Spatial-Mode Analysis of Micromachined Optical ... - IEEE Xplore
JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 15, NO. 4, AUGUST 2006
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Spatial-Mode Analysis of Micromachined Optical Cavities Using Electrothermal Mirror Actuation Wei Liu and Joseph J. Talghader, Member, IEEE
Abstract—The performance of optical microcavities is limited by spectral degradation resulting from thermal deformation and fabrication imperfections. In this paper, we study the spatial-mode properties of micromirror optical cavities with respect to commonly seen aberrations. Electrothermal actuation is used to slightly adjust the shape and position of micromirrors and study the effects this has on the spatial-mode structure of the cavity spectrum. The shapes of the micromirrors are changed using Joule heating with thermal expansion deformation. Significant differences in mirror tilt, curvature, and astigmatism are measured, but the tilt has by far the biggest impact on cavity finesse and resolution. We demonstrate that unwanted higher order spatial modes can be suppressed electrically and an amplitude reduction for the higher order modes of over 60% has been obtained with a tuning current of 5.5 mA. A fundamental mode finesse of approximately 60 is maintained throughout tuning. These tunable cavities have great potential in applications using cavity arrays or requiring dynamic mode control. [1559]
Fig. 1. Schematic graph of electrically tunable micromirrors. The mirror temperature is controlled by Joule heating with a current sent through the mirror and its supports. The Joule heating will increase the mirror temperature and in turn change the mirror shape.
I. INTRODUCTION
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PTICAL microcavities, such as might be used as filters for pressure sensing [1]–[3], chemical detection [4] or optical communication [5]–[8] are sensitive to spectral degradation. This is particularly true for cavities in integrated arrays where process nonuniformities will cause the micromirrors that form different cavities to have slightly different alignments. Without individual adjustment, when one cavity is properly aligned, tilt will be present in other cavities. Several authors have addressed the issue of mirror distortion in micromachining, but only a small number have examined their implications on optical cavities. For mirrors, Dunn et al. [9] described bimorph stress and thermal expansion effects in micromirrors and identified three general regions for mirror elastic behavior. Liu et al. [10], [11] discussed optical coating design techniques to create thermally stable mirrors and a method to control mirror curvature using Joule heating. In addition, adaptive deformable micromirrors have shown great success in correcting piston and tilt [12]–[14] and less frequently for correcting curvature and spherical aberrations [15] for systems with millimeter to centimeter-scale apertures. For optical cavities, Moon et al. [16] investigated the performance limit of parallel cavities and showed that thermal deformation and fabrication imperfections decrease cavity finesse and degrade performance. Manuscript received March 29, 2005; revised November 23, 2005. Subject Editor H. Zappe. The authors are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JMEMS.2006.878881
Fig. 2. Schematic graph of tunable micro-cavities. A tunable micromirror is placed as one mirror of a Fabry–Pérot optical cavity. The second mirror is a flat silicon wafer with a high reflectivity coating. The tunable mirror is fabricated on a silicon substrate. The substrate is then backside etched to provide an opening around the mirror area so that the mirror would not be impacted by reflection or absorption from the substrate. The two mirrors are then assembled together to form the tunable cavity.
Ideally, a fully adaptive mirror system could be used to control the aberrations in each mirror of a microcavity array. Unfortunately, the complex machinery of adaptive optics would be difficult to fit into an array where the individual apertures are . In this paper, electrothermal actuation on the order of 100 is used to slightly adjust the shape and position of micromirrors and study the effects this has on the spatial-mode structure of the cavity spectrum. We focus on microcavities built from very simple but highly aberrated micromirrors because they clearly show the relative importance of various aberrations and aberration control. The cavities were formed using electrically tunable micromirrors whose shape and orientation were controlled by an applied current. Schematic graphs of a tunable mirror and a tunable cavity are shown in Fig. 1 and Fig. 2. Typical micromirrors were composed of a structural material and deposited optical coatings. The thermal expansion mismatch between different layers led to thermally induced changes that tilted the
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mirrors when they were heated. Since the mirrors had very low thermal conductance, they were very sensitive to thermally induced shape changes and relatively little current was needed to tilt them sufficiently to control cavity mode structure and suppress unwanted higher order spatial modes. An amplitude reduction for the higher order modes of over 60% was obtained using a tuning current of 5.5 mA. A fundamental mode finesse of approximately 60 is maintained throughout tuning. II. BACKGROUND A. Optical Cavity Spectrum In order to understand the effects of fabrication imperfections on optical cavity performance, we will derive an ideal cavity spectrum and examine the effects of mirror tilt, curvature, and astigmatism on it. A schematic diagram of a tunable cavity is shown in Fig. 2. A shape tunable micromirror is placed as one mirror of a Fabry–Pérot optical cavity. The second mirror is a flat silicon wafer with a high reflectivity coating. The substrate is assumed to be backside-etched to provide an opening around the mirror area so that the mirror would not be impacted by reflection or absorption from the substrate. The two mirrors are assembled together to form the tunable cavity. (The experimental fabrication sequence will be described more fully later.) Assume that the radii of curvature of the two mirrors are and , respectively, the mirrors are at position and , respectively, and the cavity length is . The equation for the transverse electric field of a cavity mode is [17]
(1)
where the
parameters are defined as (4) (5)
These results give the field and frequency of each cavity mode. Any electric field that propagates inside the cavity can be represented as a sum of the fields of one or more modes. Any , that is incident upon the cavity from outside field, will excite modes inside the cavity. The excited modes and their relative amplitudes can be calculated using many of the same principles of as a Fourier expansion [17] (6) The normalization condition is given by
(7) where and are magnetic permeability and electric permittivity, respectively. The excited mode amplitudes are then given by
(8) and is the power coupled into mode . Note that each mode has a coefficient that represents what fraction of the original power ends up in that mode. This coupling efficiency into a given spatial mode is defined as
where is the th Hermite function, and are the beam spot size and beam radius evaluated at , respectively, and (9) (2) In this equation, is the confocal parameter representing the propagation distance over which a minimum spot size beam will expand to of its original diameter. Equation (1) is only accurate when the mode is well confined by the finite mirrors (see [18, 19.5]). For the cavities used in this study, typical values are cavity length , , curved mirror radius of , and , so the fundamental mode waist is about 38.7 , the mode size at the side mirror is about 40.8 , and the highest mode that is still well confined by the cavity is , where . In other words, this equation is a good approximation for modes up to 5. The resonant frequency of the (axial transverse) mode is given by [17] (3)
These results set the foundation for a spectrum calculation of any optical cavity with suitably high Fresnel number. Given the cavity configuration and incident field, the cavity mode can be calculated and then the coupling coefficients of each mode can be obtained. Combining the coupling coefficients with the mode frequency information, the spectrum can be computed. When the mirror shapes change, the cavity parameters change accordingly and result in cavity modes changes; this in turn changes the coupling coefficient for each mode and yields a different spectrum. As will be seen in the experimental section, under an applied current, the mirrors thermally deform and show measurable changes in tilt and curvature. The impact of a change in mirror curvature on the mode spectrum is straightforward and can be taken into account by simply making the curvature term temperature dependent. The impact from the tilting is more complicated. Consider the cavity in Fig. 3 which is formed by a spherical mirror and a flat mirror (the type of cavity used in this project). For the flat mirror, the optical axis should be
LIU AND TALGHADER: SPATIAL-MODE ANALYSIS OF MICROMACHINED OPTICAL CAVITIES
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the and higher order modes and introduces fine structure into the cavity spectrum. As an example, consider a flat-curved cavity with cavity length . Assume that the curved mirror has a curvature , the operating wavelength is , and the incident beam is an order-00 Hermite–Gaussian beam on the optical axis and has a waist of 20 at the flat mirror position. We first calculate the factors
(10) Fig. 3. Impact of mirror tilt. The cavity is formed by a spherical mirror and a flat mirror. The optical axis of this cavity is the straight line perpendicular to the flat mirror and passing through the center of the sphere. Suppose that the incident beam is on-axis originally. When mirror is at its original position, the optical axis is solid line AB (note that B is the center of the sphere in this case, which is different from the center of the mirror A). If the mirror rotates about point A (such as what happens when the mirror is heated), the sphere center moves from point B to point C , and the optical axis of the system changes to line CD . However, the incident beam is still in its original direction AB , so the incident beam is now off-axis and this increases the coupling efficiency into higher modes and decreases the coupling into fundamental mode.
