[Basic Optics Skills] Summary of key points in resonant passive cavity design

If you start designing a resonant passive cavity, the following factors need to be considered:

(1) Cavity shape

According to the classification of cavity shape, it can be divided into linear cavities and annular cavities. The most familiar two-mirror F-P cavity is the linear cavity. On this basis, adding cavity mirrors can expand the cavity structure, as shown in Figure 1 below:

Figure 1 Comparison of two cavity structures

The biggest difference between the two structures is that light propagates in a linear cavity like a "reentrant run", and the distance it takes to circulate in the cavity is twice the length of the physical cavity; while light travels in a ring "circle", and the distance it takes to circulate in the cavity is equal to the length of the physical cavity. Compared with ring cavities, linear cavities have the advantage of being more compact, but there is a problem of light return at the input mirror. How to choose the cavity structure depends mainly on the specific application.

(2) Cavity length

The cavity length determines the free spectral region (FSR) of the cavity: FSR=c/nL, where c is the speed of light, n is the refractive index, and L is the distance for one cycle of light in the cavity. Considering different cavity shapes, for the cavity determined by FSR, the required length of the linear cavity is half of that of the annular cavity. It is worth noting that if the light entering the cavity is pulsed light with a repetition frequency of f, the FSR of the cavity needs to be equal to f (or an integer multiple), that is, the time interval of the input pulse is equal to the time of one cycle of transmission in the cavity (or an integer multiple), so as to ensure that the previous pulse exactly coincides with the next cavity-entering pulse after one cycle in the cavity.

(3) Stable cavity conditions

The stability of the cavity refers to the paraxial light traveling back and forth in the cavity multiple times without escaping laterally. It is the basic condition for maintaining the stable resonance state of the cavity. The stability conditions of the resonant cavity can be expressed by the light transmission matrix, which satisfies:

The writing rule of the transmission matrix is: first select a starting point arbitrarily in the cavity, and multiply the transformations (including free space and concave mirrors) that have traveled through the cavity for a week in the form of a matrix from right to left. Among them, the transformation matrices passing through the free space of length l and the concave mirror with a radius of curvature R are respectively. Stability conditions can be used to guide the determination of the radius of curvature of a concave mirror and the setting of the distance between optical elements.

(4) Fineness

The concept of fineness originates from the F-P cavity. It is an important parameter used to characterize the spectral resolution capability of the resonator, and is generally represented by F. When the FSR (i.e., cavity length) of the cavity is constant, the higher the fineness, the better the performance of the cavity. However, excessive fineness also places higher requirements on the performance of the control system, environmental noise isolation, and cavity mirror coating damage threshold. Therefore, the precision of the cavity should be estimated based on the specific application scenario.

Taking the two-mirror F-P cavity as an example, under ideal conditions where the intra-cavity loss only comes from the imperfect reflectivity of the cavity mirror, that is, without considering the scattering, absorption and other losses of the mirror, the precision of the cavity can be estimated as:

, where R is the reflectivity of the cavity mirror. In actual situations, fineness is affected by many factors and can be effectively evaluated by cavity ring-down.

Design example:

According to our application requirements: use a passive cavity to filter and analyze the noise of a femtosecond pulse light source with a repetition frequency of 75 MHz. We designed an eight-mirror annular cavity with a cavity length of 4m and a precision of about

1500. The actual photo is shown in Figure 2 below:

Figure 2 The actual image of the eight-mirror annular cavity

Next, we analyze the eight-mirror annular cavity based on the above design points.

First of all, we choose the ring cavity mainly based on the following considerations: the linear cavity will have the problem of light return on the input mirror, which will affect the normal operation of the laser, so an optical isolator is necessary. However, the dispersion effect introduced by the optical isolator will cause the pulse time domain to broaden, so we use the structure of a ring cavity.

Since the repetition frequency of the input light source is 75 MHz, according to the requirements in design points (2), we set the total cavity length to about 4m, and the optimal resonant cavity length can be found through the electric translation stage in Figure 2.

After the cavity length is determined, we then select the appropriate concave mirror and the distance between them based on the stable cavity conditions. Here, except for the bottom pair of concave mirrors with a radius of curvature of 6m (if there is a nonlinear crystal in the cavity, a concave mirror with a smaller radius of curvature needs to be used to obtain a smaller spot size, thereby improving the nonlinear conversion efficiency), the other six are plane mirrors. The transmission matrix for a complete circle of light traveling in the cavity is:

where L is half the length of the cavity, R is the radius of curvature of the concave mirror, and z1 represents the distance between the two concave mirrors. When L=2m, R=6m, the curve of stability condition 1/2 (A+D) changing with z1 is shown in Figure

3. It can be seen from the figure that when z1 changes in (0, 1), the stable cavity condition is satisfied. According to the space limitations of the experimental platform and the consideration of as compact a structure as possible, we set z1 to 0.5m.

Figure 3 The cavity stability condition changes with the distance z1 between the two concave mirrors

Finally, determine the precision of the cavity, that is, determine the reflectivity of the input and output mirrors. According to the fineness estimation formula F≈π/(1-R) ​​in point (4), if you want to obtain a fineness of 1500, the reflectance should reach 99.8%. Considering that the eight-mirror cavity has a large number of mirrors besides the input and output mirrors, and the reflectivity cannot reach 100%, we set the reflectivity of the input and output mirrors to 99.9%. Under this condition, the precision of the measured cavity is 1467.

Passive cavities involve many related technologies, including optical, electrical, mechanical and other aspects. This article only starts from the perspective of general principles and briefly introduces several issues that need to be considered when designing passive cavities. In the future, I will also discuss some issues related to passive cavities based on my own experimental experience, such as cavity-optical mode matching skills, comparison of several different cavity locking technologies, etc.