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All Optical Tutorials on Thorlabs.comFabryPerot TutorialScanning FabryPérot InterferometersFabryPérot interferometers are optical resonators used for highresolution spectroscopy. With the ability to detect and resolve the fine features of a transmission spectrum with high precision, these devices are commonly used to determine the resonant modes of a laser cavity, which often feature closelyspaced spectral peaks with narrow line widths. Spatial Mode StructureClick to Enlarge Figure 1: Spatial mode structure for lowestorder TEM modes. Figure reproduced from Further Development of NICEOHMs.^{2} The most common configuration of a FabryPérot interferometer is a resonator consisting of two highly reflective, but partially transmitting, spherical mirrors that are facing one another. This type of resonator can be fully characterized by the following set of parameters:
Light waves entering the resonator through the input mirror will, depending on the mirror reflectivity, travel a large number of roundtrips between the two mirrors. During this time, the waves also undergo either constructive or destructive interference. Constructive interference reinforces the wave and builds up an intracavity electric field. This corresponds to the case where a standing wave pattern is formed between the resonator mirrors, which occurs when the resonator length L is equal to an integer multiple of half the wavelength, qλ/2. All other wavelengths that do not fit this criteria are not supported by the resonator and destructively interfere. By assuming a general Gaussian beam solution to the paraxial wave equation, it can be shown^{1} that only the following frequencies, ν_{qmn}, can exist inside the resonator: where q, m, and n are mode numbers that can take on positive integers or zero and c is the speed of light. The gparameters of the resonator g_{1,2} are given by where R_{1,2} > 0 for concave mirrors and R_{1,2} < 0 for convex mirrors. These frequencies are referred to as Gaussian transverse electromagnetic (TEM) modes of order (m,n), or the HermiteGaussian modes, and are typically denoted as TEM_{m,n}. The mode numbers m and n are associated with transverse modes, which describe the intensity pattern perpendicular to the optical axis, while q corresponds to the longitudinal mode. The TEM_{00} modes with m = n = 0 are called the fundamental TEM modes, or longitudinal modes, while TEM modes with m,n > 0 are called higherorder TEM modes. Figure 1 shows the spatial intensity pattern for a number of modes. Indices m and n correspond to the number of nodes in the vertical and horizontal directions respectively. For the case of light in the near infrared region, the parameter q is on the order of 10^{6}. The gparameters of a resonator are often found in the socalled stability criterion. A resonator is called stable when its gparameters fulfill the condition 0 ≤ g_{1}g_{2} ≤ 1.^{1} Transmission Spectrum of a FabryPérot InterferometerClick to Enlarge Figure 2: Mode spectrum of a FabryPérot interferometer for mirror reflectances of 99.7%, 80%, and 4%, illustrated by a blue, red, and green curve, respectively. 99.7% reflectance corresponds to the case for the SA200 series, which has a free spectral range of 1.5 GHz. The 4% reflectance corresponds to a typical "fringing effect" arising from reflections between parallel surfaces on glass plates. For the case where light is spatially mode matched to the fundamental TEM_{00} mode, i.e., when the wave fronts of the Gaussian beam perfectly match with the mirror surfaces and the incoming beam is aligned to the optical axis of the resonator, no higher order modes (m,n > 0) are evoked. The transmission spectrum of the resonator consists only of TEM_{00} modes that differ from each other by different values of the parameter q. The distance between two consecutive TEM_{00} modes ν_{q00} and ν_{q+1 00} is called the free spectral range (FSR) of the resonator and is given by This equation holds for all linear resonators comprised of two mirrors. The transmission intensity of a resonator, I_{t}, as a function of the frequency detuning from mode q, Δ_{q} = νν_{q}, is given by the wellknown Airyformula^{3} where I_{0} is the intensity of the light incident on the instrument and all other variables are as described above. Figure 2 to the right shows the typical transmission spectrum of a FabryPérot interferometer. From the above equation it can be seen that the onresonance transmission, T^{c}_{res}, is given by which also makes clear that the onresonance transmission not only depends on the transmission of a single mirror, but also on the mirror's reflection and loss coefficients, since all mirror coefficients are connected through t_{1,2} + r_{1,2} + l_{1,2} = 1 by definition. One always strives to keep absorption losses as small as possible to obtain maximum transmission for a given set of r_{1,2}. It can be seen from Equations (1) and (2) that the position of higher order transverse modes strongly depends on mirror spacing, L, and the radii of curvature of the mirrors, R_{1,2}. For the special case when the two mirrors have the same radius, i.e. R_{1} = R_{2} = R, and the distance between the mirrors is equal to the mirror radius, i.e. L= R, the