perpendicular to the mirror surface; for the spherical mirror, the optical axis should pass through the center of the sphere. Therefore, the optical axis of such a cavity is the straight line perpendicular to the flat mirror and passing through the center of the sphere. When mirror is at its original position, the optical axis is solid line (note that is the center of the sphere in this case, which is different from the center of the mirror ). Suppose that the incident beam is on-axis originally. When a current is applied to the mirror, as will be seen in the experimental section, the mirror shows no measurable translational movement, i.e. the center of the mirror stays at the same position , but the mirror rotates about point and the sphere center now moves from point to point . Recall that the optical axis of this cavity is the straight line perpendicular to the flat mirror and passing through the center of the sphere, so now the optical axis of the system changes to line . However, the incident beam is still in its original direction , so the incident beam is now off-axis and this increases the coupling efficiency into higher modes and decreases the coupling into the fundamental mode. A side effect of tilting is that the mirror is no longer symmetric about the optical axis, so the integral limit in the spectrum decomposition (8) needs to be adjusted accordingly. The cavity length is also changed under tilting, and its main effect is to change the absolute value of mode frequency. The change in cavity length can be neglected compared to other effects if only relative mode position is of concern. The micromirrors used in this study have slightly different ( 5%) curvatures along the two axes of the mirror. Attention must be paid to use the correct parameters, but the cavity mode in each direction can still be calculated in the same manner. (See [17] 2.11 and [18] 16.4 for more details.) Notice that the fundamental mode frequencies are solely determined by cavity length and are the same in both directions. However, the higher order modes frequencies are dependent on the parameters and are different in the two directions. This splits
The mode waist is at the flat mirror position [18] and the waist size of the fundamental mode can be calculated from (11) can be obtained with (1), in So the equation for mode which is the beam spot size at the left cavity mirror (assume that the incident beam coming from left) and can be calculated by
(12) because the waist is at the flat mirror. Notice that The highest mode that is still well confined by this cavity is , where . Modes that have higher than this propagate in the cavity with very high diffraction losses because the mode size is relatively large compared to the mirror size, and therefore very little power will exist in these modes. From the assumption that the incident beam is an order-00 Hermite-Gaussian beam on the optical axis and has a waist of at the flat mirror position, notice that , the incident field at the curved mirror is given by (13) where (14) and are the beam spot size and beam radius evaluated at the curved mirror, respectively. Notice that , at the curved mirror, so (15) (16) Now that we have the mode equation and the incident field, we can calculate the power coupled into each mode using (8) (17)
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Fig. 4. Simulated spectrum of (a) a flat-curved cavity and (b) with introduced curvature change. (a) The cavity length L = 1 mm, the curved mirror has a curvature R = 10 mm, the operating wavelength = 1:57 m, and the incident beam is an order-00 Hermite–Gaussian beam on the optical axis with a waist of 20 m at the flat mirror position. Note that the incident beam is not a perfect match to the cavity and higher order modes are present. (b) The cavity is the same except for that the radius of curvature of the curved mirror changes from 10 to 9 mm, and the main effect is change of position (frequency) of the higher modes.
Fig. 5. Simulated spectrum of a flat-curved cavity with introduced astigmatism or tilt. (a) The cavity is the same as the cavity used in Fig. 4 except for that the curved mirror has curvature of 9 mm and 10 mm in two perpendicular directions (e.g., astigmatism). As can be seen from the graph, the higher modes degenerate and introduce fine structure to the spectrum; this effect can be explained from the results in Fig. 4, i.e., higher mode frequency changes when mirror curvature changes. (b) The cavity is the same as the cavity used in Fig. 4 except for that the mirror is tilted 2 10 rad. As can be seen from the graph, a large amount of energy is now coupled into the first higher mode.
In realistic situations, only the lowest few modes have nonnegligible coupled power. For the cavity and incident field in this example, the normalized coupling coefficients are , , , , , . All other modes have coupling coefficients less than 0.0001 and their contributions are negligible. The frequency of each mode can be calculated with (3)
mirror is tilted , and a large amount of energy is now coupled into the first higher mode. From these results, it is possible to tune the cavity spectrum via control of mirror shape and orientation.
(18) so the distance between two adjacent higher modes is approximately 10% of the distance between two adjacent longitudinal modes. Combining the mode coupling coefficients with the mode frequency information, the cavity spectrum can be calculated and the result is shown in Fig. 4. The spectrum of cavity with aberrated mirrors can be calculated similarly. Fig. 4 and Fig. 5 show the spectrums of the same cavity in the above example except for an introduced aberration. In Fig. 4(b), the radius of curvature of the curved mirror changes from 10 mm to 9 mm, and the main effect is change of position(frequency) of the higher modes. In Fig. 5(a), the curved mirror has curvature of 9 mm and 10 mm in two perpendicular directions, therefore has astigmatism aberration. As can be seen from the graph, the higher modes are almost degenerate and introduce fine structure to the spectrum; this effect can be explained from the results in Fig. 4, i.e. higher mode frequency changes when mirror curvature changes. By far the most significant change happens when the mirror is tilted. In Fig. 5(b), the
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B. Micromirror Shape Control Micromirrors are often coated with various materials to yield desired optical properties. Typically the mirror body and coating materials have different coefficients of thermal expansion, which make the mirrors susceptible to deformation with changing temperature. Compared to their macro-scale counterparts, the thermal effects in micromirrors can be very strong because the coating of a micromirror is often comparable in thickness to the mirror body itself. The micromirrors used in this study have an elliptic shape with their radii of curvature (from here on called curvature) slightly different ( 5%) in the x and y directions. When a mirror is heated, its curvature changes by about the same magnitude in both the and directions. In addition to the change in curvature, one also sees a slight change in the tilt of the mirror, possibly due to random fabrication imperfections. Experimental results show that under the conditions used for this study, the mirror has no measurable translational movement when heated. Therefore, the main aberrations seen in these mirrors are defocus, tilt and astigmatism. Coma and spherical aberrations in the mirrors are negligible throughout the experiment. The electrothermal tuning in this work occurs due to asymmetries in the fabrication and thermal bimorph effects. The tilt arises because certain supports deflect upward relative to others causing a shift in mirror angle. While random asymmetries are used in this work, the asymmetry could of course be designed
LIU AND TALGHADER: SPATIAL-MODE ANALYSIS OF MICROMACHINED OPTICAL CAVITIES
Fig. 6. Interferometric images of a mirror under various applied currents. These images are taken without readjusting the measurement apparatus, therefore, any change shown in the image is due to the mirror itself. As can be seen from the image, when current changes and the mirror is at a different temperature, the fringe patterns in all images are mostly still concentric circles, which indicates that, the mirror is still in spherical shape with slight aberrations. On the other hand, the centers of the circles are not at the same position relative to the mirror under different currents, which indicates that the angle between the measurement beam and the mirror surface changes. In other words, the mirror is tilted relative to its original orientation. Also, it can be seen from the graph that the mirror shows no measurable translational movement.
into the structure by making one support wider than the others or adding a small amount of metal or dielectric on top of one support. The curvature tuning arises because of bimorph deformation between the reflective metal and polysilicon mirror structure. Such deformation is easily controlled [11]. III. EXPERIMENT A. Device Fabrication and Measurement Setup A schematic graph of the devices used in this study is shown in Fig. 1. The devices were fabricated at the Nanofabrication Center (NFC) at the University of Minnesota. The devices were thin circular polysilicon plates coated with thin Ti/Au for high and thickness reflectivity. The plates are of radius 100 . The thickness of the Ti and Au layers are 3 and 60 0.5 nm, respectively. An oxide sacrificial layer was used under the polysilicon structural layer. The polysilicon layer was also used to create the posts shown in Fig. 1. Openings through the backside of the wafer were etched using a deep-trench system so that the mirror would not be impacted by reflection or absorption from the substrate. A standard concentrated HF etch was used to release the mirrors. The devices were then wire-bonded so that a small current could be run through the mirror to induce heating. Fig. 6 shows interferometric images of one of the mirrors under various applied currents.
