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Fabry-Perot Interferometer Tutorial
Scanning Fabry-Perot Interferometers
The core of Thorlabs' scanning Fabry-Perot interferometers is an optical cavity formed by two, nearly identical, spherical mirrors separated by their common radius of curvature as shown in Figure 1. This configuration is known as a mode-degenerate, confocal cavity design, which is generally referred to as a confocal Fabry-Perot. The cavity is mode degenerate because the frequency of certain axial and transverse cavity modes are the same (degenerate). This degeneracy greatly simplifies the alignment of the instrument by eliminating the need to carefully mode match the input to the cavity. The confocal design offers several benefits over flat-plate interferometers such as easier alignment since the confocal interferometer is fairly insensitive to angular alignment. Additionally, the confocal design offers a unique property that at constant finesse as resolving power increases, so does the etendue (etendue is defined as the radiation from a source within a solid angle, Ω, subtended by an aperture with area A). By contrast, in a flat-plate interferometer as resolving power increases (at constant finesse) the etendue decreases; meaning that an increase in light intensity decreases resolution and vice-versa.
The inner concave surface of each mirror has a highly reflective coating, while the outer convex surface has a broadband antireflection coating. The curvature on the outer surface, which matches that of the inner surface, eliminates lensing effects. In a confocal system, the mirror spacing matches the curvature of radius (r) as depicted in Figure 1. To illustrate the operation of this confocal cavity, it is useful to follow a ray as it enters the cavity off-axis and travels one round trip through the cavity. As seen in Figure 2, the beam enters the cavity at a height H. A portion of the ray follows the path numbered 1, 2, 3, and 4; it is then reflected onto path 1 again. The dotted lines exterior to the cavity in Figure 2 represent the portion of the ray that is transmitted through the cavity mirrors when the cavity is resonant with the input (i.e., when the round trip distance equals mλ, where m is an integer, and λ is the wavelength of the input radiation). The approximate optical path length (L) of one round trip through the cavity can be expressed as
L = 4r
Resonance and Free Spectral Range (FSR)
In order to achieve a maximum in resonance from a Fabry-Perot cavity, the complete round-trip phase delay must be a multiple of 2π. For a plano-plano Fabry-Perot cavity with a round trip distance of 2r, this condition is satisfied when the frequency is mc/2r, were m is any integer, c is the speed of light in air, and r is the mirror separation. Therefore, the separation, or Free Spectral Range (FSR) between two transmission peaks is c/2r.
For a confocal Fabry-Perot cavity, we must take into account that the modes of the cavity are Gaussian. By taking into account the phase shift of a Gaussian mode in the confocal cavity, it can be shown  that resonance frequencies of the transverse modes either overlap or fall exactly halfway between the longitudinal mode resonances. Therefore, the FSR for a confocal cavity the free spectral range is c/4nd.
Total Corrected Round Trip Distance
Because the actual optical path length of the confocal Fabry-Perot cavity is dependent on H, the resonance condition will vary across the input beam. This variation across the input aperture is a critical practical consideration when using a confocal Fabry-Perot Interferometer.
In order to develop an equation that relates the resolution of the interferometer to H, the geometrical optical path length with a correction for spherical aberration must be considered. Doing so with the approximation that 0 < H << r, yields an optical path length of
L = 4r - H4/r3
As the input beam diameter increases, the second term in Eq. (2) becomes significant in comparison to λ.
Finesse and Resolution of a Confocal Cavity
For a Fabry-Perot cavity, the finesse is a measure of the interferometer's ability to resolve closely spaced spectral features. The minimum resolvable frequency increment of an interferometer is based on the Rayleigh Criterion, which stipulates that for two closely spaced lines of equal intensity to be resolved, the sum of the two individual lines at the midway point can at most be equal to the intensity of one of the original lines (see Figure 3).
The total finesse of an interferometer is defined as the ratio of the FSR to the FWHM of the resonant peak, where Δ is the FWHM. As can be seen in Figure 3, two lines separated by Δ are just resolvable according to the Rayleigh criterion. Therefore, Δ quantifies the resolution of the system.
The equation for the total finesse is given by
Ft = FSR/Δ
Note that during the manufacturing of the SA200 series interferometers, Ft is maximized in order to adjust the cavity length to the confocal condition by maximizing its value. This method provides a very precise means for setting the required length of the cavity to better than λ.
The FSR and the FWHM of a representative lineshape are shown in Figures 4 and 5, respectively. An Ft of 294 is measured using a DFB laser with a linewidth that cannot be considered infinitely small in comparison to the resolution of the cavity. Therefore, the true Ft is about 320, assuming a 2 MHz laser linewidth.
A measured finesse has a number of contributing factors: the mirror reflectivity finesse FR, the mirror surface quality finesse Fq, and the finesse due to the illumination conditions (beam alignment and diameter) of the mirrors Fi. Therefore, the total inverse of the finesse of a system can be written as
1/Ft = [(1/FR)2 + (1/Fq)2 + (1/Fi)2]1/2
where, for mirrors with a reflectivity close to 1, the effective mirror reflectivity finesse is given by
FR = pi*R/(1-R2)
Here, R is the mirror reflectivity.
While the definition for the reflectivity finesse is ambiguous, Eq. (5) is presented as an effective finesse that is defined by Eq. (3) when the other contributing factors are negligible. For the SA200 series, the reflectivity finesse dominates when operating with proper illumination .
