Optical sensitivity analysis of deformed mirrors for microcantilever array IR imaging

Optical sensitivity is a major issue to improve the sensor responsivity and the spatial resolution of uncooled optomechanical focal plane arrays (FPA). The optical sensitivity is closely related to the mirror length and the undesired mirror deformation induced from the imbalanced residual stresses in different layers. In this paper, the influences of mirror length and deformation on the optical sensitivity are discussed by Fourier Optics. Theoretical analysis and experiments demonstrate that the optical sensitivity is seriously degraded by undesired mirror deformation, and that there exists an optimal mirror length which makes the optical sensitivity achieve its maximum under a certain mirror deformation. Based on the results, an optimized mirror configuration is presented to increase the optical sensitivity of substrate-free bi-material microcantilever array (SFBMA). 2009 Optical Society of America OCIS codes: (230.4685) Optical microelectromechanical devices; (070.6110) Spatial filtering; (040.6808) Thermal IR arrays; (120.4630) Optical inspection. References and links 1. M. Kohin and N. Butler, “Performance limits of uncooled VOx microbolometer focal plane arrays,” Proc. SPIE 5406, 447-453(2004). 2. C. M. Hanson, “Barriers to background-limited performance for uncooled IR sensors,” Proc. SPIE 5406, 454-464(2004). 3. J. K. Gimzewski, Ch. Gerber, E. Meyer, and R. R. Schlittler, “Observation of a chemical reaction using a micromechanical sensor,” Chem.Phys. Lett. 217, 589-594 (1994). 4. J. R. Barnes, S. J. Stephenson, M. E. Welland, Ch. Gerber, and J. K.Gimzewski, “Photothermal spectroscopy with femtojoule sensitive using a micromechanical device,” Nature 72, 79-82 ( 1994). 5. E. A. 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Lai, T. Perazzo, Z. Shi, and A. Majumdar, “Photothermal measurements with picoWatt resolution using micro-optomechanical sensors,” Appl. Phys. Lett. 71, 306–308 (1997). 12. T. Perazzo, M. Mao, O. Kwon, and A. Majumdar, “Infrared vision uncooled micro-optomechanical camera,” Appl. Phys. Lett. 74, 3567–3569 (1999). 13. Y. Zhao, M. Y. Mao, and R. Horowitz, “Optomechanical uncooled infrared imaging system: Design, microfabrication, and performance,” J. Microelectromech. S. 11, 136-146 (2002). 14. Z. Duan, Q. Zhang, X. Wu, L. Pan, D. Chen, W. Wang, and Z. Guo, “Uncooled optically readable bimaterial micro-cantilever infrared imaging device,” Chin. Phys. Lett. 20, 2130–2132 (2003). #104140 $15.00 USD Received 18 Nov 2008; revised 20 Feb 2009; accepted 21 Feb 2009; published 4 Mar 2009 (C) 2009 OSA 16 March 2009 / Vol. 17, No. 6 / OPTICS EXPRESS 4367 15. Z. Miao, Q. Zhang, Z. Guo, X. Wu, and D. Chen, “Optical readout method for microcantilever array sensing and its sensitivity analysis,” Opt. Lett. 32, 594-596 (2007). 16. F. Dong, Q. Zhang, D. Chen, L. Pan, Z. Guo, W. Wang, Z. Duan, and X. Wu, “An uncooled optically readable infrared imaging detector,” Sens. Actuators A 133, 236-242 (2007). 17. Z. Miao, Q. Zhang, D. Chen, Z. Guo, F. Dong, Z. Xiong, X. Wu, C. Li, and B. Jiao, “Uncooled IR imaging using optomechanical detectors,” Ultramicroscopy 107, 610-616 (2007). 18. Z. Guo, Q. Zhang, F. Dong, D. Chen, Z. Xiong, Z. Miao, C. Li, B. Jiao, and X. Wu, “Performance analysis of microcantilever arrays for optical readout uncooled infrared imaging,” Sens. Actuators A 137, 13-19 (2007). 19. Z. Xiong, Q. Zhang, J. Gao, X. Wu, D. Chen, and B. Jiao, “The pressure-dependent performance of a substrate-free focal plane array in an uncooled infrared imaging system,” J. Appl. Phys. 102, 113524-1-6 (2007). 20. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. 82, 172-175 (1909). 21. Y. Min and Y. Kim, “In situ measurement of residual stress in micromachined thin films using a specimen with composite-layered cantilevers,” J. Micromech. Microeng. 10, 314-321(2000). 22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 5. 23. C. Li, B. Jiao, S. Shi, D. Chen,T. Ye, Q. Zhang, Z. Guo, F. Dong, and Z. Miao, “A novel uncooled substrate-free optical-readable infrared detector: design, fabrication and performance,” Meas. Sci. Technol. 17, 1-6 (2006). 24. S. Garcia-Blanco, P. Topart, Y. Desroches, J. S. Caron, F. Williamson, C. Alain, and H. Jerominek, “Lowtemperature vacuum hermetic wafer-level package for uncooled microbolometer FPAs,” Proc. SPIE 6884, 68840P-1-8 (2008). 25. T. Cheng and Q. Zhang, “Uncooled Infrared Imaging Using a Substrate-Free Focal-Plane Array,” IEEE Electron. Device Lett. 29, 1218-1221 (2008). 26. D. Grbovic, N. Lavrik, S. Rajic, and P. 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Introduction
Infrared (IR) imaging system plays a critical role both in military and civilian applications ranging from night vision, surveillance, fire detection to spectral imaging.Detection of the spectrum ranging from 8 m to 14 m, so called long wave infrared radiation (LWIR), is of particular interest, not only because this is the atmospheric transmission window but also because it contains the peak of the blackbody spectrum for objects at around room temperature.LWIR detectors can be broadly classified into two categories: photonic and thermal.Although photonic devices are of noise equivalent temperature difference (NETD) as low as 5-10 mK and of short response time, it requires cooling to cryogenic temperature (about 80K) in order to reduce thermal noise.The additional cooling system increases weight and cost.Thermal detectors are based on measuring the amount of heat produced in the detector upon the absorption of IR radiation and can be operated at room temperature.Uncooled vanadium oxide (VO x ) and amorphous silicon (a-Si) microbolometers are presently the technologies of choice for thermally imaging.However, these devices are