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The measurement setup is shown in Fig. 7. As can be seen, from the graph, the setup consists of a laser source, beam shaping optics, cavity, and an imaging and measurement system. The laser source used is a Tunics-Plus, a tunable external cavity laser made by Photonetics. For the data shown during in this paper, the laser wavelength was fixed at 1.57 measurement unless specified otherwise, but the change in cavity length has been converted to an equivalent change in frequency for ease in analyzing the free spectral range. The beam shaping optics are used to adjust the beam waist size and position to provide an appropriate feed into the cavity. The output from the laser source is first collimated, then magnified and then focused down before entering the cavity. The beam shaping optics were designed so that the incident beam into the . cavity has a beam waist of 19 The beam then entered a Fabry–Pérot optical cavity which is formed by a tunable curved micromirror and a flat mirror. The flat mirror is a silicon wafer coated with a high reflectivity multilayer. The curved mirror is fabricated PECVD using the aforementioned process and is wire-bonded and packaged onto a custom-made PC-board. The package is attached to a 2-D stage so that the mirror can be positioned in the plane perpendicular to the incident beam; this is needed to align the incident beam and the mirror. There are also knobs on the hosting frame which can be used to adjust the orientation of the mirror. The cavity length is tuned using a piezoelectric scanner connected to the flat mirror. The cavity spectrum is obtained by scanning the mirror through a free spectral range for a given applied current. As mentioned above, the laser wavelength was during measurement, but the change in kept fixed at 1.57 cavity length has been converted to an equivalent change in frequency for ease in analyzing the free spectral range. The rest of the apparatus are for imaging and measurement. An IR camera is utilized to display an enlarged image of the mirror and the incident beam. This helps to ensure proper alignment. The transmitted intensity was detected using an InGaAs detector and the signal was sent to a lock-in amplifier. A chopper in the light path modulates the signal and provides a reference for the lock-in amplifier. The data is then collected and analyzed with a computer. B. Testing Methodology The mirrors are first characterized using a Zygo phase-measurement interference microscope to get the relationship between mirror shape and applied current. In a typical experiment, the mirror is first thermally cycled several times between room temperature and the highest operating current. This step is necessary due to stress relaxation observed in the samples [10]. After such treatment, the thermal behavior of the mirror stabilizes as long as it is operated below the maximum cycled current. If the mirror is to be operated beyond the original limit, this thermal cycling step must be repeated. The mirror surface profile was then measured under various currents. The surface profile provides the height data across the mirror, and mirror curvature and tilting data can be extracted from the profile. These data are later used to model the tunable cavity.
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Fig. 7. Tunable cavity measurement setup. The setup consists of a laser source, beaming shaping optics, cavity, and an imaging and measurement system. The beam shaping optics are used to adjust the beam waist size and position to provide an appropriate feed into the cavity. The cavity is a Fabry–Pérot optical cavity formed by a tunable curved micromirror and a flat mirror.
The next step is to configure the measurement system. First the laser beam is characterized using a Spiricon LBA-100A laser beam analyzer to determine its waist size and waist position, the beam shaping optics are then adjusted accordingly until the incident beam into the cavity has the desired properties. Next the flat mirror is adjusted to be perpendicular to the incident beam. Next the incident beam and the curved mirror are aligned such that the beam passes through the center of the mirror. This is done with the help of the imaging system. The mirror position and orientation are then fine-tuned to get a clean spectrum. Spatial-mode testing was performed by scanning the mirror through a free spectral range under various applied currents. The current starts from zero and then increases in steps of 0.5 mA between measurement scans up to a maximum value of 5.5 mA. The spectrum was taken at each applied current, and the results are analyzed and compared with simulation results. For parameters used in the simulation, the mirror curvature and tilt were extracted from the Zygo data, and the cavity length can be determined using the following method. Change the incident laser wavelength and measure the transmitted signal intensity after the cavity to find the wavelength for several adjacent transmission peaks. The condition for transmission peaks is that
where is some integer, is the wavelength and length, therefore, for two adjacent peaks,
is the cavity
so
The change in cavity length can be controlled precisely through the positioning system, and this method was applied at several cavity lengths and the data was fitted to the above formula to determine the cavity length.
C. Experimental Results The mirrors are first characterized with a Zygo phase-measurement interference microscope. Fig. 6 shows the interferometric image of a mirror under various applied currents. These images are taken without re-adjusting the measurement apparatus, therefore, any change shown in the image is due to the mirror itself. As can be seen from the image, when the current changes and the mirror is at a different temperature, the fringe patterns in all images are mostly still concentric circles, which indicates that the mirror is still in spherical shape with slight aberrations. On the other hand, the centers of the circles are not at the same position relative to the mirror under different currents, which indicates that the angle between the measurement beam and the mirror surface changes. In other words, the mirror is tilted relative to its original orientation. Also, it can be seen from the graph is that the mirror shows no measurable translational movement. Both curvature and tilt data can be extracted from the surface profile using the mirror position under 4.5 mA applied current as a reference. When the current is 4.5 mA, the mirror curvature is 10 mm along one axis and 10.2 mm along the other. When the current is 5.0 mA, the mirror curvature changes to 9.8 and . When the current 10 mm, and the mirror is tilted is 5.5 mA, the mirror curvature changes to 9.5 mm and 9.7 mm, . and the mirror is tilted In summary, under different applied currents, the mirror has different curvature and tilt. When used in a cavity, these will alter the cavity configuration and generate different mode profiles, as evidenced by further experiments. Fig. 8 shows the cavity spectrum under different applied currents. The cavity length is 1.2 mm, measured using the method described in Section III-B. The mirror curvature and tilt are those calculated from the Zygo data. The incident beam waist is . The results shown at the flat mirror and the waist size is 19 in Fig. 8 clearly demonstrate the suppression of higher order , the cavity modes using electrical tuning. With first higher mode has an amplitude of about 32% of that of the fundamental mode, which indicates that a significant amount of
LIU AND TALGHADER: SPATIAL-MODE ANALYSIS OF MICROMACHINED OPTICAL CAVITIES
Fig. 8. Measured cavity spectrum under various applied currents. The cavity length is 1.2 mm. The incident beam waist is at the flat mirror and the waist size is 19 m. (a) With I = 4:5 mA, the first higher mode has an amplitude of about 32% of that of the fundamental mode, which indicates that a significant amount of energy is coupled into the higher mode. As the current goes up, the mirror temperature changes and the induced tilt is in favor of better alignment, as a result more energy is coupled into the fundamental mode and the amplitude of the higher mode decreases. (b) When I = 5:0 mA, the first higher mode amplitude is about 11% to that of the fundamental mode. (c) With I = 5:5 mA, the first higher mode almost disappears. This is because the first higher mode is of odd symmetry but the incident field is of even symmetry when mirror and beam are properly aligned, therefore, the coupling efficiency into the first higher mode is zero.
energy is coupled into the higher mode. As the current goes up, the mirror temperature changes and the induced tilt is in favor of better alignment; as a result more energy is coupled into the fundamental mode and the amplitude of the higher mode decreases. , the first higher mode amplitude is about 11% When , the first of that of the fundamental mode. And with higher mode almost disappears. This is because the first higher mode is of odd symmetry but the incident field is of even symmetry when mirror and beam are properly aligned, therefore, the coupling efficiency into the first higher mode is zero. On the other hand, the amplitude of the second higher mode increases to about 13% of the amplitude of the fundamental mode. IV. DISCUSSION To better understand these results, the cavity configuration data is fed into a simulation program described in the background to calculate the cavity spectrum, and the results are shown in Fig. 9. The simulation results show the same
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Fig. 9. Simulated cavity spectrum under various applied currents considering only tilt aberrations. The results show the same general trend as the experimental results in Fig. 8—when applied current increases and the mirror is heated, more and more energy is coupled into the fundamental mode and the amplitude of higher modes decrease. Moreover, the numerical results from the simulation agree with the experimental results quite well. These results reinforce the conclusion that the cavity spectrum can be tuned electrically and unwanted higher modes can be suppressed effectively.
general trend as the experimental results—when applied current increasesand the mirror is heated, more and more energy is coupled into the fundamental mode and the amplitude of higher modes decrease. Moreover, the numerical results from the simulation agree with the experimental results quite well. These results reinforce the conclusion that the cavity spectrum can be tuned electrically and unwanted higher modes can be suppressed effectively. For the difference between the experiments and simulations, one source of error is that the incident beam waist may not be at the exact position of the flat mirror, which would cause different coupling coefficients. An interesting side note is that higher modes have lower finesse, and the effect is more obvious as the cavity length increases. An example is shown in Fig. 10. The cavity length is 3.2 mm. As can be seen from the graph, the fundamental mode still has a high finesse about 60, but the finesse of the first higher order mode peak is only about 30, and the higher order mode peaks become wider and wider while the finesse becomes lower and lower. This does not come as a surprise though since higher modes have larger diffraction loss and are expected to have lower finesse. And when cavity length increases, the Fresnel number decreases and the effect become even more apparent.
