are three resonant longitudinal modes in the fluorescence spectrum region of the laser, each mode will burn out an original hole and a mirror hole on each neon gain curve. In the process of frequency tuning, whether the original hole and the mirror hole are from the same longitudinal mode or not, they are all likely to overlap. Therefore, the gain of each longitudinal mode is provided by the particles within the four-hole regions, including the overlapping parts. Note that there is no problem of the longitudinal mode competitions for gain particles when the holes on the different curves overlap because the atomic energy structures of the ${\mathrm{Ne}}^{20}$ and the ${\mathrm{Ne}}^{22}$ are discrepant^[20,22]. When the three longitudinal modes approach their symmetrical distribution with respect to the frequency $({\upsilon }_{01}+{\upsilon }_{02})/2$, the overlap of the holes occurs between the intermediate mode and the side modes. Compared to the result described in Fig. 2(b), the intermediate mode belongs to the strong mode in competition with the side modes. The former can contend for more gain particles in the competition, while the latter is further suppressed. By appropriately increasing the loss of the resonant cavity or enlarging the longitudinal mode space, the side modes will be closer to the fluorescence edge of the gain curve in the mode-symmetry case. Thereby, the oscillations of the side modes are likely to be completely suppressed.

When the laser gain medium is the gas mixture composed of dual neon isotopes, the steady-state operation condition for the three longitudinal modes in the standing-wave case is^[15]$$\left[\begin{array}{c}{G}_1({\upsilon }_1)+G_2({\upsilon }_1)-L_r\\ G_1({\upsilon }_2)+G_2({\upsilon }_2)-L_r\\ G_1({\upsilon }_3)+G_2({\upsilon }_3)-L_r\end{array}\right]=\left[\begin{array}{c}{\phi }_{11(k)}{\phi }_{12(k)}{\phi }_{13(k)}\\ {\phi }_{21(k)}{\phi }_{22(k)}{\phi }_{23(k)}\\ {\phi }_{31(k)}{\phi }_{32(k)}{\phi }_{33(k)}\end{array}\right]·\left[\begin{array}{c}{d}_{1(k)}\\ d_{2(k)}\\ d_{3(k)}\end{array}\right],$$(5)where $k=1$ and 2 denote the subscripts of the functions related to ${\mathrm{Ne}}^{20}$ and ${\mathrm{Ne}}^{22}$, respectively, $d_{1(k)}$, $d_{2(k)}$, and $d_{3(k)}$ are the hole depths of the three different modes on the same gain curve, $[{\phi }_{ij(k)}]$ is a symmetric matrix with the size of $3\times 3$, and each element ${\phi }_{ij(k)}$ in the matrix can be expressed as^[15]$${\phi }_{ij(k)}=F({\upsilon }_i-{\upsilon }_j)+F(2{\upsilon }_{0k}-{\upsilon }_i-{\upsilon }_j),$$(6)where ${\upsilon }_{0k}$ is the center frequency of the ${\mathrm{Ne}}^{20}$ or ${\mathrm{Ne}}^{22}$ gain curve. $G_k(\upsilon )$ is the frequency-dependent small-signal gain on the corresponding gain curve, of which the expression is^[15]$$G_k(\upsilon )=G_{mk}·\exp \{-{\left[2\sqrt{2}(\upsilon -{\upsilon }_{0k})/\mathrm{\Delta }{\upsilon }_{Dk}\right]}^2\},$$(7)where $\mathrm{\Delta }{\upsilon }_{Dk}$ and $G_{mk}$ are, respectively, the Doppler linewidth of the corresponding neon gain curve and the small-signal gain of the curve at the center frequency ${\upsilon }_{0k}$. $G_{m2/}{G}_{m1}$ is equivalent to the abundance ratio $R$ of the two isotopes of ${\mathrm{Ne}}^{20}$ and ${\mathrm{Ne}}^{22}$. $G_{m1}$ and $G_{m2}$ can be written as^[15]$$G_{m1}=G_m\times R/(1+R),G_{m2}=G_m/(1+R),$$(8)where $G_m$ is the total unsaturated amplification coefficient, which can be expressed as^[23]$$G_m=3\times {10}^{-4}l/\varphi ,$$(9)where $l$ and $\varphi$ are, respectively, the length and the diameter of the capillary.