still expensive, moreover, studies indicate that these technologies are reaching their performance limits (NETD 25mK) [1,2].These limitations drove research attention to exploring other IR detecting techniques with lower cost and the potential of improving performance approaching background.In 1994, Gimzewski et al indicated that bimaterial microcantilever (BM) could be used as a sensitive temperature sensor [3,4], which was demonstrated later being a suitable choice for LWIR detector [5,6].Since then the IR imaging technologies based on BM have attracted more and more attention.With the development of micro electromechanical system (MEMS) technologies, the promising IR detecting techniques based on bimaterial microcantilever (BM) structures appeared.It has been shown that IR detecting technologies based on the thermomechanical effect of BM have the potential of reaching NETD 5-9 mK [7,9].Many inspection techniques, including capacitive and optical, can be used to detect the bending of the BM sensor due to the absorption of the IR radiation [8].Capacitive sensing technique is able to increase the sensitivity of FPA to 36%/K [9,10].However, the need for electrical interconnect to each pixel makes it hard to increase the thermal isolation, and leads to fabrication complexity, which has kept the cost prohibitively high for many commercial applications.Fortunately, these problems could be solved by non-contact optical inspection methods such as optical lever method and interferometry method.The optical lever method is only suitable for a single microcantilever [3,4,7,11].The interferometry method developed the researchers in Berkeley can measure the vertical displacement of a microcantilever array simultaneously [12,13].However, the low resolution of the order of /10 ( is the wave length of inspection light) and the susceptibility to environmental vibration of this method made it unsuitable for engineering applications.We proposed an optical inspection method using knife-edge filter for large array in 2003, which is suitable for both a single cantilever and a large cantilever array.What is deserved to note is that this method solved the low resolution and the susceptibility to environmental vibration successfully [14,15].Based on this optical method, we designed a FPA named substrate-free bimaterial microcantilever array (SFBMA) and obtained room temperature IR images [16][17][18][19].In comparison with traditional FPAs, substrate-free bimaterial microcantilever array (SFBMA) shows the following advantages: (1) up to 80% of the incident IR energy can be absorbed by SFBMA [23]; (2) such a structure (SFBMA) avoids sacrificial layer during releasing process, which greatly decreases fabrication difficulty and solves the problem of easy adhesion between microcantilever and substrate; (3) SFBMA has lower vacuum requirement and longer lifetime due to elimination of the 2 m air gap conductance between the cantilever and the substrate.Even situated in the atmosphere, a room temperature IR image can be obtained using such a structure [19].(Note: A vacuum packaging below 10 mTorr should be achieved and maintained during the lifetime of the package for conventional FPAs [24]).In our following work, a SFBMA containing 100×100 pixels (pixel size: 200 m×200 m) was fabricated, and thermal images of room temperature objects were obtained with sensor responsivity 13 gray/K (Gray was considered as the unit of the digital number obtained from the CCD) and NETD 650mK [17,18].In order to further improve the sensor responsivity and the spatial resolution, a SFBMA with a smaller pixel but higher sensor responsivity should be designed.As known, the thermomechanical sensitivity of SFBMA seriously decreases with diminishing the pixel size [16][17][18].Thus, to improve the optical sensitivity becomes a major issue in order to further improve the sensor responsivity of SFBMA.In the knife-edge filter technique we proposed, changes in the inclination angle of the mirrors of microcantilever array are converted to intensity changes of mirror images on the CCD plane by placing a knife-edge on the spectrum plane.The optical resolution is determined by the spectrum width of the reflecting mirror which is associated with mirror length and undesired mirror deformation.In this paper, the influences of mirror length and undesired deformation on the optical sensitivity are discussed by Fourier Optics, and the analysis is validated experimentally.The operation of SFBMA will be described using Fig. 1(c).The infrared energy from the IR object to SFBMA is absorbed by the SiN x on each pixel.Because the conductivity and the radiation absorptivity of each pixel are constant, the larger the amount of the incident IR energy on each pixel is, the higher the temperature rising of each pixel is.Owing to the difference in thermal expansion coefficients of SiN x and Au, the temperature change results in a proportional bending of the bi-material leg.The mirror has a rigid connection with the bimaterial leg, thus the inclination angle changes with bending of the bi-material leg.As a result, the inclination angle change of each mirror is proportional to the amount of the incident IR energy on each pixel.To increase the change of mirror inclination angle in a finite size, a structure of two-fold interval metal-coated legs was designed.As temperature changes, the coated (bi-material) legs bend while the single-material legs keep unbending.Suppose there is a temperature change on the bi-material leg and 1 is the inclination angle change of the mirror for single-fold leg, then for two-fold interval metal-coated leg, 2 is the inclination angle change of the added bi-material leg.The total inclination angle change of the mirror will be 1 + 2 .Therefore, a larger deflection of the mirror can be achieved.