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mirror cavities. If the cavity length is tuned and the mirror curvature is not changed, the positions of the higher order spatial-mode frequencies will shift and the spectral resolution will degrade. A cavity with tunable curvature mirrors could be of tremendous benefit here. V. SUMMARY
Fig. 10. Measured finesse in different modes. The cavity length is 3.2 mm. The fundamental mode has a high finesse about 60, but the finesse of the first higher order mode peak is only about 30, and the higher order mode peaks become wider and wider while the finesse becomes lower and lower. This is because higher modes have larger diffraction loss.
Notice that the experimental nonidealities are apparent in this figure. Such nonidealities are also present in the experimental results shown in Fig. 8; however, the diffraction loss is lower and the magnitude of the fundamental mode is much larger in Fig. 8, so the nonidealities are not as evident when compared to the fundamental mode. The mirror shape and spectral tuning results have good short-term and long-term repeatability. Neither show significant change under continuous measurement over a few days. No significant difference was found in the data when measured again after two months. The experimental results in last section clearly demonstrated that the cavity spectrum can be electrically tuned and unwanted higher modes can be suppressed. This is particularly useful for applications using cavity arrays, such as sensor arrays used to detect toxic gases. With MEMS technology, many cavities can be fabricated on a single substrate and therefore reduce cost. However, the cavity mirror orientation can be slightly different due to non-uniformity in the fabrication process, which means that there is no guarantee that when one cavity is in good alignment other cavities will be properly aligned as well. In addition, for the array to be useful, these cavities must be independently controllable. Electrical tunability therefore is very useful in such cases. Each mirror can then be individually addressed and controlled without affecting the operation of other cavities. Another potential application is in scanning cavities that must be made confocal over a wide tuning range. In a confocal cavity, the mirror curvature equals the cavity length, which causes all higher order spatial-mode frequencies fall exactly on the longitudinal mode frequencies or halfway between them. This property allows confocal cavities to extract meaningful spectra from extended sources, a feat that is often impossible for parallel
The spatial-mode properties of micromirror optical cavities are described in this paper with respect to commonly seen aberrations. The shape of the micromirrors in the cavities can be changed using Joule heating in conjunction with thermal expansion deformation. Significant differences in mirror tilt, curvature, and astigmatism are measured, but the tilt has by far the biggest impact on cavity finesse and resolution. Unwanted higher order spatial modes can be suppressed electrically and an amplitude reduction for the higher order modes of over 60% has been obtained with a tuning current of 5.5 mA. These tunable cavities have great potential in applications that use cavity arrays and require dynamic mode control. REFERENCES [1] Y. Kim and D. P. Neikirk, “Micromachined Fabry-Perot cavity pressure transducer,” IEEE Photon. Technol. Lett., vol. 7, pp. 1471–1473, Dec. 1995. [2] J. Han, “Novel fabrication and characterization method of Fabry-Perot microcavity pressure sensors,” Sens. Actuators A, Phys., vol. A75, pp. 168–175, May 1999. [3] W. J. Wang, D, G. Guo, R. M. Lin, and X. W. Wang, “A single-chip diaphragm-type miniature Fabry-Perot pressure sensor with improved cross-sensitivity to temperature,” Meas. Sci. Technol., vol. 15, pp. 905–910, May 2004. [4] H. Alause, F. Grasdepot, J. P. Malzac, W. Knap, and J. Hermann, “Micromachined optical tunable filter for domestic gas sensors,” Sens. Actuators B, Chem., vol. B43, pp. 18–23, Sep. 1997. [5] C. Lin, J. Fu, and J. S. Harris, Jr., “Widely tunable Al O -GaAs DBR filters with variable tuning characteristics,” IEEE J. Sel. Topics Quantum Electron., vol. 10, pp. 614–621, May/Jun. 2004. [6] A. Spisser, R. Ledantec, C. Seassal, J. L. Leclercq, T. Benyattou, D. Rondi, R. Blondeau, G. Guillot, and P. Viktorovitch, “Highly selective 1.55 m InP/air gap micromachined Fabry-Perot filter for optical communications,” Electron. Lett., vol. 34, pp. 453–455, Mar. 1998. [7] S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett., vol. 15, pp. 434–436, Mar. 2003. [8] A. Tarraf, F. Riemenschneider, M. Strassner, J. Daleiden, S. Irmer, H. Halbritter, H. Hillmer, and P. Meissner, “Continuously tunable 1.55-m VCSEL implemented by precisely curved dielectric top DBR involving tailored stress,” IEEE Photon. Technol. Lett., vol. 16, pp. 720–722, Mar. 2004. [9] M. L. Dunn, Y. Zhang, and V. M. Bright, “Deformation and structural stability of layered plate microstructures subjected to thermal loading,” J. Microelectromech. Syst., vol. 11, pp. 372–384, Aug. 2002. [10] W. Liu and J. J. Talghader, “Thermally invariant dielectric coatings for micromirrors,” Applied Optics, vol. 41, pp. 3285–3293, Jun. 2002. [11] ——, “Current-controlled curvature of coated micromirrors,” Opt. Lett., vol. 28, pp. 932–934, Jun. 2003. [12] W. D. Cowan, M. K. Lee, B. M. Welsh, V. M. Bright, and M. C. Roggemann, “Surface micromachined segmented mirrors for adaptive optics,” IEEE J. Sel. Topics Quantum Electron., vol. 5, pp. 90–101, Jan./Feb. 1999. [13] G. Vdovin, S. Middlehoek, and P. M. Sarro, “Technology and applications of micromachined silicon adaptive mirrors,” Opt. Eng., vol. 36, pp. 1382–1390, May 1997. [14] T. G. Bifano, R. Krishnamoorthy, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane surface-micromachined silicon deformable mirror,” Opt. Eng., vol. 36, pp. 1354–1360, May 1997.
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[15] Y. Shao, D. L. Dickensheets, and P. Himmer, “3-D MOEMS mirror for laser beam pointing and focus control,” IEEE J. Sel. Topics Quantum Electron., vol. 10, pp. 528–535, May/Jun. 2004. [16] J. Moon and A. M. Shkel, “Analysis of imperfections in a micromachined tunable-cavity interferometer,” Proc. SPIE, vol. 4334, pp. 46–53, Mar. 2001. [17] A. Yariv, Optical Electronics in Modern Communications. Oxford, U.K.: Oxford University, 1997. [18] A. E. Siegman, Lasers. New York: University Science Books, 1986. Wei Liu the B.S. degree in physics from University of Science and Technology of China(USTC), Hefei, China, in 1995. He received the M.S. and Ph.D. degrees in electrical engineering from University of Minnesota, Minneapolis, in 2000 and 2005, respectively. He is currently with Microsoft as a software engineer.
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Joseph J. Talghader (S’93–M’95) received the B.S. degree in electrical engineering from Rice University, Houston, TX, in 1988. He was awarded an NSF Graduate Fellowship and attended the University of California at Berkeley, where he received the M.S. and Ph.D. degrees in 1993 and 1995, respectively. From 1992 to 1993, he worked at Texas Instruments as a Process Development Engineer. After graduating from Berkeley in 1995, he joined Waferscale Integration. In 1997, he joined the faculty at the University of Minnesota, Minneapolis, where he is now an Associate Professor. Dr. Talghader has received 3M Nontenured Faculty Awards on three occasions. He has served on various program committees and reviews, including service as Program chair of the 2003 IEEE/LEOS Optical MEMS Conference and as Guest Editor of the IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS 2004 Special Issue on Optical Microsystems. He will be Conference Chair of the 2006 IEEE/LEOS Optical MEMS Conference at Yellowstone/Big Sky.
