Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Experiments on the Porch Swing Bearing of Michelson Interferometer for Low Resolution FTIR

Experiments on the Porch Swing Bearing of Michelson Interferometer for Low Resolution FTIR Hindawi Publishing Corporation Advances in Optical Technologies Volume 2013, Article ID 948638, 9 pages http://dx.doi.org/10.1155/2013/948638 Research Article Experiments on the Porch Swing Bearing of Michelson Interferometer for Low Resolution FTIR Tuomas Välikylä and Jyrki Kauppinen Department of Physics and Astronomy, University of Turku, 20014 Turku, Finland Correspondence should be addressed to Tuomas Vali ¨ kyla; ¨ tuomas.valikyla@utu.fi Received 22 March 2013; Accepted 8 May 2013 Academic Editor: Joseph Rosen Copyright © 2013 T. Vali ¨ kyla¨ and J. Kauppinen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Porch swing bearing for the linear motion of the mirror in Michelson interferometer for mid-infrared low resolution Fourier transform spectrometer was studied experimentally using the modulation depth of the collimated laser beam. eTh mirror tilting was measured to be lower than 5 𝜇 rad over 3 mm mirror travel using two different bearings assemblies. Additionally, the manufacturing tolerances of the bearing type were proved to be loose enough not to limit the interferometer application. These demonstrate that the porch swing without any adjustment mechanisms provides the sufficient motion linearity. 1. Introduction ferometer. eTh shearing is the second major phenomenon determining the modulation depth. It is caused by the lateral Inexpensive and robust portable Fourier transform infrared shift of the output beams. With the plane mirror setup, the (FTIR) spectrometers for gas analysers are still needed, shearing is usually negligible because the plane mirrors do because the existing solutions have some weaknesses like notshiftthe beamswitheachother.Ifthe cube cornerswould expenses, bulky size, or sensitivity to the temperature varia- be used as the end mirrors, the output beams would not tions. The most sensitive component in FTIR spectrometer tilt but they may be shifted with each other which would is the interferometer. We have aimed to design such an decrease the modulation depth especially when the size of −1 interferometer with 25 mm beam diameter and with 2 cm the radiation source is finite. Maximizing the modulation depth sets the requirements for the movement of the mirror. resolution in the mid-infrared region. Numerous interfer- ometer designs can be found, for example, from the papers The movable mirror has to remain parallel to the image of by Jackson [1] and Kauppinen et al. [2]and from thebook the second mirror formed by the beam splitter with extreme by Grith ffi s and de Haseth [ 3]. We have selected Michelson accuracy over the desired mirror travel. This linear motion interferometer with plane mirrors, because it has only a of the mirror is achieved by a suitable bearing system which few, even inexpensive, optical components. Using Michelson is reviewed quite throughly by Jackson [1]. The air bearings, interferometer, it is possible to minimize the size and cost of both compressed air and magnetic suspensions [4, 5], provide theinterferometerstructurewhilekeepingthebeamdiameter very smooth movement and low friction but are usually bulky and expensive. Various sliding bearings, like syringes or rails and the throughput constant. Additionally, we will not use any dynamic alignment of the optics which are commonly and guide bar systems, have many advantages including low applied in most solutions. costs but, in general, they are quite sensitive to produce misalignments and can require special materials or extreme One of the key parameters in interferometer design is cleanliness in manufacturing [1]. the modulation depth of the interferogram. In Michelson interferometer built using plane mirrors, the modulation eTh porch swing bearings, presented in Figure 1,have depth is practically mainly aeff cted by the tilting of optics none of these disadvantages. eTh y have quite a simple which causes an angle between the output beams from inter- structurewhich canbemaderathersmall.Theporch swing 2 Advances in Optical Technologies Mirror (a) (b) (c) Figure 1: Side view of typical porch swing bearing assemblies for realizing the linear movement of the mirror. The drawing (a) shows flat springs as the flexure elements and the drawing (b) stiffener clamps in the springs. In the drawing (c), the flexure pivots are used. The driving force F is exerted to the upper arm. Because the linearity of the motion is essential, we Yaw have studied experimentally the tilting of our interferom- eter design which has a porch swing as in Figure 2.We Mirror will demonstrate nearly a perfect porch swing, where the modulation depth has almost no decrease over the 3 mm driving path. Additionally, we have quite good measurement results with the porch swing where all the parts of the bearing have normally achievable machining tolerances. eTh Roll tilting was estimated using a Helium-Neon laser beam and its modulation depth. In addition to the tilting, there are many other phe- nomena, like thermal expansion and stability in temperature Pitch changes, the force constant and the resonance frequencies of the mechanism, or the aging of the mechanics, which have to be considered in design. The lifetime of the flexure elements, typicallymadeofsteel,isnormallyverylongiftheyareloaded Figure 2: Type of the porch swing bearing used in the measure- correctly. However, these are out of the scope of this article, ments and the definition of the directions of the coordinate axes and the labels of the rotations. and we concentrate on the driving stability. 2. Experiments on Motion of the Porch Swing Bearing is a four bar parallelogram linkage where the pivots are usually flexure pivots, flat springs, or other flexure elements. 2.1. Requirements for the Linearity. The maximum allowed eTh y provide virtually frictionless driving and do not need decrease of the modulation depth sets the limits for the tilt lubrication. Additionally, they have no wear if they are loaded anglewhich hastobeachievedduringasingle scan andfrom correctly. scan to scan; hence, the signal strength is proportional to According to Gritffi hs and de Haseth [ 3], the porch swing the modulation depth. eTh modulation depth of the circular bearing was first used in an interferometer by Walker and output beam of the interferometer can be expressed as Rex [6]inthe 1970s. However, Jones[7] has mentioned that the porch swing mechanism for the linear motion is “well 2𝐽 (2𝜋 ]) (2𝜋 ]) (1) 𝑚= ≈1 − , known,” already in 1951. Walker and Rex used the flexure 2𝜋 ] 8 pivots as in Figure 1(c) in their bearing design. This kind where 𝐷 is the beam diameter, 𝛼 is the tilt angle, ] is of interferometer has been operated successfully in harsh wavenumber or the reciprocal of the wave length, or ] =1/𝜆 , environments like helicopters by Small et al. [8], in an air and 𝐽 is the Bessel function of the first kind, and the uniform balloon by Huppi et al. [9], or during a rocket flight by intensity distribution over the cross-section of the collimated Kemp and Huppi [10]. According to Hanel et al. [11], a kind infrared beam is assumed [2, 18]. So, the modulation depth of parallel spring suspended moving shaft was used in the is roughly proportional to the square of the tilt angle which interferometer in Nimbus III space flight in 1969. Wishnow makes the interferometer very sensitive to the changes in et al. [12] have described an interferometer with porch swing thetiltangle.Tokeepthe modulation depth above0.95, driving for visible band imaging spectrometer used in the the tilt angle should be less than about 14 𝜇 rad with the telescope. Many patents, as [13–17], are also related to the −1 porchswing design. eTh porchswing drivingisalsousedin beam diameter of 25 mm and the wavenumber of 3000 cm . a few commercial products, but they typically use a dynamic This wave number is one common convention to report the alignment system. modulation depth, and so we use it here also. 𝐷𝛼 𝐷𝛼 𝐷𝛼 Advances in Optical Technologies 3 𝑦 We have demonstrated the effect of the stiffening clamps by measuring the tilt angle as a function of the mirror displacement usingthe porchswing bearingwhich was 𝑃 (𝑧 ,𝑦 ) 2 2 2 𝑙+ Δ𝑙 constructed of two 𝑙 = 50.0 mm long arms and four steel 𝑃 (𝑧 ,𝑦 ) 0 0 0 springs with ℎ = 50.0 mm long flexible part. The springs were 12.7 mm wide and 0.30 mm thick. eTh y were mounted at the endofthe arms as pairswithdistanceof2.7mm between ℎ+ Δℎ them. eTh configuration of the spring elements was as in Figure 2. eTh stiffening clamps were two pairs of aluminum plates with length of 25 mm which was the half of the spring length ℎ. eTh y were mounted in the halfway of the arms 𝑙 𝑧 𝑃 (𝑙 ,0) as in Figure 1(b).Thelower armofthe porchswing was 𝑂 (0,0) 1 rigidly mounted on the table, and the upper arm could be Figure 3: Side view of the porch swing structure with manufactur- pushed using a rod with n fi e threads. eTh light source was ing errors Δ𝑙 in the arm and Δℎ in the flexure lengths which produce a red Helium-Neon laser whose beam was collimated to the pitch angle 𝛼 when the upper arm is pushed to 𝑃 .Theerrorsare the plane wave. eTh coherence length of the laser light was exaggerated. roughlyabout 20cm,soitwas notnecessary to scan the interferometer exactly around the zero of the optical path difference. Instead, the zero of the mirror displacement 𝑧 was the equilibrium position of the spring elements. eTh 2.2.Stieff ningClampsintheFlexuresReduceParasiticMotions. If the porch swing bearing is perfect, the arms stay parallel optical setup is depicted in Figure 4. eTh end mirror M1 of with each otherduringdriving.Themovable armfollows the Michelson type interferometer