The depth of the holes of the same longitudinal mode on different gain curves satisfies the following relationship^[15]: $$d_{q(2)}/d_{q(1)}=G_2({\upsilon }_q)/G_1({\upsilon }_q),$$(10)where $q$ is the longitudinal mode ordinal number. Hence, the output power $P_t$ of the HeNe laser with a dual-neon-isotope gain medium operating in three longitudinal modes can be expressed as^[15]$$P_t=C_0·\sum _{i=1}^3(d_{i(1)}+d_{i(2)})=C_0·\sum _{i=1}^3{d}_{i(1)}·[1+G_2({\upsilon }_i)/G_1({\upsilon }_i)],$$(11)where $C_0$ is a scale constant with the value of $2\times {10}^4$^[24]. When the laser is in single-longitudinal-mode or dual-longitudinal-mode operation, the expressions of the output power in the corresponding mode-operation case can be obtained by removing the nonoscillatory mode terms from Eqs. (5) and (11).

In addition to the hole burning effect, the output power of the laser is also affected by the cross-relaxation effect. This effect can provoke the redistribution of some excited atoms throughout the whole gain curve, making the stimulated radiation at a resonant frequency produce a certain degree of saturation over the entire gain curve^[25]. When the cross-relaxation effect is considered, Eq. (11) should be modified as^[24]$$P_t=C_0·\sum _{i=1}^3\{d_{i(1)}·[1+G_2({\upsilon }_i)/G_1({\upsilon }_i)]+H_0·[G_1({\upsilon }_i)+G_2({\upsilon }_i)]\},$$(12)where $H_0$ is a constant corresponding to the homogenous line broadening with the value of 0.35^[24].

The mode-coupling process refers to the phenomenon in which the longitudinal modes compete for gain particles when multiple laser resonant modes run near the overlap area of the burning holes. In this process, different longitudinal modes compete for limited gain media, resulting in a significant suppression on side-mode oscillations and a simultaneous reduction of the laser output power. Based on the mode-coupling model, considering both the hole burning effect and the cross-relaxation effect, we can design the laser parameters for single-longitudinal-mode output with high monomode-power conversion efficiency. As well, we can also design the intensity-temperature tuning curve for the laser frequency stabilization.

Figure 4 shows the simulation result of the hole depths $d_{1(1)}$ and $d_{3(1)}$ of the side modes varying with $\mathrm{\Delta }\upsilon$ when the laser resonant modes approach the mode symmetry region, where $\mathrm{\Delta }\upsilon$ is the difference between the intermediate-mode frequency and the frequency $({\upsilon }_{01}+{\upsilon }_{02})/2$. In the simulation, the isotope shifting of ${\upsilon }_{02}-{\upsilon }_{01}$ is 950 MHz^[15,24], the Doppler linewidths $\mathrm{\Delta }{\upsilon }_{D1}$ and $\mathrm{\Delta }{\upsilon }_{D2}$ are, respectively, 1600 and 1525 MHz^[15,24]. Taking Fig. 4(a) as an example, the solid and dashed curves of the same color, respectively, represent the variations of $d_{1(1)}$ and $d_{3(1)}$ with $\mathrm{\Delta }\upsilon$ and the cavity loss $L_r$ under the same design parameters $(l,\varphi ,L,\mathrm{\Delta }{\upsilon }_L,R)$. It can be seen from the figure that when $L_r$ increases to 0.027, a frequency interval appears where both the values of the $d_{1(1)}$ curve and the $d_{3(1)}$ curve are zero. This result implies that the side modes are completely suppressed, and only the intermediate mode can oscillate under the cavity loss. Simulation results similar to those shown in Fig. 4(a) can be observed from Figs. 4(b) to 4(d) when the variable parameters are the capillary length $l$, the capillary diameter $\varphi$, and the cavity length $L$, respectively. At the regular 1 to Torr gas filling pressure of the low-power HeNe lasers, the pressure-dependent homogeneous linewidth $\mathrm{\Delta }{\upsilon }_L$^[26] only affects the size of the monomode interval, as shown in Fig. 4(e). Figure 4(f) shows that the influence of the abundance ratio of neon isotopes $R$ on the oscillations of the side modes is most significant, and the complete suppression of the side modes only occurs when $R$ is reduced to 1.