SFBMA structure
Noting that each microcantilever is directly supported by the honeycomb-like framework instead of substrate, it allows removing substrate silicon at the area where the microcantilevers are situated to form a substrate-free structure.For traditional FPA with substrate in Fig. 2(a), the incident IR flux must get through the Si substrate and the air layer before absorbed by microcantilever.Because of the IR reflection at both of two interfaces between the substrate and the air, only about 50% of the incident IR energy can reach the microcantilever [23].The substrate-free structure in Fig. 2(b) increases the absorption of IR flux.More precisely, over 80% of the incident IR energy can be absorbed.

Knife-edge filtering method
In SFBMA, the inclination angle change of each mirror is proportional to the amount of the incident IR energy on each pixel.Figure 3 shows a schematic diagram of the optical inspection system using a knife-edge filter for simultaneously detecting the inclination angle changes of mirrors.A diverging light from a cold light source (LED) passes through a pin hole and is reflected by a Beam Splitter.This visible light is converted into parallel light, which first passes through a Fourier lens, and then irradiates the entire surface of the SFBMA.Each mirror of SFBMA pixels reflects the visible light, which passes through Fourier lens again and is then turned into a converging light (diffraction spectrum) at the spectrum plane, after which an image of each mirror is formed on the CCD plane by an imaging lens.More specifically, with n×n SFBMA pixels, for instance, n×n spectrums are formed.For a real SFBMA, every mirror of the SFBMA has the same initial inclination angle, because the residual stresses of the bi-material legs are the same due to the same fabrication process.Before IR imaging, the mirrors' spectrums form on the same position on the spectrum plane, and a knife-edge filter is placed on the spectrum plane to block half of the zeroth diffraction spectrum of each mirror.In other words, the amount of light of each spectrum passing through the knife-edge filter is equal.As a result, every mirror is imaged on the CCD plane at a consistent intensity.As an object is focused on the SFBMA, the temperature changes, each mirror tilts due to thermal deformation of the bi-material legs and its spectrum shifts.Therefore, the light flux passing through the knife-edge filter changes.The mirror images with different intensity are formed on the CCD since each mirror inclination angle of the individual pixels of SFBMA differs due to the different temperature changes on the pixels.And the infrared image is converted to a visible image on the CCD.Fig. 3. Schematic diagram of the uncooled infrared imaging system using knife-edge filtering inspection method.

