Fabry-Perot Interferometer

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1 Experimental Optics Contact: Maximilian Heck Ria Krämer Last edition: Ria Krämer, March 2017 Fabry-Perot Interferometer

2 Contents 1 Overview 3 2 Safety Issues Eye hazard Electrical hazard Theoretical Background The plane mirror resonator The confocal and concentric resonator The stability of an optical resonator Setup an equipment Equipment Practical procedure of the experiments Alignment of the scanning Fabry-Perot resonator with spherical mirrors Characterization of the resonator with spherical mirrors Alignment of the Fabry-Perot resonator using plane mirrors Goals of the experimental work Characterization of a scanning Fabry-Perot interferometer using spherical mirrors Characterization of the plane mirror Fabry Perot interferometer Additional task: Investigation of resonator stability A Preliminary questions 16 B Final questions 17

3 1 Overview The Fabry-Perot interferometer was invented in 1897 by Charles Fabry and Alfred Perot. In contrast to other, more conventional types like the Michelson or Mach-Zehnder interferometer, consists the Fabry-Perot interferometer of an optical resonator, which, based on multi-beam interference, can provide an extremely high spectral resolution power λ/ λ up to 10 7 for optical wavelengths λ. In this way, state-of-the-art Fabry-Perot cavities may exceed the resolution of classical diffraction gratings by a factor of 100 and provide an irreplaceable tool in particular for studies of the hyperfine structure in atomic spectra. The aim of this lab is to set up different Fabry-Perot resonator configurations and to characterize them. The resolution power of the different arrangements can be determined and they can be compared concerning alignment, stability and performance. Figure 1: Charles Fabry ( ), left, and Alfred Perot ( ), right, were the first physicists to construct an optical cavity for interferometry [1]. 2 Safety Issues 2.1 Eye hazard The HeNe laser used in this lab provides a cw output of 2.5mW. Please, use appropriate laser safety goggles in order to avoid damage to your eyes. It is recommended to discard any reflecting accessories like watches and jewelry. Do not look directly into the laser beam. Use the key switch at the laser power supply due to high voltage risks (20 kv). 2.2 Electrical hazard The piezo actuator is operated at 150V, its power supply may cause an electric shock. Do not remove the associated BNC cable from the back of the control unit PTC 1000! 3

4 3 Theoretical Background Fabry-Perot cavities make use of curved and plane mirrors as well, depending on their field of application. Whereas spherical mirrors are commonly employed for confocal schemes especially, the very basics of optical resonators are better studied by means of elementary plane surface reflections. We thus start with a brief description of that case and switch over afterwards to peculiarities of cavities consisting of spherical mirrors. 3.1 The plane mirror resonator We consider two plane mirrors with equal reflectivities 0 < R < 1 in a parallel arrangement separated by a distance d from each other [2]. A plane wave with an amplitude A i falls onto a mirror with an angle of incidence α. At each mirror surface, the beam is partially reflected (reflectivity R < 1) and transmitted. The scheme is sketched in Fig. 2. The phase difference φ between two adjacent beams A p and A p+1 depends on the mirror distance d and the refractive index n of the dielectric medium within the resonator: φ = 4π λ n d cos α. (1) We assume n = 1 for an air-filled resonator and an angle of incidence of α = 0 from now on. Considering the transmissivity T and the reflectivitiy R of the resonator mirrors, the amplitudes of the transmitted beams are A 1 = T A i, A 2 = TRe i φ A i, A 3 = TR 2 e i2 φ A i,..., A p = TR p e ip φ A i. (2) The output is a coherent superposition of all amplitudes A p with 1 p <, since an effectively infinite number of interfering waves is assumed for mirror reflectivities near 1. α A i A 1 A 2 A 3 A t R d R A 4 Figure 2: Multi-beam interference at two parallel mirrors with the reflectivity R, separated by the distance d. 4