was mounted on the top actually a circular trajectory but if the travel length is of the movable arm and the other optics on the table. eTh short, the motion is nearly linear. The imperfections of the interference fringes were magnified by a diverging lens to a screen with a millimeter scale attached to it. We measured porchswing structurecause theunwantedrotations of the movablearm andsotilting of themirrorduringthe travel. the distances 𝑑 between adjacent fringes on the screen as These undesired rotations are also called parasitic motions. illustrated in Figure 5. eTh magnification of the lens was considered by scaling the distance values. eTh experiment When only the plane mirrors are used in the interferometer, the modulation depth is practically aeff cted only by tilting was repeated with and without the stiffening clamps. The tilt around 𝑥 and 𝑦 axes or pitch and yaw rotations, respectively. angle can be calculated from the fringe distance 𝑑 as The rotation around the 𝑧 axis and shifting on the -plane 𝑘𝜆 𝜆 have either no or negligible effect on the modulation depth. (2) 𝛼= = , Additionally, the arm shisft always in the 𝑦 direction because 2𝐷 2𝑑 of the circular trajectory, but with small travel, this shift is where 𝑘 is thenumberofthe fringesacrossthe beam with negligible. diameter of 𝐷 and 𝜆 is the wave length of the beam. If The length differences of the arms Δ𝑙 or the flexure the fringe distances are measured orthogonally, the actual elements Δℎ cause the pitch rotation 𝛼 of the arm as distance is illustrated in Figure 3.Theangle of thenonparallelismofthe neutral axis of the flexures 𝜃 or the principal axis of the inertia 𝑑 𝑑 𝑥 𝑦 𝑑= . of the flexures 𝜙 produce the yaw rotation 𝛽 .Thetilting is (3) 2 2 √𝑑 +𝑑 𝑥 𝑦 also resulted by any deviation of the driving force F from the ideal position which lies along a line which is parallel to the 𝑧 Results of the tilt angles are presented in Figure 6.Without axis andrunsthrough thecenterofthe cross-sectiononthe the stiffening clamps the tilt angle was about 90 𝜇 rad aeft r -plane of the porch swing. eTh se errors are discussed in 4 mm translation, but with clamps the tilt always stayed well more detailinthe appendix andinthe literature [19–24]. In below 20 𝜇 rad. u Th s, the results support the proposition that addition to the above errors, many extra possible sources of the clamps increase the stability of the moving arm. the parasitic motions are mentioned in the literature. es Th e include, among others, the friction at the contact point of the driving force, the asymmetric mass distribution of the arm, 2.3. Manufacturing Tolerances. We estimated the maximum gravity orientation, and variations of the spring material. As allowed manufacturing tolerances of the porch swing bear- a summary, the porch swing is most sensitive to the unequal ings which had the typical dimensions 𝑙/ℎ = 50/50 , 𝑙/ℎ = lengths of the arms Δ𝑙 and the angle between the principal 100/50, 𝑙/ℎ = 150/50 ,and 𝑙/ℎ = 50/100 .Thewidth of axes of inertia 𝜙 . the bearings was 𝑤=50 mm. As the limit for errors, we When the flat springs are used as the flexure elements, used the conditions given in Section 2.1 or 𝑚 ≥ 0.95 and the parasitic rotations can been substantially decreased by the 𝛼≤14𝜇 rad which ensures that the bearing is suitable for stiffening clamps in the middle of the springs as in Figures FTIR interferometer. 1(b) and 2 [7, 16, 20, 25]. The clamps seem to make flexure The porch swing structure is most sensitive to the dieff r- elements more similar to each other and increase the stiffness ence in the arm lengths Δ𝑙 and the angle 𝜙 between the ends in the 𝑥 direction which decrease the sensitivity to the tilting. of the arms. eTh greatest allowed arm length dieff rences were 𝑥𝑦 𝑥𝑦 4 Advances in Optical Technologies BS Laser M2 Tilt range M1 −20 Figure 4: Measurement setup for studying the fringe pattern of the output beam of the interferometer. eTh beam from Helium-Neon Mirror displacement 𝑧 (mm) laser was collimated in C and then pointed through the Michelson interferometer built from plane mirrors M1 and M2 and a cube beam No stiffening clamps splitter BS. eTh mirror M2 was rigidly mounted, and the mirror With stiffening clamps M1 was moved using the porch swing bearing. The diverging lens Figure 6: The stiffener clamps mounted to the spring strips L was used to enlarge the fringe pattern on the screen S which had a decreased the tilt angle of the movable arm of the porch swing. The millimeter scale attached to it. shaded area is the range of allowed tilt angle 𝛼≤14𝜇 rad. The zero of the displacement was the equilibrium position of the spring elements. eTh error bars represent the maximum measurement error of the tilt angle. eTh maximum error of the mirror displacement was about ± 0.1 mm. the worst case. This positioning accuracy is well possible in practise. According to (A.5)and (1), this causes 0.015 decline in the modulation depth. However, the more common arm length in FTIR is about 150 mm which leads to a decline of about 0.002. Exerting the driving force to the movable arm is sometimes more practical than to the ideal position. According to (A.7), the force position𝑎=ℎ+6 mm produced 14 𝜇 rad tilt angle which decreased the modulation depth to about 0.94 in the worst case when the arms were 50 mm long. In the other cases, the tilt was below 8 𝜇 rad and the Figure 5: eTh tilt angle can be determined from the fringe pattern by modulation at least 0.99. Although, these tolerances were very measuring the distance 𝑑 between the adjacent fringes or counting loose, even the resulted worst case modulation depth values the number of fringes 𝑘 across the beam. Both of these can also be were small enough. measured orthogonally to obtain pitch and yaw rotations. Hatheway [21] has estimated the tilt angles over 300 𝜇 rad using the worst case tolerances. These are much greater than our estimations above. This was mainly because of calculated using (A.1a)and (A.1b). These values varied from significantly larger values for the tolerances and longer 5 mm 0.02 mm to 0.06 mm between the cases. The tolerance is loos- stroke. Incidentally, over 80% of these angle values came from ening when the arm is extending. eTh nonparallelism of the the arm length error and the nonparallelism of the arm ends. arm ends causes angle 𝜙 between the principal axes of inertia Clearly, the most essential properties of the porch swing of the flexures. The maximum allowed angles were from are the length difference of the arms and the nonparallelism 230 𝜇 rad to 680 𝜇 rad according to (A.6). This corresponds to ofthearmends.Thishasalreadybeennotedbymanyauthors, 0.01 mm . . . 0.03 mm difference in the lengths of the opposite for example, by Walker and Rex [6]and Strait [14]who have sides of the arm ends. However, by careful machining, the designed adjustment mechanism to minimize these errors. error Δ𝑙 ≤ 0.01 mm at least is achievable. So, these tolerances However, thepreviouscalculationsshowthatadjustment do not seem to limit the usefulness of the bearing. mechanisms are not necessarily needed which simpliefi s the The largest allowed length difference of the flexures Δℎ construction and may reduce manufacturing costs. was 0.6 mm in the worst case where 𝑙=50 mm, but one can attain the tolerance about Δℎ ≤ 0.1 mm which would −1 decrease the modulation depth about 0.001 units at 3000 cm 2.4. Measured Modulation Depth with Porch Swing Bearings. and with 𝐷=25 mm. If one flexure end was mounted We have studied a few porch swings by measuring the 0.2 mm away from its correct position in 𝑥 direction, the modulation depth as a function of the mirror position. angle between the neutral axes of the flexures would be 𝜃= In the following, we present two examples of them. In the 4 mrad with the porch swing 𝑙=ℎ=50 mm, which was first example, the porch swing had dimensions in millimeters fringes Tilt angle ( rad) Advances in Optical Technologies 5 −30 −60 −10 −5 −4 −3 −2 −1 012 −0.05 0 0.05 0.1 0.15 0.2 Angle 𝜙 between the arm ends (mrad) Length difference Δl of the arms (mm) Calculated pitch Measured yaw Calculated yaw Measured pitch Measured pitch Figure 8: eTh measured and calculated values of the pitch tilt as a Figure 7: Measured pitch and yaw tilt angles and the calculated yaw function of the length difference Δ𝑙 between the arms. eTh pitch was angle from (A.6)asafunction of theangle 𝜙 between the arm ends caused by the length difference of the arms. The error bars represent or the angle of nonparallelism of the inertia axes of the flexures. eTh the maximum measurement errors estimated from the readings of error bars represent the maximum measurement errors estimated the scales on the measurement equipment. from the readings of the scales on the measurement equipment. and the yaw were eliminated almost completely when the thicknesses of the spacer combinations were about 0.14 mm 𝑙 = 133.4 , ℎ = 50.0,and 𝑤 = 46.5 . Two flat springs were and 0.30 mm. This corresponds to the length difference Δ𝑙 = mounted in each end of the porch swing. We used the 30 mm 0.14 mm. eTh pitchwas also calculated using( A.1a)and stiffening clamps mounted in the halfway of flat springs. eTh (A.1b)which,among themeasuredvalues, is plottedin porch swing was a part of the Michelson interferometer in a Figure 8. eTh measured length differences are shifted so that very similar setup as in Section 2.2,but theprojectionofthe they are zero when the pitch is about zero. In addition to the fringe pattern was not enlarged. The distances between the tilt elimination, we obtained some support to the equations adjacent fringes were determined by using (2)and (3)and of the pitch. According to Figure 8,the measured pitch counting the number of fringes across the beam in vertical was increased with slightly smaller rate than the calculated and horizontal