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Spatial-Mode Analysis of Micromachined Optical Cavities Using Electrothermal Mirror Actuation Wei Liu and Joseph J. Talghader, Member, IEEE
Abstract—The performance of optical microcavities is limited by spectral degradation resulting from thermal deformation and fabrication imperfections. In this paper, we study the spatial-mode properties of micromirror optical cavities with respect to commonly seen aberrations. Electrothermal actuation is used to slightly adjust the shape and position of micromirrors and study the effects this has on the spatial-mode structure of the cavity spectrum. The shapes of the micromirrors are changed using Joule heating with thermal expansion deformation. Significant differences in mirror tilt, curvature, and astigmatism are measured, but the tilt has by far the biggest impact on cavity finesse and resolution. We demonstrate that unwanted higher order spatial modes can be suppressed electrically and an amplitude reduction for the higher order modes of over 60% has been obtained with a tuning current of 5.5 mA. A fundamental mode finesse of approximately 60 is maintained throughout tuning. These tunable cavities have great potential in applications using cavity arrays or requiring dynamic mode control. [1559]
Fig. 1. Schematic graph of electrically tunable micromirrors. The mirror temperature is controlled by Joule heating with a current sent through the mirror and its supports. The Joule heating will increase the mirror temperature and in turn change the mirror shape.
I. INTRODUCTION
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PTICAL microcavities, such as might be used as filters for pressure sensing [1]–[3], chemical detection [4] or optical communication [5]–[8] are sensitive to spectral degradation. This is particularly true for cavities in integrated arrays where process nonuniformities will cause the micromirrors that form different cavities to have slightly different alignments. Without individual adjustment, when one cavity is properly aligned, tilt will be present in other cavities. Several authors have addressed the issue of mirror distortion in micromachining, but only a small number have examined their implications on optical cavities. For mirrors, Dunn et al. [9] described bimorph stress and thermal expansion effects in micromirrors and identified three general regions for mirror elastic behavior. Liu et al. [10], [11] discussed optical coating design techniques to create thermally stable mirrors and a method to control mirror curvature using Joule heating. In addition, adaptive deformable micromirrors have shown great success in correcting piston and tilt [12]–[14] and less frequently for correcting curvature and spherical aberrations [15] for systems with millimeter to centimeter-scale apertures. For optical cavities, Moon et al. [16] investigated the performance limit of parallel cavities and showed that thermal deformation and fabrication imperfections decrease cavity finesse and degrade performance. Manuscript received March 29, 2005; revised November 23, 2005. Subject Editor H. Zappe. The authors are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JMEMS.2006.878881
Fig. 2. Schematic graph of tunable micro-cavities. A tunable micromirror is placed as one mirror of a Fabry–Pérot optical cavity. The second mirror is a flat silicon wafer with a high reflectivity coating. The tunable mirror is fabricated on a silicon substrate. The substrate is then backside etched to provide an opening around the mirror area so that the mirror would not be impacted by reflection or absorption from the substrate. The two mirrors are then assembled together to form the tunable cavity.
Ideally, a fully adaptive mirror system could be used to control the aberrations in each mirror of a microcavity array. Unfortunately, the complex machinery of adaptive optics would be difficult to fit into an array where the individual apertures are . In this paper, electrothermal actuation on the order of 100 is used to slightly adjust the shape and position of micromirrors and study the effects this has on the spatial-mode structure of the cavity spectrum. We focus on microcavities built from very simple but highly aberrated micromirrors because they clearly show the relative importance of various aberrations and aberration control. The cavities were formed using electrically tunable micromirrors whose shape and orientation were controlled by an applied current. Schematic graphs of a tunable mirror and a tunable cavity are shown in Fig. 1 and Fig. 2. Typical micromirrors were composed of a structural material and deposited optical coatings. The thermal expansion mismatch between different layers led to thermally induced changes that tilted the
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mirrors when they were heated. Since the mirrors had very low thermal conductance, they were very sensitive to thermally induced shape changes and relatively little current was needed to tilt them sufficiently to control cavity mode structure and suppress unwanted higher order spatial modes. An amplitude reduction for the higher order modes of over 60% was obtained using a tuning current of 5.5 mA. A fundamental mode finesse of approximately 60 is maintained throughout tuning. II. BACKGROUND A. Optical Cavity Spectrum In order to understand the effects of fabrication imperfections on optical cavity performance, we will derive an ideal cavity spectrum and examine the effects of mirror tilt, curvature, and astigmatism on it. A schematic diagram of a tunable cavity is shown in Fig. 2. A shape tunable micromirror is placed as one mirror of a Fabry–Pérot optical cavity. The second mirror is a flat silicon wafer with a high reflectivity coating. The substrate is assumed to be backside-etched to provide an opening around the mirror area so that the mirror would not be impacted by reflection or absorption from the substrate. The two mirrors are assembled together to form the tunable cavity. (The experimental fabrication sequence will be described more fully later.) Assume that the radii of curvature of the two mirrors are and , respectively, the mirrors are at position and , respectively, and the cavity length is . The equation for the transverse electric field of a cavity mode is [17]
(1)
where the
parameters are defined as (4) (5)
These results give the field and frequency of each cavity mode. Any electric field that propagates inside the cavity can be represented as a sum of the fields of one or more modes. Any , that is incident upon the cavity from outside field, will excite modes inside the cavity. The excited modes and their relative amplitudes can be calculated using many of the same principles of as a Fourier expansion [17] (6) The normalization condition is given by
(7) where and are magnetic permeability and electric permittivity, respectively. The excited mode amplitudes are then given by
(8) and is the power coupled into mode . Note that each mode has a coefficient that represents what fraction of the original power ends up in that mode. This coupling efficiency into a given spatial mode is defined as
where is the th Hermite function, and are the beam spot size and beam radius evaluated at , respectively, and (9) (2) In this equation, is the confocal parameter representing the propagation distance over which a minimum spot size beam will expand to of its original diameter. Equation (1) is only accurate when the mode is well confined by the finite mirrors (see [18, 19.5]). For the cavities used in this study, typical values are cavity length , , curved mirror radius of , and , so the fundamental mode waist is about 38.7 , the mode size at the side mirror is about 40.8 , and the highest mode that is still well confined by the cavity is , where . In other words, this equation is a good approximation for modes up to 5. The resonant frequency of the (axial transverse) mode is given by [17] (3)
These results set the foundation for a spectrum calculation of any optical cavity with suitably high Fresnel number. Given the cavity configuration and incident field, the cavity mode can be calculated and then the coupling coefficients of each mode can be obtained. Combining the coupling coefficients with the mode frequency information, the spectrum can be computed. When the mirror shapes change, the cavity parameters change accordingly and result in cavity modes changes; this in turn changes the coupling coefficient for each mode and yields a different spectrum. As will be seen in the experimental section, under an applied current, the mirrors thermally deform and show measurable changes in tilt and curvature. The impact of a change in mirror curvature on the mode spectrum is straightforward and can be taken into account by simply making the curvature term temperature dependent. The impact from the tilting is more complicated. Consider the cavity in Fig. 3 which is formed by a spherical mirror and a flat mirror (the type of cavity used in this project). For the flat mirror, the optical axis should be
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the and higher order modes and introduces fine structure into the cavity spectrum. As an example, consider a flat-curved cavity with cavity length . Assume that the curved mirror has a curvature , the operating wavelength is , and the incident beam is an order-00 Hermite–Gaussian beam on the optical axis and has a waist of 20 at the flat mirror position. We first calculate the factors
(10) Fig. 3. Impact of mirror tilt. The cavity is formed by a spherical mirror and a flat mirror. The optical axis of this cavity is the straight line perpendicular to the flat mirror and passing through the center of the sphere. Suppose that the incident beam is on-axis originally. When mirror is at its original position, the optical axis is solid line AB (note that B is the center of the sphere in this case, which is different from the center of the mirror A). If the mirror rotates about point A (such as what happens when the mirror is heated), the sphere center moves from point B to point C , and the optical axis of the system changes to line CD . However, the incident beam is still in its original direction AB , so the incident beam is now off-axis and this increases the coupling efficiency into higher modes and decreases the coupling into fundamental mode.