Sensor responsivity
Responsivity S sys of the sensor is a key performance parameter, which is defined as the image gray level change on CCD in response to a unit temperature change of IR object: , The IR energy conversion efficiency H of SFBMA is significantly different from that of the traditional FPAs.Heat on SFBMA should first flows to the honeycomb-like framework support structure, then substrate.Because of the isolation of the framework, the IR energy conversion efficiency H of SFBMA will be higher than that of traditional FPAs.It almost remains unchanged as its pixel size changes [25].S T is proportional to the pixel size [16][17][18][19], that is, shortening mirror (or pixel size) will seriously degrade the sensor responsivity.Then for SFBMA with small pixel size, it is very important to try to increase the optical sensitivity.Optical sensitivity is influenced by mirror length and undesired deformation.With the notation defined in Fig. 4, supposing the length of the mirror is L, and the curvature radius of the mirror along the length direction is R, the distance between the deformed mirror and the ideal mirror at coordinate x is:

Optical sensitivity analysis
Here R x , the first order approximation of z is given as: Suppose the mirror is irradiated by a normally incident plane wave with amplitude of A and wave length of ( =500 nm), the phase difference between the wave reflected by the deformed mirror and the wave reflected by the reference mirror at coordinate x is 2 2 / x R .In terms of Fourier Optics, the light intensity distribution ( ; , ) f I x R L of the diffraction spectrum on the spectrum plane can be expressed as [22]  ( ; , ) ( ) ( / )exp( 2) exp( 2) where f (=100mm) is the focal length of the Fourier lens.In order to facilitate analysis, the light intensity distribution is normalized as: ( ; , ) ( ; , ) ( ; , ) Assuming that the knife-edge filter is placed at coordinate f x on the spectrum plane to block the spectrum (Fig. 4) and the light only ranging from -to f x can pass through the knife-edge filter, the intensity on the CCD becomes: ( , , ) ( ; , ) The slope of the normalized intensity integration ( , , ) f N x R L denotes the normalized optical sensitivity corresponding to the coordinate f x of the knife-edge filter.Since the knife-edge filter is placed in the spectrum plane to block half of the diffraction spectrum in experiment, the slopes of ( , , ) f N x R L at the central position are taken as the normalized optical sensitivity.
Here, it should be noted that the normalized optical sensitivity is proportional to what is defined above.The normalized optical sensitivity is taken as the optical sensitivity : The relationship between the inclination angle change mir d of the mirror and the spectrum shift According to Parseval theorem: Then ( , ) R L can be written as: Equation ( 13) indicates the influences of mirror length and undesired deformation on the optical sensitivity.

Influence of mirror deformation and optimization
For a mirror with a definite length, its undesired deformation will seriously degrade the optical sensitivity.According to Eq. ( 6), Fig. 5(a) shows the normalized intensity distribution in the spectrum plane with different R (L=180 m), from which it can be seen that undesired mirror deformation disperses mirror spectrum, and thus affects the optical sensitivity.Figure 5(b) shows the normalized intensity integration of the curves in Fig. 5(a).As explained above, the slopes of each integration curves at the central position is taken as the optical sensitivity.
For an ideal plane mirror (R= ), the slope 0 12.57 deg -1 .For the deformed mirrors with R=25mm and 5.3 mm, the slopes are 1 2.557 deg -1 and 2 0.4643 deg -1 respectively, and the ratio of curves with respect to mirror deformation curvature (=1/R) (L= 180 m and 50 m), from which it can be apparently seen that the optical sensitivity is seriously degraded by undesired deformation.Unfortunately, the mirrors of fabricated microcantilevers are always curving, which can be seen from a 3D profile of two typical SFBMA pixels obtained by a Veeco Profiler in Fig. 6.The bending probably results from residual stresses in the thin films, which include the intrinsic stress gradient across the SiN x , Cr (Chromium layer were used to enhance adhesion between gold and silicon nitride) and Au films, as well as the imbalanced residual stresses in these films.In principle, the mirror undesired deformation caused by residual stresses can be reduced by optimizing fabricating technologies.However, it was found very difficult to completely eliminate the undesired deformation.According to the Stoney formula [20], the curvature of a bimaterial system is proportional to the thickness ratio (in our case, the ratio is the thickness of Cr and Au to the thickness of SiN x ).Hence, the bending of mirror with thin Cr and Au films is less than that with thick Cr and Au films.In contrast, bi-material legs require slight thick Au films so as to improve S T [16][17][18][19].However, bi-material legs and mirrors are metalized simultaneously in fabrication process of conventional FPAs.In other words, the thickness of Cr and Au films in mirrors is equivalent to that in bi-material legs.To solve this problem, thick metal films for bi-material legs and thin metal films for mirrors are separately deposited.With extremely thin Cr and Au films, a poor mirror reflecting effect was observed.Only when the thickness of Cr layer 5nm and the thickness of Au layer 25nm, a receivable reflectivity could be obtained.Thus, 5-nm-thick chromium and 25-nm-thick gold are used as the reflecting layers.We prepared a series of SFBMAs, whose parameters were shown in Table 1.For SFBMA1 and SFBMA2, the surface profiles along the length direction of their mirrors are measured by a Veeco Profiler, and respectively shown in Fig. 7(a) and Fig. 7(b).The deformation radiuses of these two mirrors are 25mm and 5.3mm approximately.It can be clearly seen that the bending radius was greatly reduced through thinning metal films on mirror.Because the bi-material leg structures are the same, the performance of SFBMA1 is greater than that of SFBMA2.For SFBMA1 and SFBMA3, it can be seen from Fig. 7 that pixel size has little influence on the mirror curvature.