5 The total transmitted amplitude A t results as A t = A p = T A i p=0 R p e ip φ = p=0 T 1 R e i φ Ai, (3) If there is no absorption, we have T + R = 1 and the transmitted intensity I t = A t A t is given as I t = I i (1 R) 2. (4) (1 R) R sin 2 φ 2 The intensity will be at its maximum for a phase difference of φ = 2πm, where m 1 is the order of interference. We can now normalize the transmitted intensity to the incident intensity I i : I t norm = It I i = ( ) 2 2 π F 2 R sin 2 φ 2, (5) where the parameter F R is the finesse defined by R F R = π 1 R. (6) The finesse also indicates the effective number of interfering beams within the resonator. Considering that the phase difference depends on the frequency ν (see Eq. (1)), Fig. 3 shows the normalized intensity for different values of the finesse in dependency on the frequency. The graph shows two resonance peaks, whereas the distance between two resonances is the so-called free spectral range δν. For plane mirrors, it can be calculated by using ν = c/λ with the vacuum speed of light c and Eq. (1): δν = c 2d. (7) The spectral resolution is defined by the full width at half maximum (FWHM) value of the peak. From an experimental point of view, the finesse describes the ratio between the free spectral range (FSR) δν and the spectral resolution ν of the instrument in the frequency domain: F = δν ν. (8) The finesse F R, calculated from the reflectivity of the mirrors, quantifies the optical quality of the resonator made of ideal perfectly polished and adjusted plane mirrors. However, in experiments, mirror surface irregularities will cause unwanted phase shifts at each reflection. 5

6 Figure 3: Normalized transmission of a Fabry-Perot resonator for various values of the finesse F 77 as a function of the frequency ν. A monochromatic light source is presumed. Their contribution to the total finesse F t may be denoted by F q [3]: 1 F t = ( 1 F R ) 2 ( ) (9) Since all light rays slightly diverge from their source due to the limited spatial coherence, Fabry-Perot interferometers always produce concentric interference rings rather than single on-axis spots (see Fig. 4a). From their radial intensity distribution, characteristic parameters of the setup may be obtained [2]. The rings are numbered, starting from the innermost ring, with p = 1, 2,.... From the ratio of the fringe separation δx p of ring p and p + 1 and the FWHM of peak x p the finesse can be calculated: F q F = δx p x p. (10) In order to observe the fringe pattern, the parallel transmitted light is focused to an image plane by a lens with a focal length f (see Fig. 4b). Thus, we get for small angles α p D p = 2 f tan α p 2 f α p. (11) 6

7 Fabry-Perot Interferometer (a) (b) Fabry-Perot resonator Image plane Lens αp αp Dp d Figure 4: (a) Concentric interference rings of a Fabry-Perot cavity with plane mirrors [2]. (b) Illustration of the imaging of the Fabry-Perot-Fringes on an image plane. From [2] we get α2p = λ (p 1 + e), d (12) where e is the fractional ring order at the center with e < 1. Combining Eq. (11) and (12) we get a linear dependence of the squared diameter D2p on the ring number p: λ D2p = 4 f 2 (p 1 + e). d (13) 3.2 The confocal and concentric resonator Now we consider a Fabry-Perot resonator consisting of two spherical mirrors. There are two common resonator configurations, the confocal resonator and the concentric resonator, which are both depicted in Fig. 5. In both cases, the two spherical mirrors have equal radii of curvature r. For the confocal resonator, the mirror distance d is equivalent to the mirror curvature (d = r), and for the concentric configuration, the distance is set to d = 2r. In contrast to the plane mirror resonator, where parallel off-axis beams receive the same optical path length as on-axis beams, the spherical mirrors cause various path lengths depending on the distance of the ray from the optical axis. The distance from the optical axis is denoted by the off-axis parameter ρ. For greater ρ, the optical path difference will be higher, having a negative impact on the finesse of the resonator, which is described by Fρ. Therefore, the 7

8 (a) r r (b) r r ρ ρ d = r d = 2r Figure 5: Geometry of (a) confocal and (b) concentric resonator arrangement and its beam path. total finesse is given by 1 F t = ( 1 F R ) 2 ( ) 2 ( F q F ρ ) 2 with F ρ = λ 4r 3 4 ϱ. (14) 4 The finesse F R is given by Eq. (6) and F q is the contribution of the mirror irregularities. The free spectral range and the spectral resolution (see Eq. (8)). Since the free spectral range depends on the optical path length within the resonator, it is calculated for the confocal and concentric arrangement by δν conf. = c 4d, δν conc. = c 2d. (15) 3.3 The stability of an optical resonator The stability of an optical resonator depends on the geometry of the system. Aside from confocal and concentric arrangements, several other configurations allow stable operation. The general condition for a stable cavity is given as [3] 0 g 1 g 2 1 with g i = 1 d r i (16) for two mirrors with the radii of curvature r 1 and r 2 with the stability parameters g i. This equation is illustrated in Fig. 6. Resonators are stable, if their stability parameters lie within the gray area. Resonators, which parameters lie on the black line, are still stable, however small disturbances can cause the resonator to become instable. 8