Figure 5 shows the variations of the output power $P_t$ of the semi-external-cavity HeNe laser with $\delta \upsilon$ from the excitation of a specified mode to its annihilation, where $\delta \upsilon$ is the difference between the specified mode frequency and the frequency $({\upsilon }_{01}+{\upsilon }_{02})/2$. In the simulation, $L_r=0.027$, $l=180\mathrm{mm}$, $\varphi =1.30\mathrm{mm}$, $L=270\mathrm{mm}$, $\mathrm{\Delta }{\upsilon }_L=140\mathrm{MHz}$, and $R=1$. During the process of $\delta \upsilon$ varying from $-504$ to 542 MHz, $P_t$ changes quasi-symmetrically with the power valley at $\delta \upsilon =20\mathrm{MHz}$, as depicted by the blue curve, which considers only the hole burning effect. The curve valley with frequency interval of 75 MHz in the green frame corresponds to the single-longitudinal-mode operation region of the laser, which is steep due to the overlap of the holes of the side modes and the intermediate modes. The part outside the frame corresponds with the dual-longitudinal-mode operation region of the laser. When $\delta \upsilon =-265$ or 289 MHz, $P_t$ reaches the maximum value and the laser outputs two modes with equal intensities. The red curve in the figure considers both the hole burning effect and the cross-relaxation effect. Compared with the blue curve, the power represented by the red curve is increased throughout the overall cycle, and the conversion efficiency for the monomode power is augmented from 0.7 to 0.9, while the mode distributions corresponding to feature points of the two curves are the same. The characteristics of the curves, including the high-finesse valley as well as the profile that contains no local depressions, are highly beneficial for the frequency stabilization scheme that takes power as the frequency discrimination signal.

Figure 6 shows the experimental setup of the proposed laser for the short-exposure interferometry. The central wavelength of the laser tube (QJHP-270, Aibo Laser, China) is 632.8 nm, and its equivalent cavity length is 270 mm. The effective length and diameter of the capillary of the laser are 180 and 1.30 mm, respectively. The pressure ratios of the gas mixture are ${\mathrm{P}}_{\mathrm{He}}:{\mathrm{P}}_{\mathrm{Ne}}=7:1$ and ${\mathrm{P}}_{\mathrm{Ne}20}:{\mathrm{P}}_{\mathrm{Ne}22}=1:1$. The laser is integrated into the mechanical assembly, as shown in the lower right corner of the figure. The output light is split by a BS (JING DA OPTIC, splitting ratio: T:R = 90:10). The principle of the frequency stabilization of the laser is demonstrated in Fig. 1. The transmitted light is coupled into a polarization-maintaining pigtail through a fiber collimator (MC Fiber Optics, connector type: FC/APC), and the reflected light is utilized as the reference light to stabilize the high-power monomode laser, which is received by a silicon PD (HAMAMATSU, spectral response range: 320–1100 nm). The light intensity signals from the PD are collected continuously by a data acquisition module (DAQM, ART Technology) and sent to the microprocessor. The temperature of the HeNe tube is controlled through a temperature control module (TCM, Yexian Technology, accuracy: $\pm 0.001°\mathrm{C}$) and a heater (Tengfeng Metallic Material, titanium wire, 0.3 mm diameter) wound on the tube. The longitudinal mode distribution of the HeNe output is obtained by a Fabry–Perot scanning interferometer (LEICE TECHNOLOGY, LCFPSI, center wavelength: 633 nm, FSR: 2 GHz, resolution: 10 MHz) and an oscilloscope (Tektronix, DPO2024, bandwidth: 200 MHz, sampling ratio: 1 GHz/s). The power of the laser is measured by an optical power meter (Ophir, PD300-UV, power range: 2 µW to 300 mW). The frequency stability is measured by a wave meter (HighFinesse, WS8-2, resolution: 2 MHz). The interferograms are recorded by a Fizeau phase-shifting interferometer (Interfero, Bysco 100), and the fiber pigtail of the laser source is connected to the interferometer host through an FC/APC connector.

Figure 7(a) shows the variation of the laser power throughout a complete tuning cycle as the tube temperature rises from 47.14°C to 47.61°C. The figure reveals an intensity tuning curve with a bimodal shape and a sharp valley. The valley of the curve corresponds to the single-longitudinal-mode operation area of the laser, and the peak of the curve corresponds to the laser output superimposed by two longitudinal modes with equal intensities. The variation trend is inconsistent with the simulation result. The actual power at the valley and peak of the curve obtained by the optical power meter are 2.44 and 2.5 mW, respectively, indicating that the monomode power conversion efficiency of the laser can reach 97.6%. Figure 7(b) shows the frequency changes of the intermediate mode as the temperature of the three-mode laser tube rises from 47.35°C to 47.39°C measured by the wavelength meter. Point A in the figure represents that the high-frequency side mode is nearly annihilated. Point B indicates that the laser works at the power valley during the temperature turning. Point C denotes that the low-frequency side mode is just about to oscillate. The size of the monomode interval is about 80 MHz. Figure 7(c) shows the mode distributions at different tube temperatures when laser output varies over half the tuning cycle, measured by the scanning interferometer. During this process, the longitudinal modes move with a decrease in frequency due to the thermal expansion of the resonant cavity. The laser output changes from a single longitudinal mode with ordinal number $q$ to multiple longitudinal modes and then to another single longitudinal mode with ordinal number $q+1$. It is worth noting that the longitudinal mode spectrum of the laser consists of three resonant modes when the tube temperature is 47.43°C or 47.60°C. The disparity between this observation and the simulation result may stem from the hysteresis effect^[25] or machining errors in the optical elements.