directions at the mirror travel of 2 mm from pitch. However, therewas stillsomeyaw tilt almost every the equilibrium position of the spring elements. Aeft r the rfi st measurement point which among the other measurement assembly, the porch swing caused the tilting a way too much uncertainties may have affected the results. to be used in an interferometer, thus some adjustment was Slight tilting, which was left aeft r the above ne fi tuning, required. was eliminated by replacing the 30 mm clamps with the Firstly, we removed the yaw tilt by adding thin spacers 44 mm clamps and by careful reassembly and some minor under one flexure end which effectively lengthened the other changes of the spacers. u Th s, the flexible parts shortened side of thearm andthuschanged theangle 𝜙 between the arm from 10 mm to 3 mm. The changes in the fringe pattern were ends.Thespacerthickness,which eliminated theyaw,was no more distinctive by the human eye, so the tilting was about 0.16 mm. The flexures were then probably very close estimated by measuring the modulation depth of the inter- to parallel. eTh other measured spacer thicknesses and the ferometer output beam, which was focused on a photodiode corresponding values of 𝜙 were shied ft so that the angle 𝜙=0 as depicted in Figure 9.Themovable armwas displacedby when theyaw tilt wasabout zero.Themeasuredpitch and a pushing rod with n fi e threads. eTh zero position of the yaw tilt angles and the calculated yaw angles from (A.6)are mirror displacement 𝑧 was the equilibrium position of the presented in Figure 7. eTh measured yaw angles were in quite flat springs. In each mirror position, the rod mount was good agreement with (A.6). However, thepitch didnot seem pushed carefully by hand to get a movement of a few fringes to be fully independent of the angle 𝜙 , although the opposite whichcausedafewcompletesinusoidalcyclesinthe voltage could be expected. It is probably due to some uncertainty in signal from photodiode circuit. Because the photodiode was the measurement. For example, only the upper arm ends were DC coupled, the positive minimum and maximum voltages adjusted, so some errors might have remained in the lower 𝑉 and 𝑉 could be recorded. The visibility, or the modul- arm dimensions. min max ation depth, is then about Next, the pitch tilt was removed by adding more spacers under both springs on the same end of the upper arm, while 𝑉 −𝑉 max min keeping the difference of thickness between the two spacer 𝑚≈ . (4) 𝑉 +𝑉 stacks at about 0.16 mm to get nearly zero yaw. The pitch max min Tilt angle ( rad) Tilt angle ( rad) 6 Advances in Optical Technologies 0.9 0.8 BS 0.7 Laser M2 0.6 0.5 M1 Figure 9: Measurement setup for determining the modulation 0.4 depth of the output beam of the interferometer. The beam from 0 0.5 1 1.5 2 2.5 3 Helium-Neon laser was collimated in C and then pointed through Mirror displacement 𝑧 (mm) the Michelson interferometer built from plane mirrors M1 and M2 and a cube beam splitter BS. eTh mirror M2 was rigidly mounted Initial alignment: and the mirror M1 was moved using the porch swing bearing. eTh Good Mediocre converging lens L was used to focus the beam on the photodiode D. Good Poor Figure 10: eTh modulation depth of an interferometer with well- adjusted porch swing bearing. Four different initial tilt angles were used. eTh zero of the displacement was the equilibrium position Although this is not the most accurate way to determine of the spring elements. The maximum measurement error in the the modulation depth, hence, the noise is on these voltage modulation is about 0.01 and in the position about 0.03 mm. values; our experience has shown that it gives a very good approximation especially when the signal has low noise as in this case. The results of the previous modulation depth measure- ments are presented in Figure 10. In the rfi st two measure- ments, the initial modulation depth was aligned to about 0.9 0.92 whichisasclose to 1aspossiblewithusedoptical components. Over 3.0 mm travel, the modulation depth was decreased not more than about 0.04 units. eTh result 0.8 corresponds to about 4 𝜇 rad tilt angle when the Gaussian distribution of laser intensity is considered [18]. This tilt is below the 14 𝜇 rad limit set in Section 2.1. It would cause −1 0.7 about 0.005 decrease in the modulation at 3000 cm with a uniformly distributed beam which diameter is 25 mm. In the other two measurements, an initial tilt was adjusted. eTh 0.6 decrease of the modulation depth is roughly proportional 01 2 3 4 5 to the squared tilt angle according to (1), so the initial tilt Mirror displacement 𝑧 (mm) should cause a more rapid decrease in the modulation depth. Measurement 1 However, this could not be observed which is a sign of very Measurement 5 Measurement 2 Measurement 6 low tilting. Measurement 4 Measurement 7 eTh above discussed porch swing was clearly machined poorly, because much adjustment was required. In the follow- Figure 11: eTh modulation depth of the interferometer with a ing example, the porch swing was assembled from properly properly machined and carefully assembled porch swing. eTh zero cnc machined parts without modifying or tuning the parts in of the displacement was the equilibrium position of the spring any way after the machining. eTh dimensions in millimeters elements. eTh maximum measurement error in the modulation was were 𝑙 = 110 , ℎ=53,and 𝑤=65 .Theflat steelsprings were about 0.01 and in position about 0.03 mm. 0.2mmthick and10mmwide. eTh springshad the51mm stiffening clamps in the middle. eTh drawing scale of Figure 2 corresponds to these dimensions apart from the flexible parts −1 which are exaggerated for clarity. eTh measurement setup was about 0.007 decrease in the modulation depth at 3000 cm . similar as in the previous experiment. The modulation depth The error in the determining of the modulation depth from of the interferometer was decreased about 0.05 units during the photodiode signal was about 0.01. However, the results the mirror travel of 3 mm as represented in Figure 11.This were not repeatable very well. This was probably because corresponds to about 5 𝜇 rad tilt angle when the distribution of thefrictionbetween thepushing rodand theupper arm of thelaser beam is considered.Thetiltisbelow the14 𝜇 rad although the friction was signica fi ntly reduced by a glass plate limit set in Section 2.1. Using 25 mm beam, this tilt produces betweenthe rodand thearm endasaslidebearing. Modulation depth Modulation depth Advances in Optical Technologies 7 Walker and Rex [6]usedaninterferometerwhich hadthe We used the flat springs as the flexure elements of flexure pivot bearing as in Figure 1(c).Theyhavereportedthat thebearing.Weobservedthatthe tiltingwas substantially the tilting right aeft r the assembly was about 150 𝜇 rad, but decreased and the driving stability was improved by the they achieved to decrease it to the acceptable value of 5 𝜇 rad stiffening clamps mounted in the middle of the springs. The aer ft adjusting of the pivot centers using the adjustment clamps also increased the force constant or the spring rate of mechanism they had designed. Kemp and Huppi [10]used the bearing which might help in the vibration control of the the similar interferometer and reported 5 𝜇 rad maximum tilt system. over 5 mm travel but they did not mention if any adjustments were required. However, we think that they also might had Appendix to adjust thepivotstoobtainsucffi ientlylow tilting. As notedearlier,wehaveachievedthe maximumtiltof5 𝜇 rad Equations of Parasitic Motions using only carefully machined parts without any tuning or adjustment mechanisms. eTh interferometer of Onillon et al. If the flexures are approximated as rigid links, as can be done [26] used a porch swing, which maintained the tilt below with long stieff ning clamps, and the arm lengths are 𝑙 and 5 𝜇 rad over ±2 mm motion range and apparently had no arm 𝑙+Δ𝑙 and the flexure lengths are ℎ and ℎ+Δℎ,the structure length adjustment system, but it was not actually designed for looks like a quadrilateral as in Figure 3.Theequations forthe FTIR spectrometer. Auguson and Young [27]havereported coordinates of the point 𝑃 can be derived by the intersection the tilting of vfi e fringes aer ft 1 cm travel. Their interferometer of two circles. eTh rfi st circle is centered at 𝑃 and has the was, however, for far infrared and utilizing the ball bearings. radius of 𝑙+Δ𝑙 and the second at 𝑃 with the radius of ℎ+Δℎ. With 3.75 inches beam they apparently used, the tilt was about If the upper arm is shieft d by 𝑧 , the coordinates of the corner 17 𝜇 rad, which would have been too much for mid-infrared. 𝑃 are Several authors have demonstrated the tilting of the porch swing bearings. Jones [7]has reported thetiltof 𝑎 𝑘 34 𝜇 rad with a porch swing made with spring strips and 𝑧 =𝑧 + (𝑙−𝑧 )+ 𝑦 , (A.1a) 2 0 0 0 asymmetrically mounted stieff ning clamps. Hatheway [ 21] 𝑑 𝑑 used monolithic flexures, where the flexure element and the 𝑎 𝑘 𝑦 = ( 1− ) 𝑦 + (𝑙−𝑧 ) , (A.1b) clamps pressing it to the arm were machined in one piece. 