perpendicular to the mirror surface; for the spherical mirror, the optical axis should pass through the center of the sphere. Therefore, the optical axis of such a cavity is the straight line perpendicular to the flat mirror and passing through the center of the sphere. When mirror is at its original position, the optical axis is solid line (note that is the center of the sphere in this case, which is different from the center of the mirror ). Suppose that the incident beam is on-axis originally. When a current is applied to the mirror, as will be seen in the experimental section, the mirror shows no measurable translational movement, i.e. the center of the mirror stays at the same position , but the mirror rotates about point and the sphere center now moves from point to point . Recall that the optical axis of this cavity is the straight line perpendicular to the flat mirror and passing through the center of the sphere, so now the optical axis of the system changes to line . However, the incident beam is still in its original direction , so the incident beam is now off-axis and this increases the coupling efficiency into higher modes and decreases the coupling into the fundamental mode. A side effect of tilting is that the mirror is no longer symmetric about the optical axis, so the integral limit in the spectrum decomposition (8) needs to be adjusted accordingly. The cavity length is also changed under tilting, and its main effect is to change the absolute value of mode frequency. The change in cavity length can be neglected compared to other effects if only relative mode position is of concern. The micromirrors used in this study have slightly different ( 5%) curvatures along the two axes of the mirror. Attention must be paid to use the correct parameters, but the cavity mode in each direction can still be calculated in the same manner. (See [17] 2.11 and [18] 16.4 for more details.) Notice that the fundamental mode frequencies are solely determined by cavity length and are the same in both directions. However, the higher order modes frequencies are dependent on the parameters and are different in the two directions. This splits
The mode waist is at the flat mirror position [18] and the waist size of the fundamental mode can be calculated from (11) can be obtained with (1), in So the equation for mode which is the beam spot size at the left cavity mirror (assume that the incident beam coming from left) and can be calculated by
(12) because the waist is at the flat mirror. Notice that The highest mode that is still well confined by this cavity is , where . Modes that have higher than this propagate in the cavity with very high diffraction losses because the mode size is relatively large compared to the mirror size, and therefore very little power will exist in these modes. From the assumption that the incident beam is an order-00 Hermite-Gaussian beam on the optical axis and has a waist of at the flat mirror position, notice that , the incident field at the curved mirror is given by (13) where (14) and are the beam spot size and beam radius evaluated at the curved mirror, respectively. Notice that , at the curved mirror, so (15) (16) Now that we have the mode equation and the incident field, we can calculate the power coupled into each mode using (8) (17)
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Fig. 4. Simulated spectrum of (a) a flat-curved cavity and (b) with introduced curvature change. (a) The cavity length L = 1 mm, the curved mirror has a curvature R = 10 mm, the operating wavelength = 1:57 m, and the incident beam is an order-00 Hermite–Gaussian beam on the optical axis with a waist of 20 m at the flat mirror position. Note that the incident beam is not a perfect match to the cavity and higher order modes are present. (b) The cavity is the same except for that the radius of curvature of the curved mirror changes from 10 to 9 mm, and the main effect is change of position (frequency) of the higher modes.
Fig. 5. Simulated spectrum of a flat-curved cavity with introduced astigmatism or tilt. (a) The cavity is the same as the cavity used in Fig. 4 except for that the curved mirror has curvature of 9 mm and 10 mm in two perpendicular directions (e.g., astigmatism). As can be seen from the graph, the higher modes degenerate and introduce fine structure to the spectrum; this effect can be explained from the results in Fig. 4, i.e., higher mode frequency changes when mirror curvature changes. (b) The cavity is the same as the cavity used in Fig. 4 except for that the mirror is tilted 2 10 rad. As can be seen from the graph, a large amount of energy is now coupled into the first higher mode.
In realistic situations, only the lowest few modes have nonnegligible coupled power. For the cavity and incident field in this example, the normalized coupling coefficients are , , , , , . All other modes have coupling coefficients less than 0.0001 and their contributions are negligible. The frequency of each mode can be calculated with (3)
mirror is tilted , and a large amount of energy is now coupled into the first higher mode. From these results, it is possible to tune the cavity spectrum via control of mirror shape and orientation.
(18) so the distance between two adjacent higher modes is approximately 10% of the distance between two adjacent longitudinal modes. Combining the mode coupling coefficients with the mode frequency information, the cavity spectrum can be calculated and the result is shown in Fig. 4. The spectrum of cavity with aberrated mirrors can be calculated similarly. Fig. 4 and Fig. 5 show the spectrums of the same cavity in the above example except for an introduced aberration. In Fig. 4(b), the radius of curvature of the curved mirror changes from 10 mm to 9 mm, and the main effect is change of position(frequency) of the higher modes. In Fig. 5(a), the curved mirror has curvature of 9 mm and 10 mm in two perpendicular directions, therefore has astigmatism aberration. As can be seen from the graph, the higher modes are almost degenerate and introduce fine structure to the spectrum; this effect can be explained from the results in Fig. 4, i.e. higher mode frequency changes when mirror curvature changes. By far the most significant change happens when the mirror is tilted. In Fig. 5(b), the
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B. Micromirror Shape Control Micromirrors are often coated with various materials to yield desired optical properties. Typically the mirror body and coating materials have different coefficients of thermal expansion, which make the mirrors susceptible to deformation with changing temperature. Compared to their macro-scale counterparts, the thermal effects in micromirrors can be very strong because the coating of a micromirror is often comparable in thickness to the mirror body itself. The micromirrors used in this study have an elliptic shape with their radii of curvature (from here on called curvature) slightly different ( 5%) in the x and y directions. When a mirror is heated, its curvature changes by about the same magnitude in both the and directions. In addition to the change in curvature, one also sees a slight change in the tilt of the mirror, possibly due to random fabrication imperfections. Experimental results show that under the conditions used for this study, the mirror has no measurable translational movement when heated. Therefore, the main aberrations seen in these mirrors are defocus, tilt and astigmatism. Coma and spherical aberrations in the mirrors are negligible throughout the experiment. The electrothermal tuning in this work occurs due to asymmetries in the fabrication and thermal bimorph effects. The tilt arises because certain supports deflect upward relative to others causing a shift in mirror angle. While random asymmetries are used in this work, the asymmetry could of course be designed
LIU AND TALGHADER: SPATIAL-MODE ANALYSIS OF MICROMACHINED OPTICAL CAVITIES
Fig. 6. Interferometric images of a mirror under various applied currents. These images are taken without readjusting the measurement apparatus, therefore, any change shown in the image is due to the mirror itself. As can be seen from the image, when current changes and the mirror is at a different temperature, the fringe patterns in all images are mostly still concentric circles, which indicates that, the mirror is still in spherical shape with slight aberrations. On the other hand, the centers of the circles are not at the same position relative to the mirror under different currents, which indicates that the angle between the measurement beam and the mirror surface changes. In other words, the mirror is tilted relative to its original orientation. Also, it can be seen from the graph that the mirror shows no measurable translational movement.
into the structure by making one support wider than the others or adding a small amount of metal or dielectric on top of one support. The curvature tuning arises because of bimorph deformation between the reflective metal and polysilicon mirror structure. Such deformation is easily controlled [11]. III. EXPERIMENT A. Device Fabrication and Measurement Setup A schematic graph of the devices used in this study is shown in Fig. 1. The devices were fabricated at the Nanofabrication Center (NFC) at the University of Minnesota. The devices were thin circular polysilicon plates coated with thin Ti/Au for high and thickness reflectivity. The plates are of radius 100 . The thickness of the Ti and Au layers are 3 and 60 0.5 nm, respectively. An oxide sacrificial layer was used under the polysilicon structural layer. The polysilicon layer was also used to create the posts shown in Fig. 1. Openings through the backside of the wafer were etched using a deep-trench system so that the mirror would not be impacted by reflection or absorption from the substrate. A standard concentrated HF etch was used to release the mirrors. The devices were then wire-bonded so that a small current could be run through the mirror to induce heating. Fig. 6 shows interferometric images of one of the mirrors under various applied currents.