Figure 8(a) shows the frequency-locking process of the frequency-stabilized linearly polarized HeNe laser when the preheating of its tube is completed. The three curves depicted in the figure show the synchronized variations in the tube temperature, laser intensity $I_t$ measured by the PD, and the resonant frequency of the intermediate mode with time. Due to the presence of a high-fineness monomode valley on the intensity tuning curve, the intensity of the reference light can be adjusted to the power valley value in approximately 200 s by modifying the laser tube temperature. This action efficiently locks the operating longitudinal mode and stabilizes its frequency. Figure 8(b) shows the laser output variations with time after the tube preheating. The long-term tube temperature stability is shown by the purple curve if the operating temperature is not changed; the maximum fluctuation is $\pm 0.0015°\mathrm{C}$. The red curve and yellow curve show the synchronized changes of the intensity and the operating frequency, respectively. Due to the hysteresis problem in the frequency stabilization control system, the delayed response of temperature modulation to the laser power will cause the latter to deviate from the valley value gradually. Thereby, the frequency drift value reaches 10 MHz within the first 2 h when the ambient temperature is controlled within 22°C–24°C and 20 MHz during 5-h continuous operation. However, the frequency stability is still much better than that without power locking, which is represented by the green curve, and meets the measurement accuracy requirement of the interferometer in the 10-m-length testing cavity^[27,28].

Figure 9 compares the contrast of the interferograms generated by our proposed laser, the commercial frequency-stabilized HeNe laser (Thorlabs, HRS015B, 1.2 mW output). Figures 9(a) and 9(b) show the interferograms generated, respectively, by the two lasers, where the interferometer is disturbed by a motor with 4000 rotations per minute. Figure 9(c) shows the interferogram generated by the commercial laser in the nonvibrating environment. Compared with Fig. 9(b), the contrast of Fig. 9(c) can be maintained at a fairly high level, and there is no interferogram ambiguity phenomenon. The exposure time of Fig. 9(a) is set as 1 ms, and that of Fig. 9(b) is set as 2.5 ms. The difference is caused by the laser power of the two lasers. The red solid curve in Fig. 9(d) represents the intensity on Line_1 of Fig. 9(a), while the blue solid curve shows the intensity on Line_2. The figure also shows the corresponding contrast curves, which are calculated based on the Fourier transform analysis^[29]. Compared with the commercial laser, our power-enhanced laser can improve the interferogram contrast from about 0.26 to above 0.70 by shortening the exposure time.

In conclusion, we proposed a high-power and frequency-stabilized HeNe laser for short-exposure interferometry. A mode-coupling model for the semi-external-cavity HeNe laser was established. Both theory and experiments show that by optimizing the design parameters such as the abundance ratio of neon isotopes and the loss of the resonant cavity, the laser could form an intensity tuning curve with a high-finesse valley. When the three longitudinal modes were tuned to the mode symmetry region, the side modes were completely suppressed and the laser output lay exactly at the valley floor of the intensity tuning curve. A laser frequency stabilization device based on locating the power valley was constructed. A linearly polarized laser with the monomode power of about 2.5 mW and a frequency stability of $\pm 5\mathrm{MHz}$ (within 1 h) was achieved. The conversion efficiency for monomode power of the laser reached 97.6%. Compared with the commercial frequency-stabilized HeNe laser, this laser can improve the interferogram contrast from 0.26 to 0.70 in the vibration environment, thereby enhancing the antivibration performance of the interferometer. Since the thermal frequency stabilization system used for the HeNe laser is a typical hysteresis system, the delayed response of temperature modulation to the laser power will cause the latter to deviate from the valley value gradually. In our next work, we will focus on the improvement of the long-term frequency stability by optimizing the algorithm and thermal coupling structure of the control system.