2 0 0 𝑑 𝑑 The smallest mentioned tilt values were pitch of 5 𝜇 rad and yaw of 39 𝜇 rad. However, he has noted that these values may where not always be repeatable because reassembling increased the 1/2 tilt significantly. It seems that, both, Jones and Hatheway 𝑑=[(𝑙 − 𝑧 ) +𝑦 ] , 0 0 did not use the arm length adjustments. Muranaka et al. −1 2 2 2 [20] have built a porch swing with adjustable arm lengths 𝑎= 2𝑑 [ 𝑙+Δ𝑙 − ℎ+Δℎ +𝑑 ], (A.2) ( ) ( ) ( ) and have achieved the tilt angle less than about 0.5 𝜇 rad 1/2 with the maximum stroke of ± 3 mm. However, their device 2 2 𝑘=[ (𝑙+Δ𝑙 ) −𝑎 ] . is more appropriate for the demonstrations of the parasitic motions than as an actual bearing in an interferometer. Sizes Calculating the coordinates of 𝑃 at initial position 𝑧 =0 and 2 0 of all above mentioned interferometers and demonstration at displacement 𝑧 and assuming that the interferometer was bearings were comparable to example cases used in this perfectly aligned at its initial position, the pitch angle 𝛼 can article. be calculated using basic geometry. Figure 12 illustrates pitch angles in several cases with typical porch swing dimensions andwiththe dimensionerror of 0.2mmwhich canbe 3. Conclusion regarded as an practical upper limit. The pitch seems to be proportional to 𝑧 with the Δ𝑙 but the dependence is We have demonstrated experimentally that sufficient motion approximately quadratic with Δℎ.Italsoseems that theerror linearity of the mirror in Michelson interferometer is well 0.2 mm in the arm length 𝑙 causes about 30 times bigger tilt achievable by usingaporchswing bearingwhich hasno angle compared with the equal error in the flexure length ℎ. adjustment mechanism which is oen ft used. We defined the Jones and Young [19] have presented approximations for sufficient linearity by the maximum allowed decrease of the thepitch angleas modulation depth which was 0.05 units over 3 mm mirror −1 travel with 25 mm beam at 3000 cm . eTh corresponding 𝑧 Δ𝑙 (A.3) decrease was achieved experimentally using Helium-Neon 𝛼≈− , if Δℎ = 0, ℎ𝑙 laser and a porch swing which was manufactured using normal machining and assembly tolerances. eTh estimated 𝑧 Δℎ (A.4) 𝛼≈+ , if Δ𝑙 = 0. manufacturing tolerances for the porch swing were proven to 2ℎ 𝑙 be loose enough not to limit the application of the bearing in the FTIR interferometer. Additionally, the equations of The approximations are valid if the displacement 𝑧 is small. the parasitic motions explained the tuning of the poorly With relatively large displacements, the actual flexure lengths machined porch swing. should be replaced with the eeff ctive lengths of (5/6)ℎ [20, 8 Advances in Optical Technologies 300 alignment of this force causes the parasitic rotations which Δ𝑙 = 0.2 mm have to be usually analyzed numerically [21]. However, some Δℎ = 0 𝜇 approximations are presented by Jones and Young [19]and Muranaka et al. [20], whose model is −150 2 (ℎ−2𝑎 ) 𝑡 3 (A.7) 𝛼=𝑧 [1 + ( ) ], −300 2 2 𝑙 ℎ 175 𝑡 −3 −2 −1 0 123 where 𝑡 is the thickness of the flat spring and 𝑎 is the distance Δ𝑙 = 0 of the force F from the lower arm as in Figure 1(a).Itcan be Δℎ = 0.2 mm readilyseenthatthe pitchangle is zero if 𝑎=ℎ/2 .Thetilting from the misaligned driving force is, however, usually much 2 smallerthanthe othererrorsofthe movement as notedby Jones and Young [19]and by us. −3 −2 −1 0 123 References Mirror displacement 𝑧 (mm) 𝑙/ℎ = 50/50 𝑙/ℎ = 100/50 [1] R. R. Jackson, “Continuous scanning interferometers for mid- 𝑙/ℎ = 150/50 𝑙/ℎ = 50/100 infrared spectrometry, chapter: Instrumentation for Mid- and Far-infrared Spectroscopy,” in Hand-Book of Vibrational Spec- Figure 12: eTh pitch angle 𝛼 as a function of mirror displacement troscopy,J.M.Chalmersand P. R. Gritffi hs, Eds.,vol.1,pp. 264– 𝑧 calculated using (A.1a)and (A.1b). In the upper graph, only the 282, John Wiley & Sons, 2002. arm length error is considered, and in the lower one, only the flexure [2] J. Kauppinen, J. Heinonen, and I. Kauppinen, “Interferometers length has error. based on the rational motion,” Applied Spectroscopy Reviews, vol. 39, no. 1, pp. 99–130, 2004. [3] P.R.Gritffi hs andJ.A.deHaseth, Fourier Transform Infrared 22]. If we compare (A.3)and (A.4), we obtain the ratio of the Spectrometry, John Wiley & Sons, 2nd edition, 2007. length errors Δℎ/Δ𝑙 = 2ℎ/𝑧 . Using dimensions of a typical [4] D.R.Nohavec,L.S.Schwartz, andD.L.Trumper,“Super- design, the ratio is about (2 × 50 mm)/(3 mm) ≈30,which Hybrid Magnetic Suspensions for Interferometric Scanners,” agrees with the values in Figure 12 calculated from (A.1a)and JSME International Journal Series C,vol.40, no.4,pp. 570–583, (A.1b). 1997. We compared these approximations numerically with [5] H. Kobayashi, “Interferometric monitor for greenhouse gases (IMG)—project technical report,” Tech. Rep., IMG Mission our model using the same cases as in Figure 12 and noted Operation & Verification Committee CRIEPI, 1999. that if there is only Δ𝑙 = 0.2 mm or both Δ𝑙=Δℎ= [6] R. P. Walker and J. D. Rex, “Interferometer design and data han- 0.2 mm, the pitch from (A.3)dieff rsfrom( A.1a)and (A.1b) dling in a high vibration environment—part I: interferometer less than about 1%. However, (A.4) differed even tens of design,” in Multiplex and/or High-rTh oughput Spectroscopy ,G. percent which was a result mainly from very small numerical A. Vanasse, Ed., vol. 191 of Proceedings of SPIE,pp.88–91,August values. eTh measurement results of Jones and Young [ 19]and Muranaka et al. [20] have tte fi d quite well to both the pitch [7] R. V. Jones, “Parallel and rectilinear spring movements,” Journal approximations and our model. of Scientific Instruments , vol. 28, pp. 38–41, 1951. The nonparallelism of the neutral axes of the flexure [8]G.W.Small,R.T.Kroutil,J.T.Ditillo, andW.R.Loerop, elements may happen, for example, if the flexures are rotated “Detection of atmospheric pollutants by direct analysis of to each other by the angle 𝜃 around the 𝑧 axis. According to passive fourier transform infrared interferograms,” Analytical Hatheway [21], the yaw angle is Chemistry, vol. 60, no. 3, pp. 264–269, 1988. [9] R.J.Huppi,R.B.Shipley,and E. R. Huppi, “Balloon-borne fourier spectrometer using a focal plane detector array,” in (A.5) 𝛽= . 2𝑙ℎ Multiplex and/or High-rTh oughput Spectroscopy ,G.A.Vanasse, Ed., vol. 191 of Proceedings of SPIE, pp. 26–32, August 1979. The effective length of the flexure has to be considered in this [10] J. C. Kemp and R. J. Huppi, “Rocket-borne cryogenic Michelson equation when necessary. The principal axes of inertia of the interferometer,” in Multiplex and/or High-rTh oughput Spec- flexures become nonparallel when the flexure elements are troscopy,G.A.Vanasse,Ed.,vol.191 of Proceedings of SPIE,pp. rotated to each other by the angle 𝜙 around the 𝑦 axis which 135–142, August 1979. is the case if the ends of the arms are not parallel. Hatheway [11] R. A. Hanel, B. Schlacman, F. D. Clark et al., “eTh Nimbus [21] has given the yaw angle as III Michelson interferometer,” in Proceedings of the Aspen International Conference on Fourier Spectroscopy,G.A.Vanasse, A. S. Jn,and D. J. Baker, Eds.,pp. 231–241, AirForce Cambridge (A.6) 𝛽= . Research Laboratories, Optical Physics Laboratory, United States Air Force, January 1971. Ideally, thedriving forceliesalong alinewhich is parallel [12] E. H. Wishnow, R. Wurtz, S. Blais-Ouellette et al., “Visible to the 𝑧 axis andrunsthrough thecenterofthe cross-section imaging Fourier transform spectrometer: design and calibra- on the -plane of the porch swing [20, 21]. The nonideal tion,” in Instrument Design and Performance for Optical/Infrared Tilt angle ( rad) Tilt angle ( rad) 𝑥𝑦 𝜙𝑧 𝜃𝑧 Advances in Optical Technologies 9 Ground-based Telescopes,M.Iye andA.F.M.Moorwood,Eds., vol. 4841 of Proceeding of SPIE, pp. 1067–1077, August 2002. [13] G. L. Auth, “Ruggedized compact interferometer requiring minimum isolation from mechanical vibrations,” US Patent 4,693,603, 1987. [14] D. R. Strait, “Moving mirror tilt adjust mechanism in an inter- ferometer,” US Patent 4,991,961, February 1991. [15] R. F. Lacey, “Support for a moving mirror in an interferometer,” US Patent 4,710,001, December 1987. [16] G. R. Walker, “Precision frictionless flexure based linear translation mechanism insensitive to thermal and vibrational environments,” US Patent 6,836,968, 2005. [17] K. C. Schreiber, “Support for a movable mirror in an interfer- ometer,” Patent Application Publication 2002/0149777, 2002. [18] T. Vali ¨ kyla¨ and J. Kauppinen, “Modulation depth of Michelson interferometer with gaussian beam,” Applied Optics,vol.50, pp. 6671–6677, 2011. [19] R. V. Jones and I. R. Young, “Some parasitic deflexions in paral- lel spring movements,” Journal of Scienticfi Instruments ,vol.33, no.1,article 305, pp.11–15,1956. [20] Y. Muranaka,M.Inaba,T.Asano,and E. Furukawa,“Parasitic rotations in parallel spring movements,” Bulletin of the Japan Society of Precision Engineering,vol.25, no.3,pp. 208–213, 1991. [21] A. E. Hatheway, “Alignment of flexure stages for best rectilinear performance,” in Optomechanical and Precision Instrument Design,A.E.Hatheway, Ed., vol. 2542 of Proceedings of SPIE, pp.70–80,July1995. [22] A. E. Hatheway, “The kinetic center of the cantilever beam,” in Optomechanical Design and Precision Instruments,A.E. Hatheway, Ed., vol. 3132 of Proceedings of SPIE, pp. 218–222, [23] S. T. Smith, Flexures-Elements of Elastic Mechanisms, Gordon and Breach Science Publishers, 1st edition, 2000. [24] S. Awtar, Synthesis and analysis of parallel kinematic XY efl xure mechanisms [Ph.D. thesis], Massachusetts Institute of Technol- ogy, 2003. [25] S. Awtar, A. H. Slocum, and E. Sevincer, “Characteristics of beam-based flexure modules,” Journal of Mechanical Design, Transactions of the ASME,vol.129,no. 6, pp.625–639,2007. [26] E. Onillon, S. Henein, P. eTh urillat, J. Krauss, and I. Kjelberg, “Interferometer scanning mirror mechanism,” in Mechatronic Systems 2002: A Proceedings Volume From the 2nd Ifac Confer- ence, M. Tomizuka, Ed., Ifac Proceedings Series, International Federation of Automatic Control, Elsevier, 2003. [27] G. C. Auguson and N. O. Young, “A liquid-helium-cooled Michelson interferometer,” in Aspen International Conference on Fourier Spectroscopy,G.A.Vanasse,A.S.Jn, andD.J.Baker, Eds., pp. 281–288, Air Force Cambridge Research Laboratories, Optical Physics Laboratory, United States Air Force, January 1971. International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Optical Technologies Hindawi Publishing Corporation

Experiments on the Porch Swing Bearing of Michelson Interferometer for Low Resolution FTIR

Loading next page...