9 4 Setup an equipment 4.1 Equipment With an output power of 2.5 mw, the He-Ne laser is classified as a class 3B laser. The laser wavelength is around nm, whereat actually two longitudinal modes with a small wavelength difference are amplified within the gain curve of the He-Ne laser. The two modes are linearly polarized in perpendicular directions. Three pairs of mirrors are provided, two spherical sets with radii of curvature r of 75 mm and 100 mm, respectively; and one set of plane mirrors. The reflectivity of the mirrors is 96% for each set. Please, do not touch the mirror surfaces! The coatings are extremely damageable and expensive. Handle the mirrors with care and wear single-use gloves whenever you mount and replace the mirrors. For cleaning - even from dust etc. - special one-way lens tissues must be used. Ask your supervisor! In each mirror set, one of them is mounted on an axial piezo translator. The piezo ceramic actuator is driven by a maximum voltage amplitude of 150 V and provides an open loop sensitivity of approx nm for 5 mv noise and a maximum stroke of 8 µm. The length of the actuator is 26 mm. The piezo actuator may be controlled with respect to the amplitude, an offset and the frequency when moved back and forth according to a periodical delta voltage. Figure 7 illustrates the front panel of the control unit PTC The high voltage connector (BNC) for the piezo is located on the back, as well as the piezo voltage monitor and the trigger signal output. Figure 6: Stability of optical resonators consisting of two mirrors with the radii of curvature r 1 and r 2 separated by the distance d. The stable regions are gray underlayed and include the zero point. 9

10 Figure 7: Front panel of the control unit PTC In the "photo amplifier" unit, the gain may be varied within 0.1 and The frequency of the piezo triangle voltage can be set between 50 Hz and 100 Hz. [4] The output signal of the silicon photo diode (Siemens BPX 61) is amplified using the control unit PTC The device has an active area of 2.65 x 2.65 mm 2 which is sufficient to collect the unfocused laser light (FWHM 1 mm) without excessive losses. Its spectral range of sensitivity is given as 400 nm λ 1100 nm, with a maximum around 850 nm. Although all measurements can be performed under daylight conditions, no direct sunlight or artificial room illumination should hit the photo diode in order to ensure an optimized contrast! We use a Tektronix TDS 2012B oscilloscope with a bandwidth of 100 MHz. The amplified signal of the photo diode should be displayed on one channel, the second channel is reserved for the piezo triangle voltage. The oscilloscope is triggered externally by the control unit PTC Screen shots can be saved by using the oscilloscope software on the PC and the " get the screen" function. A polarizer mounted in a rotation stage is provided for measurements of the mode spectrum of the HeNe laser. For beam expansion and imaging there are different lenses available: two convex lenses with focal lengths of f = 60 mm, f = 150 mm and one concave lens with f = -5 mm. The CMOS camera (Thorlabs DCC1545M) has a resolution of 1280 x 1024 pixels with a pixel size of 5.2µm x 5.2µm. 10

11 Figure 8: Basic setup of the Fabry-Perot resonator: (1) optical rail, (2) laser with mounting, (3) laser power supply, (4) fixed resonator mirror, (5) piezo-driven resonator mirror, (6) control unit PTC 1000, (7) photo diode, not displayed: polarizer. (A) marks the adjustment screws of one of the mirrors, (B) marks the knob for distance adjustment. [4] 4.2 Practical procedure of the experiments Alignment of the scanning Fabry-Perot resonator with spherical mirrors A thorough alignment of the resonator components along the optical axis is essential for successful measurements. In Fig. 8 an overview of the basic resonator setup is shown. It consists of the laser, the two resonator mirrors and the photo diode. Follow the following steps for the alignment of the resonator: Remove all tabs from the optical rail (1), except for the laser (3). Mount now the piezo-driven resonator mirror on the tab, make sure to place on the pinion gear. The clamping screw has to be slightly loosened in order to use the distance adjustment. Align now the back reflection using the adjustment screws (A). In order to align the left resonator mirror (4), first, place it the other way around (pointing towards the laser instead of the second resonator mirror), and adjust the back reflection. Then, turn it around to normal position. The mirror distance should be coarsely set. Mount the photo diode (7) at the right end of the optical rail. The signal of the photo diode is put on channel 1 of the oscilloscope. On the control unit (6), the photo 11