Influence of mirror length and optimization
Although the mirror deformation can be reduced by optimizing mirror structure, it was extremely difficult to avoid the deformation.Based on the fact that the same mirror structure has the same deformation radius [20,21], we investigated the influence of mirror length on the optical sensitivity under the same deformation radius.
Figure 8a shows the curves of the normalized intensity integration distribution with different L (R =25mm) according to Eq. ( 7), and the slopes at the central position of each curves is taken as the optical sensitivity.For planes with L=180 m, 115 m, and 50 m, the slopes are 1 2.557 deg -1 , 4 6.254 deg -1 , and Contrary to our intuition, the optical sensitivity doesn't monotonically increase with the length of mirror.There exists a certain mirror length (optimal length) that makes the optical sensitivity achieve its maximum.This can be easily seen from Fig. 8(b), which is obtained from Eq. ( 13).Compared with SFBMA1, we designed a SFBMA4 with a thinner SiN x film (=0.7 m) by optimization to improve the thermo-mechanical sensitivity S T .The deformation radius of mirror with 0.7-m-thick SiN x film is measured to be 6 mm (see Fig. 10), which is much less than that of mirrors with 2-m-thick SiN x film.Variation of the normalized optical sensitivity with respect to mirror length under 6-mm mirror deformation radius is shown in Fig. 8(b).In order to gain a high optical sensitivity and a comparatively small pixel size, the mirror length and the corresponding pixel size of SFBMA4 are designed to be 55 m and 60 m×60 m respectively.In summary, the optimized SFBMA4 design parameters are shown in Table 1.

Microfabrication
The optimized SFBMA containing 80×80 pixels was fabricated using bulk micromachining technology.As shown schematically in Fig. 9 (In order to see the process more clearly, a portion of the substrate is removed imaginarily), the process consisted typically of only four masking steps (step 2-5)-wet-etch window patterning, micro-cantilever definition, thickmetal patterning for bi-material legs, and thin-metal patterning for mirrors.Sacrificial layer releasing process was not necessary.As a result, the overall level of fabrication complexity was significantly reduced as compared to the fabrication of IR detectors reported previously.The detailed fabrication is as follows.Firstly, a low-pressure chemical vapor (LPCVD) 0.7m-thick SiN x layer was deposited on a double-sided polished Si substrate.Secondly, the wetetch window was patterned in the backside of the wafer using a reactive ion etching method to form the region of releasing the sensor.Thirdly, the topside silicon nitride layer was etched by a reactive ion etching method to define microcantilever.Fourthly, 10-nm-thick chromium and 0.2-m-thick gold were evaporated sequentially and then patterned by lift-off technique to form bi-material legs.Followed by mirror metallization, 5-nm-thick chromium and 25-nmthick gold were evaporated sequentially and then patterned by lift-off technique.Finally, the substrate, Si, was etched by a wet etching method using KOH solution.A microscopic photo of the fabricated SFBMA4 is shown in Fig. 10.

Validation
The optical sensitivity analysis can be validated through testing the optical performance of three fabricated SFBMAs (SFBMA1, SFBMA2, and SFBMA3), whose parameters are shown in Table 1.Using the optical inspection platform in Fig. 3, each SFBMA is located on a stepping motor, and a 12 bit CCD (70 dB) is placed in the imaging plane.The relationship between the image intensity on the CCD and the inclination angle of the mirror is obtained by rotating the stepping motor with a precision of 0.02°, which is equivalent to the shift of the knife-edge filter.For SFBMA1 and SFBMA2, whose mirrors have the same L (=180 m) but different R ( 25mm and 5.3mm) respectively.The experimental results are shown in Fig. 11(a).The slopes of the two curves at the central position are 8410 gray/deg and 1610 gray/deg respectively, and the ratio is approximately 5.2.It could be easily seen that the experimental curves and ratio (5.2) well match the theoretical analysis curves and ratio (5.5) as shown in Fig. 5   With an f/0.8 IR lens, a 12-bit CCD and the background temperature around 25 C , a thermal image of a human hand obtained by SFBMA4 is shown in Fig. 12.There appear to be some blind spots, which failed in the fabrication process.