 
/lp/hindawi-publishing-corporation/experiments-on-the-porch-swing-bearing-of-michelson-interferometer-for-RklW0Qzt4y
Publisher
Hindawi Publishing Corporation
Copyright
Copyright © 2013 Tuomas Välikylä and Jyrki Kauppinen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN
1687-6393
DOI
10.1155/2013/948638
Publisher site
See Article on Publisher Site

Abstract

Hindawi Publishing Corporation Advances in Optical Technologies Volume 2013, Article ID 948638, 9 pages http://dx.doi.org/10.1155/2013/948638 Research Article Experiments on the Porch Swing Bearing of Michelson Interferometer for Low Resolution FTIR Tuomas Välikylä and Jyrki Kauppinen Department of Physics and Astronomy, University of Turku, 20014 Turku, Finland Correspondence should be addressed to Tuomas Vali ¨ kyla; ¨ tuomas.valikyla@utu.fi Received 22 March 2013; Accepted 8 May 2013 Academic Editor: Joseph Rosen Copyright © 2013 T. Vali ¨ kyla¨ and J. Kauppinen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Porch swing bearing for the linear motion of the mirror in Michelson interferometer for mid-infrared low resolution Fourier transform spectrometer was studied experimentally using the modulation depth of the collimated laser beam. eTh mirror tilting was measured to be lower than 5 𝜇 rad over 3 mm mirror travel using two different bearings assemblies. Additionally, the manufacturing tolerances of the bearing type were proved to be loose enough not to limit the interferometer application. These demonstrate that the porch swing without any adjustment mechanisms provides the sufficient motion linearity. 1. Introduction ferometer. eTh shearing is the second major phenomenon determining the modulation depth. It is caused by the lateral Inexpensive and robust portable Fourier transform infrared shift of the output beams. With the plane mirror setup, the (FTIR) spectrometers for gas analysers are still needed, shearing is usually negligible because the plane mirrors do because the existing solutions have some weaknesses like notshiftthe beamswitheachother.Ifthe cube cornerswould expenses, bulky size, or sensitivity to the temperature varia- be used as the end mirrors, the output beams would not tions. The most sensitive component in FTIR spectrometer tilt but they may be shifted with each other which would is the interferometer. We have aimed to design such an decrease the modulation depth especially when the size of −1 interferometer with 25 mm beam diameter and with 2 cm the radiation source is finite. Maximizing the modulation depth sets the requirements for the movement of the mirror. resolution in the mid-infrared region. Numerous interfer- ometer designs can be found, for example, from the papers The movable mirror has to remain parallel to the image of by Jackson [1] and Kauppinen et al. [2]and from thebook the second mirror formed by the beam splitter with extreme by Grith ffi s and de Haseth [ 3]. We have selected Michelson accuracy over the desired mirror travel. This linear motion interferometer with plane mirrors, because it has only a of the mirror is achieved by a suitable bearing system which few, even inexpensive, optical components. Using Michelson is reviewed quite throughly by Jackson [1]. The air bearings, interferometer, it is possible to minimize the size and cost of both compressed air and magnetic suspensions [4, 5], provide theinterferometerstructurewhilekeepingthebeamdiameter very smooth movement and low friction but are usually bulky and expensive. Various sliding bearings, like syringes or rails and the throughput constant. Additionally, we will not use any dynamic alignment of the optics which are commonly and guide bar systems, have many advantages including low applied in most solutions. costs but, in general, they are quite sensitive to produce misalignments and can require special materials or extreme One of the key parameters in interferometer design is cleanliness in manufacturing [1]. the modulation depth of the interferogram. In Michelson interferometer built using plane mirrors, the modulation eTh porch swing bearings, presented in Figure 1,have depth is practically mainly aeff cted by the tilting of optics none of these disadvantages. eTh y have quite a simple which causes an angle between the output beams from inter- structurewhich canbemaderathersmall.Theporch swing 2 Advances in Optical Technologies Mirror (a) (b) (c) Figure 1: Side view of typical porch swing bearing assemblies for realizing the linear movement of the mirror. The drawing (a) shows flat springs as the flexure elements and the drawing (b) stiffener clamps in the springs. In the drawing (c), the flexure pivots are used. The driving force F is exerted to the upper arm. Because the linearity of the motion is essential, we Yaw have studied experimentally the tilting of our interferom- eter design which has a porch swing as in Figure 2.We Mirror will demonstrate nearly a perfect porch swing, where the modulation depth has almost no decrease over the 3 mm driving path. Additionally, we have quite good measurement results with the porch swing where all the parts of the bearing have normally achievable machining tolerances. eTh Roll tilting was estimated using a Helium-Neon laser beam and its modulation depth. In addition to the tilting, there are many other phe- nomena, like thermal expansion and stability in temperature Pitch changes, the force constant and the resonance frequencies of the mechanism, or the aging of the mechanics, which have to be considered in design. The lifetime of the flexure elements, typicallymadeofsteel,isnormallyverylongiftheyareloaded Figure 2: Type of the porch swing bearing used in the measure- correctly. However, these are out of the scope of this article, ments and the definition of the directions of the coordinate axes and the labels of the rotations. and we concentrate on the driving stability. 2. Experiments on Motion of the Porch Swing Bearing is a four bar parallelogram linkage where the pivots are usually flexure pivots, flat springs, or other flexure elements. 2.1. Requirements for the Linearity. The maximum allowed eTh y provide virtually frictionless driving and do not need decrease of the modulation depth sets the limits for the tilt lubrication. Additionally, they have no wear if they are loaded anglewhich hastobeachievedduringasingle scan andfrom correctly. scan to scan; hence, the signal strength is proportional to According to Gritffi hs and de Haseth [ 3], the porch swing the modulation depth. eTh modulation depth of the circular bearing was first used in an interferometer by Walker and output beam of the interferometer can be expressed as Rex [6]inthe 1970s. However, Jones[7] has mentioned that the porch swing mechanism for the linear motion is “well 2𝐽 (2𝜋 ]) (2𝜋 ]) (1) 𝑚= ≈1 − , known,” already in 1951. Walker and Rex used the flexure 2𝜋 ] 8 pivots as in Figure 1(c) in their bearing design. This kind where 𝐷 is the beam diameter, 𝛼 is the tilt angle, ] is of interferometer has been operated successfully in harsh wavenumber or the reciprocal of the wave length, or ] =1/𝜆 , environments like helicopters by Small et al. [8], in an air and 𝐽 is the Bessel function of the first kind, and the uniform balloon by Huppi et al. [9], or during a rocket flight by intensity distribution over the cross-section of the collimated Kemp and Huppi [10]. According to Hanel et al. [11], a kind infrared beam is assumed [2, 18]. So, the modulation depth of parallel spring suspended moving shaft was used in the is roughly proportional to the square of the tilt angle which interferometer in Nimbus III space flight in 1969. Wishnow makes the interferometer very sensitive to the changes in et al. [12] have described an interferometer with porch swing thetiltangle.Tokeepthe modulation depth above0.95, driving for visible band imaging spectrometer used in the the tilt angle should be less than about 14 𝜇 rad with the telescope. Many patents, as [13–17], are also related to the −1 porchswing design. eTh porchswing drivingisalsousedin beam diameter of 25 mm and the wavenumber of 3000 cm . a few commercial products, but they typically use a dynamic This wave number is one common convention to report the alignment system. modulation depth, and so we use it here also. 𝐷𝛼 𝐷𝛼 𝐷𝛼 Advances in Optical Technologies 3 𝑦 We have demonstrated the effect of the stiffening clamps by measuring the tilt angle as a function of the mirror displacement usingthe porchswing bearingwhich was 𝑃 (𝑧 ,𝑦 ) 2 2 2 𝑙+ Δ𝑙 constructed of two 𝑙 = 50.0 mm long arms and four steel 𝑃 (𝑧 ,𝑦 ) 0 0 0 springs with ℎ = 50.0 mm long flexible part. The springs were 12.7 mm wide and 0.30 mm thick. eTh y were mounted at the endofthe arms as pairswithdistanceof2.7mm between ℎ+ Δℎ them. eTh configuration of the spring elements was as in Figure 2. eTh stiffening clamps were two pairs of aluminum plates with length of 25 mm which was the half of the spring length ℎ. eTh y were mounted in the halfway of the arms 𝑙 𝑧 𝑃 (𝑙 ,0) as in Figure 1(b).Thelower armofthe porchswing was 𝑂 (0,0) 1 rigidly mounted on the table, and the upper arm could be Figure 3: Side view of the porch swing structure with manufactur- pushed using a rod with n fi e threads. eTh light source was ing errors Δ𝑙 in the arm and Δℎ in the flexure lengths which produce a red Helium-Neon laser whose beam was collimated to the pitch angle 𝛼 when the upper arm is pushed to 𝑃 .Theerrorsare the plane wave. eTh coherence length of the laser light was exaggerated. roughlyabout 20cm,soitwas notnecessary to scan the interferometer exactly around the zero of the optical path difference. Instead, the zero of the mirror displacement 𝑧 was the equilibrium position of the spring elements. eTh 2.2.Stieff ningClampsintheFlexuresReduceParasiticMotions. If the porch swing bearing is perfect, the arms stay parallel optical setup is depicted