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The measurement setup is shown in Fig. 7. As can be seen, from the graph, the setup consists of a laser source, beam shaping optics, cavity, and an imaging and measurement system. The laser source used is a Tunics-Plus, a tunable external cavity laser made by Photonetics. For the data shown during in this paper, the laser wavelength was fixed at 1.57 measurement unless specified otherwise, but the change in cavity length has been converted to an equivalent change in frequency for ease in analyzing the free spectral range. The beam shaping optics are used to adjust the beam waist size and position to provide an appropriate feed into the cavity. The output from the laser source is first collimated, then magnified and then focused down before entering the cavity. The beam shaping optics were designed so that the incident beam into the . cavity has a beam waist of 19 The beam then entered a Fabry–Pérot optical cavity which is formed by a tunable curved micromirror and a flat mirror. The flat mirror is a silicon wafer coated with a high reflectivity multilayer. The curved mirror is fabricated PECVD using the aforementioned process and is wire-bonded and packaged onto a custom-made PC-board. The package is attached to a 2-D stage so that the mirror can be positioned in the plane perpendicular to the incident beam; this is needed to align the incident beam and the mirror. There are also knobs on the hosting frame which can be used to adjust the orientation of the mirror. The cavity length is tuned using a piezoelectric scanner connected to the flat mirror. The cavity spectrum is obtained by scanning the mirror through a free spectral range for a given applied current. As mentioned above, the laser wavelength was during measurement, but the change in kept fixed at 1.57 cavity length has been converted to an equivalent change in frequency for ease in analyzing the free spectral range. The rest of the apparatus are for imaging and measurement. An IR camera is utilized to display an enlarged image of the mirror and the incident beam. This helps to ensure proper alignment. The transmitted intensity was detected using an InGaAs detector and the signal was sent to a lock-in amplifier. A chopper in the light path modulates the signal and provides a reference for the lock-in amplifier. The data is then collected and analyzed with a computer. B. Testing Methodology The mirrors are first characterized using a Zygo phase-measurement interference microscope to get the relationship between mirror shape and applied current. In a typical experiment, the mirror is first thermally cycled several times between room temperature and the highest operating current. This step is necessary due to stress relaxation observed in the samples [10]. After such treatment, the thermal behavior of the mirror stabilizes as long as it is operated below the maximum cycled current. If the mirror is to be operated beyond the original limit, this thermal cycling step must be repeated. The mirror surface profile was then measured under various currents. The surface profile provides the height data across the mirror, and mirror curvature and tilting data can be extracted from the profile. These data are later used to model the tunable cavity.
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Fig. 7. Tunable cavity measurement setup. The setup consists of a laser source, beaming shaping optics, cavity, and an imaging and measurement system. The beam shaping optics are used to adjust the beam waist size and position to provide an appropriate feed into the cavity. The cavity is a Fabry–Pérot optical cavity formed by a tunable curved micromirror and a flat mirror.
The next step is to configure the measurement system. First the laser beam is characterized using a Spiricon LBA-100A laser beam analyzer to determine its waist size and waist position, the beam shaping optics are then adjusted accordingly until the incident beam into the cavity has the desired properties. Next the flat mirror is adjusted to be perpendicular to the incident beam. Next the incident beam and the curved mirror are aligned such that the beam passes through the center of the mirror. This is done with the help of the imaging system. The mirror position and orientation are then fine-tuned to get a clean spectrum. Spatial-mode testing was performed by scanning the mirror through a free spectral range under various applied currents. The current starts from zero and then increases in steps of 0.5 mA between measurement scans up to a maximum value of 5.5 mA. The spectrum was taken at each applied current, and the results are analyzed and compared with simulation results. For parameters used in the simulation, the mirror curvature and tilt were extracted from the Zygo data, and the cavity length can be determined using the following method. Change the incident laser wavelength and measure the transmitted signal intensity after the cavity to find the wavelength for several adjacent transmission peaks. The condition for transmission peaks is that
where is some integer, is the wavelength and length, therefore, for two adjacent peaks,
is the cavity
so
The change in cavity length can be controlled precisely through the positioning system, and this method was applied at several cavity lengths and the data was fitted to the above formula to determine the cavity length.
C. Experimental Results The mirrors are first characterized with a Zygo phase-measurement interference microscope. Fig. 6 shows the interferometric image of a mirror under various applied currents. These images are taken without re-adjusting the measurement apparatus, therefore, any change shown in the image is due to the mirror itself. As can be seen from the image, when the current changes and the mirror is at a different temperature, the fringe patterns in all images are mostly still concentric circles, which indicates that the mirror is still in spherical shape with slight aberrations. On the other hand, the centers of the circles are not at the same position relative to the mirror under different currents, which indicates that the angle between the measurement beam and the mirror surface changes. In other words, the mirror is tilted relative to its original orientation. Also, it can be seen from the graph is that the mirror shows no measurable translational movement. Both curvature and tilt data can be extracted from the surface profile using the mirror position under 4.5 mA applied current as a reference. When the current is 4.5 mA, the mirror curvature is 10 mm along one axis and 10.2 mm along the other. When the current is 5.0 mA, the mirror curvature changes to 9.8 and . When the current 10 mm, and the mirror is tilted is 5.5 mA, the mirror curvature changes to 9.5 mm and 9.7 mm, . and the mirror is tilted In summary, under different applied currents, the mirror has different curvature and tilt. When used in a cavity, these will alter the cavity configuration and generate different mode profiles, as evidenced by further experiments. Fig. 8 shows the cavity spectrum under different applied currents. The cavity length is 1.2 mm, measured using the method described in Section III-B. The mirror curvature and tilt are those calculated from the Zygo data. The incident beam waist is . The results shown at the flat mirror and the waist size is 19 in Fig. 8 clearly demonstrate the suppression of higher order , the cavity modes using electrical tuning. With first higher mode has an amplitude of about 32% of that of the fundamental mode, which indicates that a significant amount of
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Fig. 8. Measured cavity spectrum under various applied currents. The cavity length is 1.2 mm. The incident beam waist is at the flat mirror and the waist size is 19 m. (a) With I = 4:5 mA, the first higher mode has an amplitude of about 32% of that of the fundamental mode, which indicates that a significant amount of energy is coupled into the higher mode. As the current goes up, the mirror temperature changes and the induced tilt is in favor of better alignment, as a result more energy is coupled into the fundamental mode and the amplitude of the higher mode decreases. (b) When I = 5:0 mA, the first higher mode amplitude is about 11% to that of the fundamental mode. (c) With I = 5:5 mA, the first higher mode almost disappears. This is because the first higher mode is of odd symmetry but the incident field is of even symmetry when mirror and beam are properly aligned, therefore, the coupling efficiency into the first higher mode is zero.
energy is coupled into the higher mode. As the current goes up, the mirror temperature changes and the induced tilt is in favor of better alignment; as a result more energy is coupled into the fundamental mode and the amplitude of the higher mode decreases. , the first higher mode amplitude is about 11% When , the first of that of the fundamental mode. And with higher mode almost disappears. This is because the first higher mode is of odd symmetry but the incident field is of even symmetry when mirror and beam are properly aligned, therefore, the coupling efficiency into the first higher mode is zero. On the other hand, the amplitude of the second higher mode increases to about 13% of the amplitude of the fundamental mode. IV. DISCUSSION To better understand these results, the cavity configuration data is fed into a simulation program described in the background to calculate the cavity spectrum, and the results are shown in Fig. 9. The simulation results show the same
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Fig. 9. Simulated cavity spectrum under various applied currents considering only tilt aberrations. The results show the same general trend as the experimental results in Fig. 8—when applied current increases and the mirror is heated, more and more energy is coupled into the fundamental mode and the amplitude of higher modes decrease. Moreover, the numerical results from the simulation agree with the experimental results quite well. These results reinforce the conclusion that the cavity spectrum can be tuned electrically and unwanted higher modes can be suppressed effectively.
general trend as the experimental results—when applied current increasesand the mirror is heated, more and more energy is coupled into the fundamental mode and the amplitude of higher modes decrease. Moreover, the numerical results from the simulation agree with the experimental results quite well. These results reinforce the conclusion that the cavity spectrum can be tuned electrically and unwanted higher modes can be suppressed effectively. For the difference between the experiments and simulations, one source of error is that the incident beam waist may not be at the exact position of the flat mirror, which would cause different coupling coefficients. An interesting side note is that higher modes have lower finesse, and the effect is more obvious as the cavity length increases. An example is shown in Fig. 10. The cavity length is 3.2 mm. As can be seen from the graph, the fundamental mode still has a high finesse about 60, but the finesse of the first higher order mode peak is only about 30, and the higher order mode peaks become wider and wider while the finesse becomes lower and lower. This does not come as a surprise though since higher modes have larger diffraction loss and are expected to have lower finesse. And when cavity length increases, the Fresnel number decreases and the effect become even more apparent.
