When an optical cavity is used to stabilize the frequency of a laser, fluctuations in the length of the cavity must be kept to a minimum.1 However, cavity length is itself a defining parameter: it impacts various performance parameters that, in turn, affect the stability of cavity length. One can intuitively suppose that the most critical of these parameters include thermal and vibration sensitivity, Brownian motion, Doppler effects, optical coupling, and the cavity's optical Q (a unitless ‘quality’ factor). As cavity size decreases, some of these parameters tend to improve, whereas others deteriorate.
The typical optical cavity shown in Figure 1 is a Fabry–Pérot etalon consisting of two optically coupled mirrors separated by a cylindrical elastic spacer. The spacer and the mirror substrates are often made of Ultra Low Expansion Glass (ULE®), although fused silica is sometimes used, which can offer slightly better performance. To reduce the effect of environmental perturbations, the entire assembly is housed in a thermal and vibration isolation enclosure,2 giving a mechanical fractional stability ∼10−12.
Figure 1. Concept of the isolated optical cavity.
Although the thermal and acceleration sensitivity of the resulting system are extremely low, residual environmental thermal noise can propagate to the cavity spacer, causing low-frequency instabilities.3 Thermal sensitivity (see Figure 2) increases with the cube of the reduction in length. And indeed, even with a zero coefficient of thermal expansion, stability at nonzero temperatures is fundamentally limited by mechanical thermal fluctuations in the cavity's constituent materials.2 Instability associated with Brownian motion can be calculated by applying the fluctuation dissipation theorem, using the mechanical losses of the cavity spacer, (finite-size) mirror substrate, and reflective coating materials. Thermal noise in the spacer is related to cavity length in the first order, whereas the relationship between cavity length and thermal noise in the mirror substrate and coating is more complex and involves the cavity waist, as discussed below.
Figure 2. Cavity thermal instability (frequency shift) in the laboratory environment.
Vibration/acoustic sensitivity is essentially independent of cavity length. Shorter cavities provide better vibration isolation but have a greater sensitivity to manufacturing tolerances, leading to an increase in noise. These two effects tend to cancel each other out. At frequencies close to the carrier, therefore, the crossover between vibration-limited and thermal-limited performance occurs at ∼5cm.
For adequate laser stabilization, the discriminator transfer function of the stabilizing control loop must be kept steep. This requires a narrow cavity linewidth. Linewidth represents a range of frequencies that propagate as a single cavity mode and is defined as Δf = FSR/finesse, where FSR is the cavity's free spectral range and finesse is a measure of the mirrors' reflectivity. Since FSR is inversely related to cavity length, shortening the cavity widens its linewidth.
Another parameter affecting optical cavity stability is the cavity waist size, which represents the minimum size of the cavity mode. As shown in Figure 3, the cavity waist is proportional to the square root of the cavity length. For efficient cavity coupling, the beam waist must match the cavity waist, whereas the spot size on the cavity mirrors (which determines energy density absorbed) will be slightly larger. Two factors related to the beam waist affect cavity length stability: optical power density (the sensitivity of the mirrors to fluctuations in laser power) and imprecision in the optical coupling into the cavity. As the cavity size decreases, the waist approaches the coupling tolerances and more power is absorbed, increasing stabilization system noise. In summary, small optical cavities remain a good choice for applications with stringent size and weight requirements. However, reducing their size beyond a certain point can be impractical, as thermal Brownian noise and other inescapable fundamentals of physics become limiting factors.
Figure 3. Optimum coupling beam waist, given cavity length.
The author wishes to acknowledge work in this field by George Gigioli, David Bope, and Marc Heffes of Northrop Grumman Corporation, and by Matthew Taubman of Pacific Northwest National Laboratory, and thanks them for many fruitful discussions.
NASA Goddard Space Flight Center
Edward Luzhansky is a physicist in the Laser and Electro-Optics Branch at NASA Goddard Space Flight Center, where he specializes in ladars and laser communications. He holds a master's degree in electrical engineering from the University of Connecticut and one in physics from the Institute of Fine Mechanics and Optics, St. Petersburg, Russia.
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