resonator is called a confocal resonator. Figure 3 shows a typical ray trace for a beam entering the resonator parallel to the optical axis at a distance H. All Thorlabs FabryPérot interferometer models are based on such a confocal resonator design. For such a configuration, Equation (1) above simplifies to Two important conclusions can be drawn from this equation. First, all modes are degenerate, i.e., there exist higher order TEM modes that share the same frequency as the fundamental TEM_{00} (e.g. TEM_{q' 00}, TEM_{q'1 02}, TEM_{q'1 11}, TEM_{q'1 20}, TEM_{q'2 40}, TEM_{q'2 31}, TEM_{q'2 22}, … all share the same frequency). Second, the spectrum shows a regular, equidistant mode structure and the spacing between two consecutive modes is given by c/4L. If special care is not taken to obtain spatial mode matching, it is highly unlikely that higher order modes are suppressed. As a result, several higher order modes exist between two consecutive fundamental modes (TEM_{q00} and TEM_{q+1 00}) and the equidistant spacing between modes makes it appear that this FSR is equal to c/4L. To account for the existence of higher order modes, all values given for the FSR of the Thorlabs FabryPérot interferometers are referring to the socalled confocal free spectral range, ν_{FSR,conf} = c/4L. The arrows in Figure 3 highlight the difference between ν_{FSR} and ν_{FSR,conf}. With careful alignment along the resonator's optical axis and nearperfect spatial mode matching of the incident light, it is possible to extinguish every other mode in the spectrum. Figure 4 below shows a configuration with near perfect mode matching. Higher order modes still exist, but they are smaller than the fundamental modes. Further tweaking of the alignment will eventually distinguish every other mode in the spectrum. Click to Enlarge Figure 3: Schematic of a confocal FabryPérot resonator. Mirrors with radius R_{1 }=_{ }R_{2} (brown arrows) are separated by a distance L that is equal to the mirror radius. The solid green lines show a raytrace of an offaxis input beam entering the resonator at a height H. The dashed light green lines represent the beam transmitted through the second mirror; light transmitted through the first mirror is not pictured. Click to Enlarge Figure 4: Spectrum of a confocal resonator with near perfect spatial mode matching, where every other mode is extinguished as only the fundamental mode is excited. The TEM_{qmn} labels represent only one mode contained at that particular frequency; all modes are degenerate and, as described in the text, there are other modes sharing the same frequency. Finesse and Mode Width (Resolution)Click to Enlarge Figure 5: When two Lorentzian lineshapes are separated by their FWHM, they meet the Rayleigh criteria for being resolvable. A FabryPérot interferometer's performance, to a large extent^{a}, depends on the mirror reflectivity. Lowreflectivity mirrors will yield broader transmission peaks, while highreflectivity mirrors will yield narrower transmission peaks. The mirror reflectivity plays a significant role in how well the interferometer can distinguish features of the transmission spectrum. Beyond the free spectral range, two more important quantities of a FabryPérot interferometer are its finesse and mode width. The finesse, F, for mirrors having identical reflection coefficients, r, is given by For the case of a confocal interferometer, it can sometimes be convenient to express the finesse as An interferometer with a higher finesse will produce narrower transmission peaks than one with a lower finesse. Thus, a higher finesse increases the resolution of the interferometer, allowing it to more easily distinguish closely spaced transmission peaks from one another. According to the Rayleigh criterion (see Figure 5), two identical Lorentzian line shapes are resolvable when the peaks are separated by the full width at half maximum (FWHM) of each peak, denoted as Γ^{FWHM}. The FWHM mode width, also called resolution, is related to the finesse and the FSR of a confocal resonator by which is a measure of the minimum allowed spacing between two peaks to be resolved. For example, an interferometer with an FSR of 1.5 GHz and a finesse of 250 will have a FWHM of 6 MHz, and thus it will be able to distinguish features of the transmission spectrum as long as the peak values are at least 6 MHz apart. For visible light, this corresponds to a wavelength resolution of about 10 fm (10^{14} m). Further Considerations on the FinesseIn fact, a measured finesse has a number of contributing factors: the mirror reflectivity finesse, simply denoted F above, the mirror surface quality finesse F_{Q}, and the finesse due to the illumination conditions (beam alignment and diameter) of the mirrors F_{i}. The overall finesse of a system, F_{t}, is given by the relation^{4} Often, the reflectivity finesse, Equation (8), is presented as an effective finesse, which is true for the case when the other contributing factors are negligible. For Thorlabs' interferometers, the reflectivity finesse dominates when operating with proper illumination.