Using Eq. (5), the reflective coatings in the SA200 series of interferometers have been designed so that the minimum FR is better than 1.5 times the minimum specified finesse across their entire operating wavelength range for each model (see the table on the product page). This fixes the first term of Eq. (4).
The second term in Eq. (4) involves Fq, which accounts for mirror irregularities that cause a symmetric broadening of the lineshape. The effect of these irregularities is a random position-dependent path length difference that blurs the lineshape. The manufacturing process that is used to produce the cavity mirrors ensures that the contribution from Fq is negligible in comparison to our specified total finesse for each model.
The final term in Eq. (4), which deals with the illumination finesse Fi, will reduce the resolution as the beam diameter is increased or as the input beam is offset. When the finesse is limited by the Fi term, the measured lineshape will appear asymmetric. The asymmetry is due to the path length difference between an on-axis beam and an off-axis beam, resulting in different mirror spacings to satisfy the maximum transmission criteria. The approximate decrease in path length for a beam at a distance H off axis is given by the second term in Eq. (2).
To quantify the effects of the variable path length on Fi, consider an ideal monochromatic input, a delta function in wavelength with unit amplitude, entering the Fabry-Perot cavity coaxial to the optic axis and having a beam radius a. The light entering the interferometer at H = +e, where e is infinitesimally small but not zero, will negligibly contribute to a deviation in the transmitted spectrum. Light entering the cavity at H = +a will cause a shift in the transmitted output spectrum, since the optical path length of the cavity will be less by an approximate distance of a4/4r3. Assuming the input beam has a uniform intensity distribution, the transmitted spectrum will appear uniform in intensity and broader due to the shifts in the optical path length. As a result, the wavelength input delta function will produce an output peak with a FWHM of H4/4r3.
Assuming that only Fi contributes significantly to the total finesse, then Eq. (3) can be used to calculate Fi for the idealized input beam:
Fi = FSR/FWHM
Substituting λ/4 for the FSR, and (H4/4r3) for FWHM, yields
Fi = (λ/4)/(H4/4r3)
The λ/4 substitution for the FSR is understood by considering that the cavity expands by λ/4 to change from one longitudinal mode to the next. For an input beam with a real spectral distribution, the effect of the shift will be a continuous series of shifted lineshapes.
It should be noted that the shift is always in one direction, leading to a broadened or assymmetric lineshape due to the over-sized or misaligned beam.
Now, using Eq. (4), the total finesse, which includes significant contributions from both FR and Fi can be found (Note: Fq is still considered to have a negligible effect on Ft):
Ft = [ (1/FR)2 + (1/Fi)2 ]-1/2
Replacing Fi and FR yields:
Ft(H, R) = [ ((1-R2)/piR)2 + (H4/λr3)2]-1/2
Eq. (8) is used to provide an estimate (albeit an overestimate) of effects of beam diameter effects on the total finesse of a Fabry-Perot Interferometer. Several assumptions lead to the overestimation of finesse. One is that the diameter of the beam is the same as the diameter of the mirror, in practice the diameter of the beam is typically significantly smaller than that of the mirror (this also helps to reduce spherical aberration) . Another assumption is that the light is focused down to an infinitesimally small waist size, even for monochromatic light the minimum waist size is limited by diffraction, and in multimode applications the waist size can be quite large at the focus. Figure 6 provides a plot of Eq. 8 for the two cavity designs offered (r = 50 mm and r = 7.5 mm). The traces in the plot were made with the assumption that the reflectivity finesse is equal to 300, which is the typical value obtained for mirrors used in the SA200 series interferometers.
Spectral Resolving Power and Etendue
The spectral resolving power of an interferometer is a metric to quantify the spectral resolution of an interferometer, and is an extention of the Rayleigh criterion. The spectral resolving power, SR, is defined as:
SR = v/Δv = λ/Δλ
In Equation (9), v is the frequency of light and λ is its wavelength. It can be shown that for a confocal Fabry-Perot interferometer, the SR is given by:
SR = 4rF/λ
In Equation (10), F is the finesse of the interferometer, r is the radius of curvature of the mirrors, and λ is the wavelength. However, to achieve this maximum instrumental profile while the interferometer is in scanning mode, the aperture of the detector would need to be infinitesimally small; as the aperture is opened wide enough, the spectral resolving power begins to decrease. The spectral resolving power must be balanced with the etendue of the interferometer. The etendue (U) is the metric for the net light-gathering power of the interferometer. When the light source is a laser beam, the etendue provides a measure of the alignment tolerance between the interferometer and the laser beam. The etendue is defined as the product of the maximum allowed solid angle divergance (Ω) and the maximum allowed aperture area (A). For the confocal system the etendue is given by:
U = pi2λd/F
In Equation (11), F is the finesse of the interferometer, λ is the wavelength, and d is the mirror spacing. For proper use of the interferometer the spectral resolving power and etendue need to be balanced such that enough light is allowed to enter the system without significantly reducing the resolution of the interferometer. The accepted compromise for this balance is to increase the mirror aperture until the the spectral resolving power is decreased by 70% (0.7*SR) . Under this condition the "ideal" etendue becomes π2λr/F, where r is the mirror's radius.