in Figure 4. eTh end mirror M1 of with each otherduringdriving.Themovable armfollows the Michelson type interferometer was mounted on the top actually a circular trajectory but if the travel length is of the movable arm and the other optics on the table. eTh short, the motion is nearly linear. The imperfections of the interference fringes were magnified by a diverging lens to a screen with a millimeter scale attached to it. We measured porchswing structurecause theunwantedrotations of the movablearm andsotilting of themirrorduringthe travel. the distances 𝑑 between adjacent fringes on the screen as These undesired rotations are also called parasitic motions. illustrated in Figure 5. eTh magnification of the lens was considered by scaling the distance values. eTh experiment When only the plane mirrors are used in the interferometer, the modulation depth is practically aeff cted only by tilting was repeated with and without the stiffening clamps. The tilt around 𝑥 and 𝑦 axes or pitch and yaw rotations, respectively. angle can be calculated from the fringe distance 𝑑 as The rotation around the 𝑧 axis and shifting on the -plane 𝑘𝜆 𝜆 have either no or negligible effect on the modulation depth. (2) 𝛼= = , Additionally, the arm shisft always in the 𝑦 direction because 2𝐷 2𝑑 of the circular trajectory, but with small travel, this shift is where 𝑘 is thenumberofthe fringesacrossthe beam with negligible. diameter of 𝐷 and 𝜆 is the wave length of the beam. If The length differences of the arms Δ𝑙 or the flexure the fringe distances are measured orthogonally, the actual elements Δℎ cause the pitch rotation 𝛼 of the arm as distance is illustrated in Figure 3.Theangle of thenonparallelismofthe neutral axis of the flexures 𝜃 or the principal axis of the inertia 𝑑 𝑑 𝑥 𝑦 𝑑= . of the flexures 𝜙 produce the yaw rotation 𝛽 .Thetilting is (3) 2 2 √𝑑 +𝑑 𝑥 𝑦 also resulted by any deviation of the driving force F from the ideal position which lies along a line which is parallel to the 𝑧 Results of the tilt angles are presented in Figure 6.Without axis andrunsthrough thecenterofthe cross-sectiononthe the stiffening clamps the tilt angle was about 90 𝜇 rad aeft r -plane of the porch swing. eTh se errors are discussed in 4 mm translation, but with clamps the tilt always stayed well more detailinthe appendix andinthe literature [19–24]. In below 20 𝜇 rad. u Th s, the results support the proposition that addition to the above errors, many extra possible sources of the clamps increase the stability of the moving arm. the parasitic motions are mentioned in the literature. es Th e include, among others, the friction at the contact point of the driving force, the asymmetric mass distribution of the arm, 2.3. Manufacturing Tolerances. We estimated the maximum gravity orientation, and variations of the spring material. As allowed manufacturing tolerances of the porch swing bear- a summary, the porch swing is most sensitive to the unequal ings which had the typical dimensions 𝑙/ℎ = 50/50 , 𝑙/ℎ = lengths of the arms Δ𝑙 and the angle between the principal 100/50, 𝑙/ℎ = 150/50 ,and 𝑙/ℎ = 50/100 .Thewidth of axes of inertia 𝜙 . the bearings was 𝑤=50 mm. As the limit for errors, we When the flat springs are used as the flexure elements, used the conditions given in Section 2.1 or 𝑚 ≥ 0.95 and the parasitic rotations can been substantially decreased by the 𝛼≤14𝜇 rad which ensures that the bearing is suitable for stiffening clamps in the middle of the springs as in Figures FTIR interferometer. 1(b) and 2 [7, 16, 20, 25]. The clamps seem to make flexure The porch swing structure is most sensitive to the dieff r- elements more similar to each other and increase the stiffness ence in the arm lengths Δ𝑙 and the angle 𝜙 between the ends in the 𝑥 direction which decrease the sensitivity to the tilting. of the arms. eTh greatest allowed arm length dieff rences were 𝑥𝑦 𝑥𝑦 4 Advances in Optical Technologies BS Laser M2 Tilt range M1 −20 Figure 4: Measurement setup for studying the fringe pattern of the output beam of the interferometer. eTh beam from Helium-Neon Mirror displacement 𝑧 (mm) laser was collimated in C and then pointed through the Michelson interferometer built from plane mirrors M1 and M2 and a cube beam No stiffening clamps splitter BS. eTh mirror M2 was rigidly mounted, and the mirror With stiffening clamps M1 was moved using the porch swing bearing. The diverging lens Figure 6: The stiffener clamps mounted to the spring strips L was used to enlarge the fringe pattern on the screen S which had a decreased the tilt angle of the movable arm of the porch swing. The millimeter scale attached to it. shaded area is the range of allowed tilt angle 𝛼≤14𝜇 rad. The zero of the displacement was the equilibrium position of the spring elements. eTh error bars represent the maximum measurement error of the tilt angle. eTh maximum error of the mirror displacement was about ± 0.1 mm. the worst case. This positioning accuracy is well possible in practise. According to (A.5)and (1), this causes 0.015 decline in the modulation depth. However, the more common arm length in FTIR is about 150 mm which leads to a decline of about 0.002. Exerting the driving force to the movable arm is sometimes more practical than to the ideal position. According to (A.7), the force position𝑎=ℎ+6 mm produced 14 𝜇 rad tilt angle which decreased the modulation depth to about 0.94 in the worst case when the arms were 50 mm long. In the other cases, the tilt was below 8 𝜇 rad and the Figure 5: eTh tilt angle can be determined from the fringe pattern by modulation at least 0.99. Although, these tolerances were very measuring the distance 𝑑 between the adjacent fringes or counting loose, even the resulted worst case modulation depth values the number of fringes 𝑘 across the beam. Both of these can also be were small enough. measured orthogonally to obtain pitch and yaw rotations. Hatheway [21] has estimated the tilt angles over 300 𝜇 rad using the worst case tolerances. These are much greater than our estimations above. This was mainly because of calculated using (A.1a)and (A.1b). These values varied from significantly larger values for the tolerances and longer 5 mm 0.02 mm to 0.06 mm between the cases. The tolerance is loos- stroke. Incidentally, over 80% of these angle values came from ening when the arm is extending. eTh nonparallelism of the the arm length error and the nonparallelism of the arm ends. arm ends causes angle 𝜙 between the principal axes of inertia Clearly, the most essential properties of the porch swing of the flexures. The maximum allowed angles were from are the length difference of the arms and the nonparallelism 230 𝜇 rad to 680 𝜇 rad according to (A.6). This corresponds to ofthearmends.Thishasalreadybeennotedbymanyauthors, 0.01 mm . . . 0.03 mm difference in the lengths of the opposite for example, by Walker and Rex [6]and Strait [14]who have sides of the arm ends. However, by careful machining, the designed adjustment mechanism to minimize these errors. error Δ𝑙 ≤ 0.01 mm at least is achievable. So, these tolerances However, thepreviouscalculationsshowthatadjustment do not seem to limit the usefulness of the bearing. mechanisms are not necessarily needed which simpliefi s the The largest allowed length difference of the flexures Δℎ construction and may reduce manufacturing costs. was 0.6 mm in the worst case where 𝑙=50 mm, but one can attain the tolerance about Δℎ ≤ 0.1 mm which would −1 decrease the modulation depth about 0.001 units at 3000 cm 2.4. Measured Modulation Depth with Porch Swing Bearings. and with 𝐷=25 mm. If one flexure end was mounted We have studied a few porch swings by measuring the 0.2 mm away from its correct position in 𝑥 direction, the modulation depth as a function of the mirror position. angle between the neutral axes of the flexures would be 𝜃= In the following, we present two examples of them. In the 4 mrad with the porch swing 𝑙=ℎ=50 mm, which was first example, the porch swing had dimensions in millimeters fringes Tilt angle ( rad) Advances in Optical Technologies 5 −30 −60 −10 −5 −4 −3 −2 −1 012 −0.05 0 0.05 0.1 0.15 0.2 Angle 𝜙 between the arm ends (mrad) Length difference Δl of the arms (mm) Calculated pitch Measured yaw Calculated yaw Measured pitch Measured pitch Figure 8: eTh measured and calculated values of the pitch tilt as a Figure 7: Measured pitch and yaw tilt angles and the calculated yaw function of the length difference Δ𝑙 between the arms. eTh pitch was angle from (A.6)asafunction of theangle 𝜙 between the arm ends caused by the length difference of the arms. The error bars represent or the angle of nonparallelism of the inertia axes of the flexures. eTh the maximum measurement errors estimated from the readings of error bars represent the maximum measurement errors estimated the scales on the measurement equipment. from the readings of the scales on the measurement equipment. and the yaw were eliminated almost completely when the thicknesses of the spacer combinations were about 0.14 mm 𝑙 = 133.4 , ℎ = 50.0,and 𝑤 = 46.5 . Two flat springs were and 0.30 mm. This corresponds to the length difference Δ𝑙 = mounted in each end of the porch swing. We used the 30 mm 0.14 mm. eTh pitchwas also calculated using( A.1a)and stiffening clamps mounted in the halfway of flat springs. eTh (A.1b)which,among themeasuredvalues, is plottedin porch swing was a part of the Michelson interferometer in a Figure 8. eTh measured length differences are shifted so that very similar setup as in Section 2.2,but theprojectionofthe they are zero when the pitch is about zero. In addition to the fringe pattern was not enlarged. The distances between the tilt elimination, we obtained some support to the equations adjacent fringes were determined by using (2)and (3)and of the pitch. According to Figure 8,the measured pitch counting the number of fringes across the