^{b} The second term in Equation (10) involves, F_{Q}, which accounts for mirror irregularities that cause a symmetric broadening of the lineshape. The effect of these irregularities is a random positiondependent path length difference that blurs the lineshape. The manufacturing process that is used to produce the cavity mirror substrates always has to ensure that the contribution from F_{Q} is negligible in comparison to the specified total finesse of a resonator; in other words, the substrates surface figure should never be the limiting factor for the finesse. The final term in Equation (10), which deals with the illumination finesse, F_{i}, will reduce the resolution as the beam diameter is increased or as the input beam is offset. When the finesse is limited by the F_{i} term, the measured lineshape will appear asymmetric. The asymmetry is due to the path length difference between an onaxis beam and an offaxis beam, resulting in different mirror spacings to satisfy the maximum transmission criteria. To quantify the effects of the variable path length on F_{i}, consider an ideal monochromatic input, a delta function in wavelength with unit amplitude, entering the FabryPérot 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 a^{4}/R^{3}. 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 H^{4}/R^{3} (Ref. 6). Click to Enlarge Figure 6: The total finesse F_{t} as a function of beam diameter, 2H, for the SA200 and SA210 FabryPérot Interferometers using Equation (12). The finesse is calculated for a wavelength λ of 633 nm. Assuming that only F_{i} contributes significantly to the total finesse, then Eq. (9) can be used to calculate F_{i} for the idealized input beam. Substituting λ/4 for the FSR, and (H^{4}/R^{3}) for FWHM, yields: 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 oversized or misaligned beam. Now, the total finesse for the case of high reflectivity mirrors (r ≈ 1) can be found using Equation (10), which includes significant contributions from both F and F_{i} (Note: F_{q} is still considered to have a negligible effect on F_{t}): Equation (12) is used to provide an estimate, in general an overestimate, of beam diameter effects on the total finesse of a FabryPérot Interferometer, and several assumptions have been made. The first assumption 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, which also helps to reduce spherical aberration.^{5} A second 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 Equation (12) evaluated at 633 nm for the SA200 and SA210 FabryPérot Interferometers, which have cavity lengths of 50 mm and 7.5 mm respectively. The traces in the plot assume that the reflectivity finesse is equal to 250 for the SA200 and 180 for the SA210, which are typical values obtained for mirrors used in our interferometers. Cavity RingDown Time and Intracavity Power BuildUpAs a light wave travels many roundtrips inside the resonator, the light is stored inside for a certain amount of time, and only a small portion of its energy leaks out as it impinges on either the input or the output mirror. In other words, the light wave has a certain lifetime inside the resonator. This time is called the cavity ringdown time or cavity storage time, τ_{cav}, and is given by This relation can show that τ_{cav} increases with the cavity finesse, i.e., the higher the finesse and the mirror reflectivity, the longer light is stored inside the resonator. Inline with this, another important quantity is the socalled intracavity power buildup, defined by the ratio of the intracavity intensity, I_{c}, and the incident intensity according to which is given by F/π for a impedancematched cavity (i.e., with a vanishing onresonance reflection). This relates the intracavity intensity to the finesse by The fact that the power stored inside the cavity increases with finesse has to be kept in mind, when beams with high incident power are evaluated with a FabryPérot interferometer. Spectral Resolving Power and ÉtendueThe spectral resolving power of an interferometer is a metric to quantify the spectral resolution of an interferometer, and is an extension of the Rayleigh criterion. The spectral resolving power, SR, is defined as: where ν is the frequency of light and λ is its wavelength. It can be shown that for a confocal FabryPérot interferometer, the SR is given by: where 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 size of the aperture is increased, the spectral resolving power begins to decrease. The spectral resolving power must be balanced with the étendue of the interferometer. The étendue (U) is the metric for the net lightgathering power of the interferometer. When the light source is a laser beam, the étendue provides a measure of the alignment tolerance between the interferometer and the laser beam. The étendue 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 étendue is given by: where F is the finesse of the interferometer, λ is the wavelength, and L is the mirror spacing. The spectral resolving power and étendue need to be balanced for the interferometer to work correctly. 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) (Ref. 4). Under this condition the "ideal" étendue becomes π2λR/F, where R is the mirror's radius. For more information on our FabryPérot interferometers, including typical application examples, please see the full web presentation here. References
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