12 (a) (b) piezo voltage Mode 1 photo diode 50% τ Mode 2 Figure 9: Oscilloscope display for two different cases: (a) controller switch on (no laser light reaches the photo diode). (b) Observable traces for an aligned resonator with a high finesse. [4] δτ amplifier should be set to "DC coupling", the gain factor should be set to 50, and the gain variation to a center position. Switch on the piezo translator by setting the amplitude on the control unit (6) to a medium value. Frequency and offset should be set to a medium value, and the profile is set to a triangle function. The piezo voltage is put on the second channel of the oscilloscope. Slightly re-align, if needed, the back reflection of the resonator at the laser output. Now optimize your alignment using knob (B) for fine tuning the mirror distance and (A) for the angle of the mirrors until you reach the highest finesse. An ideal result is shown in Fig. 9. The amplitude of the piezo translator should be adjusted so that on the oscilloscope only two sets of resonance peaks (consisting each of two modes) are within the rising slope of the piezo voltage. In case of any clipping of the resonance peaks, reduce the gain variation Characterization of the resonator with spherical mirrors For the characterization of the resonator, the distance of the interference peaks as well as the FWHM of the peaks has to be measured (see Fig. 9). This is done by using the CURSER function of the oscilloscope. For reference, save the image of the oscilloscope screen showing the measurement results (use the "get screen" function on the computer). Do the same for the measurement of the piezo expansion rate. For the investigation of the modal spectrum of the laser use the provided polarizer, and mount it on the optical rail between laser (3) and the first resonator mirror (4). Take care, during the warm-up time of the laser tube, the mode spectrum of the HeNe emission varies. 12

13 Figure 10: Setup for the plane mirror resonator: (1) optical rail, (2) laser with mounting, (3) laser power supply, (4) fixed resonator mirror, (5) piezo driven resonator mirror, (6) control unit PTC 1000, (8) & (9) beam expander, (10) focusing lens, (8) CMOS camera, not displayed: polarizer. [4] In particular, the relative intensities of the closely neighbored lines can oscillate on time scales of several seconds. You should wait with your measurements until the equilibrium is reached Alignment of the Fabry-Perot resonator using plane mirrors For the plane mirror resonator the setup needs to be adjusted. To allow for an easier alignment, the beam needs to be expanded. Also, in order to measure the Fabry-perot fringes, a camera instead of the photo diode is used. In Fig. 10 the setup for the plane mirror resonator is shown. The following steps are required for alignment: Remove all tabs from the optical rail (1), except for the laser (2). Expand the beam, building a telescope using a concave lens (8) ( f = -5 mm) and a convex lens (9) ( f = 150 mm) with a distance of 145 mm. By adjusting the distance of both lenses, make sure the output beam has a constant diameter. Compare the beam diameter directly behind the telescope and at a further distance (at the wall). Now mount and align the resonator mirrors similar to the resonator with spherical mirrors: mount first the piezo-driven mirror (5), align the back reflection. Afterwards, 13

14 mount the left resonator mirror (4) the other way around, and again align the back reflection. Set the mirror in normal position, optionally, correct the back reflection. Set the mirror distance to a value around d = 25 mm. Mount the CMOS camera (11) at the end of the optical rail, and mount the focusing lens ( f = 60 mm) (10) in front of it. Set the distance to approximately to the focal length of the lens. Using the "uc480" software, display the camera image on the computer monitor. By adjusting the mirror angle (adjustment screws (A)), adjust the resonator so that you can see a concentric fringe pattern on the screen. The position of the focusing lens in front of the camera might need some further adjustment. Set the polarizer into the laser beam (between beam expander optics (9) and the resonator (4)) and adjust the angular position so that you see only one mode (the two modes should have different ring diameters). Try to optimize the fringe pattern in symmetry, sharpness and interference contrast. For alignment, you can use the option for visualizing horizontal and vertical image cross section of the camera software. Finally, save the image for further analysis. Fig. 11 shows an example of a recorded camera image and the respective cross section. Figure 11: Analysis of the Fabry-perot fringes: (a) recorded camera image, (b) cross section through the center of the ring system reveals the peaks and minima. 14