beam in vertical was increased with slightly smaller rate than the calculated and horizontal directions at the mirror travel of 2 mm from pitch. However, therewas stillsomeyaw tilt almost every the equilibrium position of the spring elements. Aeft r the rfi st measurement point which among the other measurement assembly, the porch swing caused the tilting a way too much uncertainties may have affected the results. to be used in an interferometer, thus some adjustment was Slight tilting, which was left aeft r the above ne fi tuning, required. was eliminated by replacing the 30 mm clamps with the Firstly, we removed the yaw tilt by adding thin spacers 44 mm clamps and by careful reassembly and some minor under one flexure end which effectively lengthened the other changes of the spacers. u Th s, the flexible parts shortened side of thearm andthuschanged theangle 𝜙 between the arm from 10 mm to 3 mm. The changes in the fringe pattern were ends.Thespacerthickness,which eliminated theyaw,was no more distinctive by the human eye, so the tilting was about 0.16 mm. The flexures were then probably very close estimated by measuring the modulation depth of the inter- to parallel. eTh other measured spacer thicknesses and the ferometer output beam, which was focused on a photodiode corresponding values of 𝜙 were shied ft so that the angle 𝜙=0 as depicted in Figure 9.Themovable armwas displacedby when theyaw tilt wasabout zero.Themeasuredpitch and a pushing rod with n fi e threads. eTh zero position of the yaw tilt angles and the calculated yaw angles from (A.6)are mirror displacement 𝑧 was the equilibrium position of the presented in Figure 7. eTh measured yaw angles were in quite flat springs. In each mirror position, the rod mount was good agreement with (A.6). However, thepitch didnot seem pushed carefully by hand to get a movement of a few fringes to be fully independent of the angle 𝜙 , although the opposite whichcausedafewcompletesinusoidalcyclesinthe voltage could be expected. It is probably due to some uncertainty in signal from photodiode circuit. Because the photodiode was the measurement. For example, only the upper arm ends were DC coupled, the positive minimum and maximum voltages adjusted, so some errors might have remained in the lower 𝑉 and 𝑉 could be recorded. The visibility, or the modul- arm dimensions. min max ation depth, is then about Next, the pitch tilt was removed by adding more spacers under both springs on the same end of the upper arm, while 𝑉 −𝑉 max min keeping the difference of thickness between the two spacer 𝑚≈ . (4) 𝑉 +𝑉 stacks at about 0.16 mm to get nearly zero yaw. The pitch max min Tilt angle ( rad) Tilt angle ( rad) 6 Advances in Optical Technologies 0.9 0.8 BS 0.7 Laser M2 0.6 0.5 M1 Figure 9: Measurement setup for determining the modulation 0.4 depth of the output beam of the interferometer. The beam from 0 0.5 1 1.5 2 2.5 3 Helium-Neon laser was collimated in C and then pointed through Mirror displacement 𝑧 (mm) the Michelson interferometer built from plane mirrors M1 and M2 and a cube beam splitter BS. eTh mirror M2 was rigidly mounted Initial alignment: and the mirror M1 was moved using the porch swing bearing. eTh Good Mediocre converging lens L was used to focus the beam on the photodiode D. Good Poor Figure 10: eTh modulation depth of an interferometer with well- adjusted porch swing bearing. Four different initial tilt angles were used. eTh zero of the displacement was the equilibrium position Although this is not the most accurate way to determine of the spring elements. The maximum measurement error in the the modulation depth, hence, the noise is on these voltage modulation is about 0.01 and in the position about 0.03 mm. values; our experience has shown that it gives a very good approximation especially when the signal has low noise as in this case. The results of the previous modulation depth measure- ments are presented in Figure 10. In the rfi st two measure- ments, the initial modulation depth was aligned to about 0.9 0.92 whichisasclose to 1aspossiblewithusedoptical components. Over 3.0 mm travel, the modulation depth was decreased not more than about 0.04 units. eTh result 0.8 corresponds to about 4 𝜇 rad tilt angle when the Gaussian distribution of laser intensity is considered [18]. This tilt is below the 14 𝜇 rad limit set in Section 2.1. It would cause −1 0.7 about 0.005 decrease in the modulation at 3000 cm with a uniformly distributed beam which diameter is 25 mm. In the other two measurements, an initial tilt was adjusted. eTh 0.6 decrease of the modulation depth is roughly proportional 01 2 3 4 5 to the squared tilt angle according to (1), so the initial tilt Mirror displacement 𝑧 (mm) should cause a more rapid decrease in the modulation depth. Measurement 1 However, this could not be observed which is a sign of very Measurement 5 Measurement 2 Measurement 6 low tilting. Measurement 4 Measurement 7 eTh above discussed porch swing was clearly machined poorly, because much adjustment was required. In the follow- Figure 11: eTh modulation depth of the interferometer with a ing example, the porch swing was assembled from properly properly machined and carefully assembled porch swing. eTh zero cnc machined parts without modifying or tuning the parts in of the displacement was the equilibrium position of the spring any way after the machining. eTh dimensions in millimeters elements. eTh maximum measurement error in the modulation was were 𝑙 = 110 , ℎ=53,and 𝑤=65 .Theflat steelsprings were about 0.01 and in position about 0.03 mm. 0.2mmthick and10mmwide. eTh springshad the51mm stiffening clamps in the middle. eTh drawing scale of Figure 2 corresponds to these dimensions apart from the flexible parts −1 which are exaggerated for clarity. eTh measurement setup was about 0.007 decrease in the modulation depth at 3000 cm . similar as in the previous experiment. The modulation depth The error in the determining of the modulation depth from of the interferometer was decreased about 0.05 units during the photodiode signal was about 0.01. However, the results the mirror travel of 3 mm as represented in Figure 11.This were not repeatable very well. This was probably because corresponds to about 5 𝜇 rad tilt angle when the distribution of thefrictionbetween thepushing rodand theupper arm of thelaser beam is considered.Thetiltisbelow the14 𝜇 rad although the friction was signica fi ntly reduced by a glass plate limit set in Section 2.1. Using 25 mm beam, this tilt produces betweenthe rodand thearm endasaslidebearing. Modulation depth Modulation depth Advances in Optical Technologies 7 Walker and Rex [6]usedaninterferometerwhich hadthe We used the flat springs as the flexure elements of flexure pivot bearing as in Figure 1(c).Theyhavereportedthat thebearing.Weobservedthatthe tiltingwas substantially the tilting right aeft r the assembly was about 150 𝜇 rad, but decreased and the driving stability was improved by the they achieved to decrease it to the acceptable value of 5 𝜇 rad stiffening clamps mounted in the middle of the springs. The aer ft adjusting of the pivot centers using the adjustment clamps also increased the force constant or the spring rate of mechanism they had designed. Kemp and Huppi [10]used the bearing which might help in the vibration control of the the similar interferometer and reported 5 𝜇 rad maximum tilt system. over 5 mm travel but they did not mention if any adjustments were required. However, we think that they also might had Appendix to adjust thepivotstoobtainsucffi ientlylow tilting. As notedearlier,wehaveachievedthe maximumtiltof5 𝜇 rad Equations of Parasitic Motions using only carefully machined parts without any tuning or adjustment mechanisms. eTh interferometer of Onillon et al. If the flexures are approximated as rigid links, as can be done [26] used a porch swing, which maintained the tilt below with long stieff ning clamps, and the arm lengths are 𝑙 and 5 𝜇 rad over ±2 mm motion range and apparently had no arm 𝑙+Δ𝑙 and the flexure lengths are ℎ and ℎ+Δℎ,the structure length adjustment system, but it was not actually designed for looks like a quadrilateral as in Figure 3.Theequations forthe FTIR spectrometer. Auguson and Young [27]havereported coordinates of the point 𝑃 can be derived by the intersection the tilting of vfi e fringes aer ft 1 cm travel. Their interferometer of two circles. eTh rfi st circle is centered at 𝑃 and has the was, however, for far infrared and utilizing the ball bearings. radius of 𝑙+Δ𝑙 and the second at 𝑃 with the radius of ℎ+Δℎ. With 3.75 inches beam they apparently used, the tilt was about If the upper arm is shieft d by 𝑧 , the coordinates of the corner 17 𝜇 rad, which would have been too much for mid-infrared. 𝑃 are Several authors have demonstrated the tilting of the porch swing bearings. Jones [7]has reported thetiltof 𝑎 𝑘 34 𝜇 rad with a porch swing made with spring strips and 𝑧 =𝑧 + (𝑙−𝑧 )+ 𝑦 , (A.1a) 2 0 0 0 asymmetrically mounted stieff ning clamps. Hatheway [ 21] 𝑑 𝑑 used monolithic flexures, where the flexure element and the 𝑎 𝑘 𝑦 = ( 1− ) 𝑦 + (𝑙−𝑧 ) , (A.1b) clamps pressing it to the arm were machined in one piece. 