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mirror cavities. If the cavity length is tuned and the mirror curvature is not changed, the positions of the higher order spatial-mode frequencies will shift and the spectral resolution will degrade. A cavity with tunable curvature mirrors could be of tremendous benefit here. V. SUMMARY
Fig. 10. Measured finesse in different modes. The cavity length is 3.2 mm. The fundamental mode has a high finesse about 60, but the finesse of the first higher order mode peak is only about 30, and the higher order mode peaks become wider and wider while the finesse becomes lower and lower. This is because higher modes have larger diffraction loss.
Notice that the experimental nonidealities are apparent in this figure. Such nonidealities are also present in the experimental results shown in Fig. 8; however, the diffraction loss is lower and the magnitude of the fundamental mode is much larger in Fig. 8, so the nonidealities are not as evident when compared to the fundamental mode. The mirror shape and spectral tuning results have good short-term and long-term repeatability. Neither show significant change under continuous measurement over a few days. No significant difference was found in the data when measured again after two months. The experimental results in last section clearly demonstrated that the cavity spectrum can be electrically tuned and unwanted higher modes can be suppressed. This is particularly useful for applications using cavity arrays, such as sensor arrays used to detect toxic gases. With MEMS technology, many cavities can be fabricated on a single substrate and therefore reduce cost. However, the cavity mirror orientation can be slightly different due to non-uniformity in the fabrication process, which means that there is no guarantee that when one cavity is in good alignment other cavities will be properly aligned as well. In addition, for the array to be useful, these cavities must be independently controllable. Electrical tunability therefore is very useful in such cases. Each mirror can then be individually addressed and controlled without affecting the operation of other cavities. Another potential application is in scanning cavities that must be made confocal over a wide tuning range. In a confocal cavity, the mirror curvature equals the cavity length, which causes all higher order spatial-mode frequencies fall exactly on the longitudinal mode frequencies or halfway between them. This property allows confocal cavities to extract meaningful spectra from extended sources, a feat that is often impossible for parallel
The spatial-mode properties of micromirror optical cavities are described in this paper with respect to commonly seen aberrations. The shape of the micromirrors in the cavities can be changed using Joule heating in conjunction with thermal expansion deformation. Significant differences in mirror tilt, curvature, and astigmatism are measured, but the tilt has by far the biggest impact on cavity finesse and resolution. Unwanted higher order spatial modes can be suppressed electrically and an amplitude reduction for the higher order modes of over 60% has been obtained with a tuning current of 5.5 mA. These tunable cavities have great potential in applications that use cavity arrays and require dynamic mode control. REFERENCES [1] Y. Kim and D. P. Neikirk, “Micromachined Fabry-Perot cavity pressure transducer,” IEEE Photon. Technol. Lett., vol. 7, pp. 1471–1473, Dec. 1995. [2] J. Han, “Novel fabrication and characterization method of Fabry-Perot microcavity pressure sensors,” Sens. Actuators A, Phys., vol. A75, pp. 168–175, May 1999. [3] W. J. Wang, D, G. Guo, R. M. Lin, and X. W. Wang, “A single-chip diaphragm-type miniature Fabry-Perot pressure sensor with improved cross-sensitivity to temperature,” Meas. Sci. Technol., vol. 15, pp. 905–910, May 2004. [4] H. Alause, F. Grasdepot, J. P. Malzac, W. Knap, and J. Hermann, “Micromachined optical tunable filter for domestic gas sensors,” Sens. Actuators B, Chem., vol. B43, pp. 18–23, Sep. 1997. [5] C. Lin, J. Fu, and J. S. Harris, Jr., “Widely tunable Al O -GaAs DBR filters with variable tuning characteristics,” IEEE J. Sel. Topics Quantum Electron., vol. 10, pp. 614–621, May/Jun. 2004. [6] A. Spisser, R. Ledantec, C. Seassal, J. L. Leclercq, T. Benyattou, D. Rondi, R. Blondeau, G. Guillot, and P. Viktorovitch, “Highly selective 1.55 m InP/air gap micromachined Fabry-Perot filter for optical communications,” Electron. Lett., vol. 34, pp. 453–455, Mar. 1998. [7] S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett., vol. 15, pp. 434–436, Mar. 2003. [8] A. Tarraf, F. Riemenschneider, M. Strassner, J. Daleiden, S. Irmer, H. Halbritter, H. Hillmer, and P. Meissner, “Continuously tunable 1.55-m VCSEL implemented by precisely curved dielectric top DBR involving tailored stress,” IEEE Photon. Technol. Lett., vol. 16, pp. 720–722, Mar. 2004. [9] M. L. Dunn, Y. Zhang, and V. M. Bright, “Deformation and structural stability of layered plate microstructures subjected to thermal loading,” J. Microelectromech. Syst., vol. 11, pp. 372–384, Aug. 2002. [10] W. Liu and J. J. Talghader, “Thermally invariant dielectric coatings for micromirrors,” Applied Optics, vol. 41, pp. 3285–3293, Jun. 2002. [11] ——, “Current-controlled curvature of coated micromirrors,” Opt. Lett., vol. 28, pp. 932–934, Jun. 2003. [12] W. D. Cowan, M. K. Lee, B. M. Welsh, V. M. Bright, and M. C. Roggemann, “Surface micromachined segmented mirrors for adaptive optics,” IEEE J. Sel. Topics Quantum Electron., vol. 5, pp. 90–101, Jan./Feb. 1999. [13] G. Vdovin, S. Middlehoek, and P. M. Sarro, “Technology and applications of micromachined silicon adaptive mirrors,” Opt. Eng., vol. 36, pp. 1382–1390, May 1997. [14] T. G. Bifano, R. Krishnamoorthy, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane surface-micromachined silicon deformable mirror,” Opt. Eng., vol. 36, pp. 1354–1360, May 1997.
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[15] Y. Shao, D. L. Dickensheets, and P. Himmer, “3-D MOEMS mirror for laser beam pointing and focus control,” IEEE J. Sel. Topics Quantum Electron., vol. 10, pp. 528–535, May/Jun. 2004. [16] J. Moon and A. M. Shkel, “Analysis of imperfections in a micromachined tunable-cavity interferometer,” Proc. SPIE, vol. 4334, pp. 46–53, Mar. 2001. [17] A. Yariv, Optical Electronics in Modern Communications. Oxford, U.K.: Oxford University, 1997. [18] A. E. Siegman, Lasers. New York: University Science Books, 1986. Wei Liu the B.S. degree in physics from University of Science and Technology of China(USTC), Hefei, China, in 1995. He received the M.S. and Ph.D. degrees in electrical engineering from University of Minnesota, Minneapolis, in 2000 and 2005, respectively. He is currently with Microsoft as a software engineer.
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Joseph J. Talghader (S’93–M’95) received the B.S. degree in electrical engineering from Rice University, Houston, TX, in 1988. He was awarded an NSF Graduate Fellowship and attended the University of California at Berkeley, where he received the M.S. and Ph.D. degrees in 1993 and 1995, respectively. From 1992 to 1993, he worked at Texas Instruments as a Process Development Engineer. After graduating from Berkeley in 1995, he joined Waferscale Integration. In 1997, he joined the faculty at the University of Minnesota, Minneapolis, where he is now an Associate Professor. Dr. Talghader has received 3M Nontenured Faculty Awards on three occasions. He has served on various program committees and reviews, including service as Program chair of the 2003 IEEE/LEOS Optical MEMS Conference and as Guest Editor of the IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS 2004 Special Issue on Optical Microsystems. He will be Conference Chair of the 2006 IEEE/LEOS Optical MEMS Conference at Yellowstone/Big Sky.