15 5 Goals of the experimental work The experiment consits of two different parts: first, a scanning Fabry-Perot resonator using spherical mirrors is set up and characterized using a photo diode. In the second part, a Fabry-Perot resonator consisting of two plane mirrors is built and the Fabry-Perot fringes are observed and analyzed. The following mirror arrangements are used: (a) r = 75 mm, d = r, (b) r = 100 mm, d = r, (c) r = 100 mm, d = 2r, (d) r = inf, d = 25 mm, 5.1 Characterization of a scanning Fabry-Perot interferometer using spherical mirrors 1. Set up and align the Fabry-Perot interferometer using arrangement (a). Determine the polarization dependence of the modes. Take screen shots of three different angular positions (do not forget to write down the positions!), documenting the extremal settings (both modes visible, only one mode (each!) visible). 2. Determine the piezo expansion rate d/ V. Measure the voltage difference V of the piezo for two adjacent resonance peaks of the same laser mode. The distance d can be obtained from Eq. (1). Compare with the theoretical value obtained from maximum stroke and maximum voltage amplitude of the piezo actuator. 3. Measure the time distance δτ Mode between the two modes. 4. For the determination of the characteristics of the Fabry-Perot interferometer, measure for each configuration (a),(b) and (c) the time distance δτ between two adjacent peaks of the same laser mode and the FWHM τ of one of the peaks. Don t forget to measure the actual mirror distance d meas after alignment! Calculate for each configuration (a),(b) and (c) the finesse F, the free spectral range δν and the spectral resolution ν. Use the relation δτ/ τ = δν/ ν. Calculate the wavelength separation δλ Mode of the two longitudinal modes of the He-Ne laser. Use the relations δτ mode /δτ = δν mode /δν, λ/δλ mode = ν/δν mode, and c = λν. 5.2 Characterization of the plane mirror Fabry Perot interferometer 1. Set up and align the plane mirror resonator. Use a mirror distance of d = 25mm. Add the camera to the setup in order to observe the Fabry-Perot fringes. Use the polarizer to block one mode and to optimize your fringe pattern. Record an image of 15

16 the Fabry-Perot fringes. Don t forget to measure the actual mirror distance d meas of the mirrors! Calculate the mirror distance d from the fringe pattern: Determine the ring diameter D p in mm of four to five rings. For this, use an appropriate software (e.g. ImageJ, Matlab,...) and extract the intensity distribution for a cross section (horizontal or vertical) through the center of the rings from the recorded image in order to reveal the intensity distribution and position of the rings. Plot D 2 p over the ring number p and perform a linear fit of the data. Using Eq. (13), the mirror distance d can be calculated. Determine the finesse F using Eq. (10), as well as the free spectral range δν and the spectral resolution ν. 5.3 Additional task: Investigation of resonator stability 1. Investigate the stability of other mirror/distance combinations such as (e) r = 75 mm, d = 2r, (f) r 1 = 75 mm, r 2 = 100 mm, d = (r 1 + r 2 )/2, (g) r 1 = 75 mm, r 2 = 100 mm, d = r 1 + r 2. Set up and align the resonator and verify its stability. Take a screen shot of the oscilloscope showing the resonances. Do not forget to measure the actual mirror distance d. Calculate the stability parameters g 1 and g 2. A Preliminary questions What is multiple-beam interference and how does a Fabry-Perot interferometer work? What are the characteristic parameters? Why is the surface roughness of the mirrors limiting the finesse of the Fabry-Perot interferometer? What is the influence of the mirror distance d on the spectral resolution and free spectral range? Are other configurations apart from the confocal and concentric resonators possible (stability criterion)? What is the influence of the beam diameter in the spherical mirrors setup on the finesse? Why is a polarizer used in the experiments? Include the following calculations also into your result part of your report! 16

17 Calculate the theoretical finesse F R,calc,, free spectral range δν calc and spectral resolution ν calc and λ calc for the resonator arrangements listed under section 5. Calculate the stability parameters for the resonator arrangements listed under section 5. B Final questions (To be answered within either the theory or discussion part.) What is the advantage of a Fabry-Perot Interferometer in comparison to a Michelson or a Mach-Zehnder interferometer? What is the meaning of the FSR? Compare the performance of the resonator arrangements used in the experiments concerning the finesse, free spectral range, resolution and stability. Are there deviations to the theoretical values? References [1] "Optoelectronics Photographs from Textbook", photonics.usask.ca/photos (2017). [2] M. Born, E. Wolf, "Principles of Optics", 6th (corrected) Edition, Pergamon Press (1984). [3] W. Demtröder, "Laser Spectroscopy Vol. 1: - Basic Principles", 3rd Edition, Springer (2003). [4] MICOS GmbH, "Laser Education Kit CA-1140 Fabry Perot Resonator", MICOS User Manual, Eschbach (2009). 17

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