2 0 0 𝑑 𝑑 The smallest mentioned tilt values were pitch of 5 𝜇 rad and yaw of 39 𝜇 rad. However, he has noted that these values may where not always be repeatable because reassembling increased the 1/2 tilt significantly. It seems that, both, Jones and Hatheway 𝑑=[(𝑙 − 𝑧 ) +𝑦 ] , 0 0 did not use the arm length adjustments. Muranaka et al. −1 2 2 2 [20] have built a porch swing with adjustable arm lengths 𝑎= 2𝑑 [ 𝑙+Δ𝑙 − ℎ+Δℎ +𝑑 ], (A.2) ( ) ( ) ( ) and have achieved the tilt angle less than about 0.5 𝜇 rad 1/2 with the maximum stroke of ± 3 mm. However, their device 2 2 𝑘=[ (𝑙+Δ𝑙 ) −𝑎 ] . is more appropriate for the demonstrations of the parasitic motions than as an actual bearing in an interferometer. Sizes Calculating the coordinates of 𝑃 at initial position 𝑧 =0 and 2 0 of all above mentioned interferometers and demonstration at displacement 𝑧 and assuming that the interferometer was bearings were comparable to example cases used in this perfectly aligned at its initial position, the pitch angle 𝛼 can article. be calculated using basic geometry. Figure 12 illustrates pitch angles in several cases with typical porch swing dimensions andwiththe dimensionerror of 0.2mmwhich canbe 3. Conclusion regarded as an practical upper limit. The pitch seems to be proportional to 𝑧 with the Δ𝑙 but the dependence is We have demonstrated experimentally that sufficient motion approximately quadratic with Δℎ.Italsoseems that theerror linearity of the mirror in Michelson interferometer is well 0.2 mm in the arm length 𝑙 causes about 30 times bigger tilt achievable by usingaporchswing bearingwhich hasno angle compared with the equal error in the flexure length ℎ. adjustment mechanism which is oen ft used. We defined the Jones and Young [19] have presented approximations for sufficient linearity by the maximum allowed decrease of the thepitch angleas modulation depth which was 0.05 units over 3 mm mirror −1 travel with 25 mm beam at 3000 cm . eTh corresponding 𝑧 Δ𝑙 (A.3) decrease was achieved experimentally using Helium-Neon 𝛼≈− , if Δℎ = 0, ℎ𝑙 laser and a porch swing which was manufactured using normal machining and assembly tolerances. eTh estimated 𝑧 Δℎ (A.4) 𝛼≈+ , if Δ𝑙 = 0. manufacturing tolerances for the porch swing were proven to 2ℎ 𝑙 be loose enough not to limit the application of the bearing in the FTIR interferometer. Additionally, the equations of The approximations are valid if the displacement 𝑧 is small. the parasitic motions explained the tuning of the poorly With relatively large displacements, the actual flexure lengths machined porch swing. should be replaced with the eeff ctive lengths of (5/6)ℎ [20, 8 Advances in Optical Technologies 300 alignment of this force causes the parasitic rotations which Δ𝑙 = 0.2 mm have to be usually analyzed numerically [21]. However, some Δℎ = 0 𝜇 approximations are presented by Jones and Young [19]and Muranaka et al. [20], whose model is −150 2 (ℎ−2𝑎 ) 𝑡 3 (A.7) 𝛼=𝑧 [1 + ( ) ], −300 2 2 𝑙 ℎ 175 𝑡 −3 −2 −1 0 123 where 𝑡 is the thickness of the flat spring and 𝑎 is the distance Δ𝑙 = 0 of the force F from the lower arm as in Figure 1(a).Itcan be Δℎ = 0.2 mm readilyseenthatthe pitchangle is zero if 𝑎=ℎ/2 .Thetilting from the misaligned driving force is, however, usually much 2 smallerthanthe othererrorsofthe movement as notedby Jones and Young [19]and by us. −3 −2 −1 0 123 References Mirror displacement 𝑧 (mm) 𝑙/ℎ = 50/50 𝑙/ℎ = 100/50 [1] R. R. Jackson, “Continuous scanning interferometers for mid- 𝑙/ℎ = 150/50 𝑙/ℎ = 50/100 infrared spectrometry, chapter: Instrumentation for Mid- and Far-infrared Spectroscopy,” in Hand-Book of Vibrational Spec- Figure 12: eTh pitch angle 𝛼 as a function of mirror displacement troscopy,J.M.Chalmersand P. R. Gritffi hs, Eds.,vol.1,pp. 264– 𝑧 calculated using (A.1a)and (A.1b). In the upper graph, only the 282, John Wiley & Sons, 2002. arm length error is considered, and in the lower one, only the flexure [2] J. Kauppinen, J. Heinonen, and I. Kauppinen, “Interferometers length has error. based on the rational motion,” Applied Spectroscopy Reviews, vol. 39, no. 1, pp. 99–130, 2004. [3] P.R.Gritffi hs andJ.A.deHaseth, Fourier Transform Infrared 22]. If we compare (A.3)and (A.4), we obtain the ratio of the Spectrometry, John Wiley & Sons, 2nd edition, 2007. length errors Δℎ/Δ𝑙 = 2ℎ/𝑧 . Using dimensions of a typical [4] D.R.Nohavec,L.S.Schwartz, andD.L.Trumper,“Super- design, the ratio is about (2 × 50 mm)/(3 mm) ≈30,which Hybrid Magnetic Suspensions for Interferometric Scanners,” agrees with the values in Figure 12 calculated from (A.1a)and JSME International Journal Series C,vol.40, no.4,pp. 570–583, (A.1b). 1997. We compared these approximations numerically with [5] H. Kobayashi, “Interferometric monitor for greenhouse gases (IMG)—project technical report,” Tech. Rep., IMG Mission our model using the same cases as in Figure 12 and noted Operation & Verification Committee CRIEPI, 1999. that if there is only Δ𝑙 = 0.2 mm or both Δ𝑙=Δℎ= [6] R. P. Walker and J. D. Rex, “Interferometer design and data han- 0.2 mm, the pitch from (A.3)dieff rsfrom( A.1a)and (A.1b) dling in a high vibration environment—part I: interferometer less than about 1%. However, (A.4) differed even tens of design,” in Multiplex and/or High-rTh oughput Spectroscopy ,G. percent which was a result mainly from very small numerical A. Vanasse, Ed., vol. 191 of Proceedings of SPIE,pp.88–91,August values. eTh measurement results of Jones and Young [ 19]and Muranaka et al. [20] have tte fi d quite well to both the pitch [7] R. V. Jones, “Parallel and rectilinear spring movements,” Journal approximations and our model. of Scientific Instruments , vol. 28, pp. 38–41, 1951. The nonparallelism of the neutral axes of the flexure [8]G.W.Small,R.T.Kroutil,J.T.Ditillo, andW.R.Loerop, elements may happen, for example, if the flexures are rotated “Detection of atmospheric pollutants by direct analysis of to each other by the angle 𝜃 around the 𝑧 axis. According to passive fourier transform infrared interferograms,” Analytical Hatheway [21], the yaw angle is Chemistry, vol. 60, no. 3, pp. 264–269, 1988. [9] R.J.Huppi,R.B.Shipley,and E. R. Huppi, “Balloon-borne fourier spectrometer using a focal plane detector array,” in (A.5) 𝛽= . 2𝑙ℎ Multiplex and/or High-rTh oughput Spectroscopy ,G.A.Vanasse, Ed., vol. 191 of Proceedings of SPIE, pp. 26–32, August 1979. The effective length of the flexure has to be considered in this [10] J. C. Kemp and R. J. Huppi, “Rocket-borne cryogenic Michelson equation when necessary. The principal axes of inertia of the interferometer,” in Multiplex and/or High-rTh oughput Spec- flexures become nonparallel when the flexure elements are troscopy,G.A.Vanasse,Ed.,vol.191 of Proceedings of SPIE,pp. rotated to each other by the angle 𝜙 around the 𝑦 axis which 135–142, August 1979. is the case if the ends of the arms are not parallel. Hatheway [11] R. A. Hanel, B. Schlacman, F. D. Clark et al., “eTh Nimbus [21] has given the yaw angle as III Michelson interferometer,” in Proceedings of the Aspen International Conference on Fourier Spectroscopy,G.A.Vanasse, A. S. Jn,and D. J. Baker, Eds.,pp. 231–241, AirForce Cambridge (A.6) 𝛽= . Research Laboratories, Optical Physics Laboratory, United States Air Force, January 1971. Ideally, thedriving forceliesalong alinewhich is parallel [12] E. H. Wishnow, R. Wurtz, S. Blais-Ouellette et al., “Visible to the 𝑧 axis andrunsthrough thecenterofthe cross-section imaging Fourier transform spectrometer: design and calibra- on the -plane of the porch swing [20, 21]. The nonideal tion,” in Instrument Design and Performance for Optical/Infrared Tilt angle ( rad) Tilt angle ( rad) 𝑥𝑦 𝜙𝑧 𝜃𝑧 Advances in Optical Technologies 9 Ground-based Telescopes,M.Iye andA.F.M.Moorwood,Eds., vol. 4841 of Proceeding of SPIE, pp. 1067–1077, August 2002. [13] G. L. Auth, “Ruggedized compact interferometer requiring minimum isolation from mechanical vibrations,” US Patent 4,693,603, 1987. [14] D. R. Strait, “Moving mirror tilt adjust mechanism in an inter- ferometer,” US Patent 4,991,961, February 1991. [15] R. F. Lacey, “Support for a moving mirror in an interferometer,” US Patent 4,710,001, December 1987. [16] G. R. Walker, “Precision frictionless flexure based linear translation mechanism insensitive to thermal and vibrational environments,” US Patent 6,836,968, 2005. [17] K. C. Schreiber, “Support for a movable mirror in an interfer- ometer,” Patent Application Publication 2002/0149777, 2002. [18] T. Vali ¨ kyla¨ and J. Kauppinen, “Modulation depth of Michelson interferometer with gaussian beam,” Applied Optics,vol.50, pp. 6671–6677, 2011. [19] R. V. Jones and I. R. Young, “Some parasitic deflexions in paral- lel spring movements,” Journal of Scienticfi Instruments ,vol.33, no.1,article 305, pp.11–15,1956. [20] Y. Muranaka,M.Inaba,T.Asano,and E. Furukawa,“Parasitic rotations in parallel spring movements,” Bulletin of the Japan Society of Precision Engineering,vol.25, no.3,pp. 208–213, 1991. [21] A. E. Hatheway, “Alignment of flexure stages for best rectilinear performance,” in Optomechanical and Precision Instrument Design,A.E.Hatheway, Ed., vol. 2542 of Proceedings of SPIE, pp.70–80,July1995. [22] A. E. Hatheway, “The kinetic center of the cantilever beam,” in Optomechanical Design and Precision Instruments,A.E. Hatheway, Ed., vol. 3132 of Proceedings of SPIE, pp. 218–222, [23] S. T. Smith, Flexures-Elements of Elastic Mechanisms, Gordon and Breach Science Publishers, 1st edition, 2000. [24] S. Awtar, Synthesis and analysis of parallel kinematic XY efl xure mechanisms [Ph.D. thesis], Massachusetts Institute of Technol- ogy, 2003. [25] S. Awtar, A. H. Slocum, and E. Sevincer, “Characteristics of beam-based flexure modules,” Journal of Mechanical Design, Transactions of the ASME,vol.129,no. 6, pp.625–639,2007. [26] E. Onillon, S. Henein, P. eTh urillat, J. Krauss, and I. Kjelberg, “Interferometer scanning mirror mechanism,” in Mechatronic Systems 2002: A Proceedings Volume From the 2nd Ifac Confer- ence, M. Tomizuka, Ed., Ifac Proceedings Series, International Federation of Automatic Control, Elsevier, 2003. [27] G. C. Auguson and N. O. Young, “A liquid-helium-cooled Michelson interferometer,” in Aspen International Conference on Fourier Spectroscopy,G.A.Vanasse,A.S.Jn, andD.J.Baker, Eds., pp. 281–288, Air Force Cambridge Research Laboratories, Optical Physics Laboratory, United States Air Force, January 1971. International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal

Advances in Optical TechnologiesHindawi Publishing Corporation

Published: May 30, 2013

References