Imaging results using optimized SFBMA
In Fig. 8(b), the normalized optical sensitivity of SFBMA4 and SFBMA1 are 3.061 deg -1 and 2.557 deg -1 respectively.Since the experimental optical sensitivity of SFBMA1 is 8410 gray/deg, the optical sensitivity of SFBMA4 is prognosticated to be 8410 3.061/ 2.557 10068 gray/deg.Actually, the optical sensitivity of SFBMA4 is approximately 9460 gray/ deg measured by experiment and there is a relative difference of about 6 % from the theoretical prediction (10068 gray/deg).
A temperature controllable IR source with a precision of 0.01 K was used to test the sensor responsivity S sys .Fifty frames of thermal images were captured at each temperature ranging from 26 C to 46 C by a step of 2 C .The sensor responsivity of each pixel in the selected test region (shown in Fig. 13b) was calculated by the relationship between the gray level (intensity) on the CCD and the temperature of the IR source.Figure 13c shows the sensor responsivity distribution.The mean sensor responsivity S sys of SFBMA4 was straightly calculated to be about 22 gray/K.
To estimate a noise equivalent temperature difference (NETD) for the system using SFBMA4, We captured 200 images of 25 C -background with inexistence of any IR objects.The gray fluctuations for the CCD pixels of the selected region are statistically shown in Fig. 13(d).The mean noise fluctuation was found to be about 7 gray.In the IR imaging system using SFBMA4, NETD could be calculated by [18] #104140 -$15.Compared with SFBMA1 (13gray/K, 200 m× 200 m) [17,18], SFBMA4 (22gray/K, 60 m× 600 m) is more sensitive but the pixel size is much smaller.It is demonstrated that a SFBMA with higher sensor responsivity and smaller pixel size was achieved through the optimization in this paper.
When the metal films on mirror are constant (5-nm-thick chromium, 25-nm-thick gold), the decrease of SiN x thickness brings the reduction of R, which sequentially corresponds to the shorter optimal length implying the smaller pixel size.Consequently, a potential improvement of the sensor responsivity and spatial resolution of SFBMA is to reduce the thickness of SiN x properly and then make mirror length to be the corresponding optimal length.

Simple relation between the optimal length and the deformation radius
Equation ( 5) can be rewritten as follows: where erf is the error function, and i 2 =-1.This equation can also be expressed with the parameters R and scaled by the mirror length L, and shows that the important scaling parameters are R / L and / L. Consequently, Eq. ( 13) can be solved with the result: When R is treated as a constant, Eq. ( 16) has a maximum value for 2 / R L 1, in other words: Equation ( 17) fits well with the calculations above.The physical meaning of Eq. ( 17) can be explained as below.
According to the theory of Fourier's optics, the complex amplitude of the mirror spectrum is contributed by all the points along the length direction of the mirror.As showed in Fig. 4, when the length of the mirror is short enough, namely the max delaying phase induced by max Z ( max Z is the distance between the bending mirror and the ideal mirror at the edge (x=L/2) of the mirror.) is less than / 2 .All the points show a positive impact on the center of the spectrum and the central intensity of the spectrum increases with the mirror length.If the length of the mirror is too long, the points whose delaying phases are greater than / 2 will give a negative impact on the center of the spectrum, which will decrease the central intensity of the spectrum.As a result, the central intensity of the spectrum has a maximum for the delaying phase 2 / 2 L R at the edge of the mirror, namely / 2 as L R .

Non-uniformity analysis
The difference in each individual unit, resulting from microfabrication process, inevitably leads to non-uniformity in IR imaging using infrared focal plane array.For SFBMA, like the sensor responsivity, the sensor non-uniformity can be divided into two parts: (1) thermomechanical sensitivity non-uniformity, which results from the non-uniformity of the film thickness, and (2) the optical sensitivity non-uniformity.The non-uniformity of residual stress causes some pixels to have different initial inclination angles from the rest of the array, which means that the knife-edge can't block half of the diffraction spectrums of all the mirrors simultaneously.Actually, the normalized intensity as a function of x f indicates the optical sensitivity of knife-edge being placed at any position.Under a certain non-uniformity of initial inclination angles, the sharper the normalized intensity carve is, the more dispersive optical sensitivity distribution will be obtained.That is, the optical sensitivity non-uniformity increases with the optical sensitivity, which can be validated but not strictly by the following calculations.Assume that the position of the knife-edge related to each mirror spectrum obeys the normal distribution with a mean of zero and a standard deviation of (Here, =0.1mm is used).The optical sensitivity non-uniformity is defined as the standard deviation of optical sensitivity distribution divided by the mean optical sensitivity.Let L=180 m, for the mirrors with R= , R=25mm and R=5.3 mm, the optical sensitivity non-uniformities are respectively calculated to be 36.03%,7.27% and 5.29 % by pseudorandom statistical method, which indicates that undesired mirror deformation can decrease optical sensitivity non-uniformity.When R=25 mm, for planes with L=180 m, 115 m, and 50 m, the optical sensitivity nonuniformities are 7.27%, 15.47% and 5.04% respectively.These results show that decreasing non-uniformity in microfabrication process and increasing the optical sensitivity are equally important.
As shown in Fig. 10, since the area of the mirrors is only 60 % of the SFBMA and the image size of the mirror is larger than the pixel size of the CCD, most of the CCD pixels are not or only partly covered by the mirror images, which causes the responsivities close to zero in Fig. 13(c) are more than others and thus the experimental sensor responsivity is less than the actual responsivity.In theory, the CCD pixels not or only partly covered by the mirror images should not be considered.However, because it is hard to determine these CCD pixels experimentally, as a conservative estimation of the sensor responsivity, they are considered statistically in this paper.This shortcoming can be eliminated by one-to-one correspondence between the pixels of CCD and those of SFBMA.

Alternatives to knife-edge method
Alternatives to the knife-edge method are to use slits with different gap length.The slit can be taken as consisting of two counter-orientated knife-edges.Consequently, the sensitivity using a slit equals the subtraction of the sensitivities of knife-edges at two different positions, whose distance is the gap length of the slit.Therefore, the relation between the optical sensitivity using a slit and the two mirror parameters (Length and deformation) will be similar to that using a knife-edge.The non-uniformity will also increase with the optical sensitivity in the inspection techniques using slits.In order to maximize the subtraction, as shown in Fig. 5(a), one knife-edge should be placed at the position where the normalized intensity equals zero and the other at the position where the normalized intensity is maximal.It should be noted that, for a typical mirror, the normalized intensity has a maximum value at x f = 0 and approaches zero as x f tends to infinity, which elucidates the reason why a knife-edge filter being placed at the centre of the spectrum is used in this paper.

Natural frequency of the SFBMA
When the substrate is removed, the framework that holds the individual pixels will have lower resonance frequency compared to the case when the substrate is present.And such structure is susceptible to mechanical noise.Usually, the influence of mechanical noise should be considered.The resonance of the framework (which is arranged in a crisscross pattern with numbers of two-end fixed beams) can be taken as that of a two-end fixed beam structure.Due to the crisscross constraint, the resonance frequency of framework will be higher than that of a single two-end fixed beam.Here, the resonance frequency of a two-end fixed beam is taken as a lower limit estimation of SFBMA.The first order resonance frequency res f of a two-end fixed beam can be calculated to be 270 Hz, and the mean noise fluctuation of SFBMA4 is measured to be 7 gray (see Fig. 13(d)).To decrease the influence of mechanical noise, a possible solution is to increase the resonance frequency by increasing the thickness of the framework.And such structure has been reported recently [26].A 10-um thick framework can be obtained when the width of the framework is 2 m by this technique.In this case, the first order resonance frequency can be calculated to be 3.88 KHz, thus mechanical noise can be neglected.

Conclusion
For optical inspection technique of bi-material microcantilever array, the optical sensitivity function with respect to mirror length L and mirror deformation radius R of microcantilever is established and validated in this paper.Theoretical analysis shows that the optical sensitivity is seriously degraded by undesired mirror deformation and, under a certain inevitable mirror deformation, it doesn't monotonically increase with the length of mirror, instead it has a maximum value at an optimal mirror length.These analysis results suggest that reducing mirror deformation and making mirror length to be the corresponding optimal length can greatly heighten the optical sensitivity, which are validated experimentally in this paper.An optimized SFBMA is proposed based on the analysis and, successfully fabricated by bulk micromachining techniques.It is demonstrated that the performance of SFBMA, including the sensor responsivity and resolution, can be greatly improved by the optimization.Further improvement of SFBMA's performance can be achieved by reducing the thickness of SiN x properly, and then making mirror length to be the corresponding optimal length.

Figure 1 (
Figure 1(a) shows the overall structure of this SFBMA, while Fig. 1(b) and Fig. 1(c) are both close-ups of an individual microcantilever (or pixel) where the basic design of SFBMA can be seen clearly.The unit microcantilever of SFBMA consists of three components: a) Single-material (SiN x ) leg isolated from framework.b) Bi-material leg made of two materials (SiN x and Au) of evidently different thermal expansion coefficient and connected to the single-material leg.c) A mirror made of SiN x and Au (SiN x used for absorption of IR flux, Au used for reflecting visible light), and connected to the tip of the bi-material leg.The operation of SFBMA will be described using Fig.1(c).The infrared energy from the IR object to SFBMA is absorbed by the SiN x on each pixel.Because the conductivity and the radiation absorptivity of each pixel are constant, the larger the amount of the incident IR energy on each pixel is, the higher the temperature rising of each pixel is.Owing to the difference in thermal expansion coefficients of SiN x and Au, the temperature change results in a proportional bending of the bi-material leg.The mirror has a rigid connection with the bimaterial leg, thus the inclination angle changes with bending of the bi-material leg.As a result, the inclination angle change of each mirror is proportional to the amount of the incident

Fig. 1 .
Fig. 1.Schematic diagram of a SFBMA structure: (a) An overall view of the SFBMA; (b) An overall view of an individual pixel; (c) A close-up of a half individual pixel where the microcantilever structure and the operation principle of SFBMA with two-fold interval metalcoated leg structure can be seen clearly.

Fig. 2 .
Fig. 2. Transmissivity of IR radiation in FPAs, side cross section of the traditional FPA structure with substrate (a) and the SFBMA (b).

T
is the temperature change of IR object, and CCD N is the image gray level change on CCD.S sys can be split into three terms: IR energy conversion efficiency H (the temperature change in the microcantilever FPA T due to the temperature change of the target IR object obj T ), thermo-mechanical sensitivity S T (the inclination angle change mir of the mirror due to temperature change FPA T ), and optical sensitivity (the grey change of image CCD N on the CCD due to the inclination angle change mir of the mirror).They are defined as: #104140 -$15.00USD Received 18 Nov 2008; revised 20 Feb 2009; accepted 21 Feb 2009; published 4 Mar 2009 (C) 2009 OSA 16 March 2009 / Vol.17, No. 6 / OPTICS EXPRESS 4371

Fig. 4 .
Fig. 4. Concerned coordinates (the scale is magnified to show the mirror deformation).The dashed and the solid spectrums respectively correspond to ideal mirror and deformed mirror which have the same length. :

Fig. 5 .
Fig. 5. Influence of mirror deformation on the optical sensitivity.(a) The normalized intensity distribution in the spectrum plane with different mirror deformation radius (Mirror length L=180 m).(b) Normalized intensity integrated by curves in (a).(c) Variations of the normalized optical sensitivity with respect to mirror deformation curvature under different mirror lengths (L=180 m and 50 m).

Fig. 8 .
Fig. 8. Influence of mirror length on the optical sensitivity.(a)The normalized intensity integration distribution with different L but with the same mirror deformation(R =25mm).(b) Variations of the normalized optical sensitivity with respect to mirror length under different mirror deformations(R= 25mm and 6 mm).

Fig. 9 .
Fig. 9. Micro-fabrication process sequence of optimized SFBMA (a portion of the substrate is removed imaginarily): 1) 0.7-m-thick SiNx was deposited on a double-polished Si wafer.2) Back SiNx was etched to form wet-etch window patterning.3) The pixel geometry was patterned using RIE.4) Lift-off thick metallization formed bi-material legs.5) Lift-off thin metallization formed mirrors.6) Wet etch of the substrate Si resulted in a membrane structure.

Fig. 10 .
Fig. 10.A micrograph image of a portion of the fabricated SFBMA4.The inset shows a surface profile along the length direction of the mirror measured by a Veeco Profiler.
(b).For SFBMA1 and SFBMA3, whose mirrors have the same R ( 25mm) but different L (=180 m and 50 m), the experimental results are shown in Fig. 11(b).The slopes of the two curves at the central position are 8410 gray/deg and 11420gray/deg, and the ratio approximates 0.74.Compared with Fig. 8(a), the experimental results well agree with the #104140 -$15.00USD Received 18 Nov 2008; revised 20 Feb 2009; accepted 21 Feb 2009; published 4 Mar 2009 (C) 2009 OSA theoretical curves, and the experimental ratio (0.74) also well accords with the theoretical analysis (0.78).

Fig. 11 .
Fig. 11.Relationship between the gray level (intensity) on CCD (12 bit) and the inclination angle of the mirrors.(a) The experimental curves and ratio (5.2) of SFBMA1 and SFBMA2 well agree with the theoretical analysis curves and ratio (5.5) in Fig. 5(b).(b) The experimental curves and ratio (0.74) of SFBMA1 and SFBMA3 well agree with the theoretical analysis curves and ratio (0.78) in Fig. 8(a).

Fig. 12 .
Fig. 12.With an f/0.8 IR lens, thermal image of a human hand was obtained using the fabricated SFBMA4.

Fig. 13 .
Fig. 13.The measuring experiments of the sensor responsivity and noise of SFBMA4.(a) IR source.(b) A thermal image of the source and the selected region for analysis.(c) Experimental grey responses of the CCD pixels in the selected region to temperature changes are statistically calculated and the average response, 22 gray/K, is taken as the sensor responsivity.(d) About 200 background images were serially captured and the grey fluctuations for the CCD pixels in the selected region are statistically calculated.The average fluctuation (7 grey levels) is taken as the sensor noise.