Youngmin Kim and Dean P. Neikirk

Microelectronics Research Center

Department of Electrical and Computer Engineering

The University of Texas at Austin

Austin, TX 78712

In these devices the cavity mirrors can be either dielectric layers or metal layers deposited or evaporated during the manufacturing process. The thickness of each layer must be tightly controlled to achieve the target performance of a sensor. However, there are unavoidable errors in thickness even though techniques of thickness control for thin films have rapidly improved [6].

In this paper, we consider a Fabry-Perot "gap" sensor which is used to measure the separation between the cavity mirrors. Variations in the response of the Fabry-Perot sensor resulting from random errors in mirror thickness are calculated. It is shown that larger processing tolerance (and hence, yield) can be achieved by designing the Fabry-Perot cavity with proper initial gap and mechanical travel. In addition, designs that maximize the linearity of the response of the sensor are also discussed.

Under the above conditions, the reflectance of a Fabry-Perot sensor coupled to a single mode fiber is equivalent to the reflectance of a plane wave (free space wavelength [[lambda]]o) with propagation direction normal to the surface of the sensor. Figure 1 shows a schematic view of the Fabry-Perot sensor consisting of q layers, where for convenience the cavity gap (of length g) is always labeled as the k

(1)

where

(2)

and and are the admittances of the environment outside the sensor and of the medium from which the light comes, respectively. The characteristic matrix

. (3)

where

(4)

with [11].

Using the characteristic matrix method to analyze a multilayer system, for example the spectral analysis of an optical filter, is easily done, but the inverse problem (e.g., design of an optical filter which has arbitrary spectral shape) is extremely difficult. For Fabry-Perot sensors, the gap g must be inferred from some measurement of reflected light intensity

(5)

between the actual gap g and the fitted gap gfit.

To choose a fitting function, the design space for the device must be specified. Here we assume two primary design variables: i) the initial gap gi of the Fabry-Perot cavity; and ii) the maximum mechanical travel t of the moving mirror. The second variable is determined by the range of the sensed quantity (e.g., the maximum pressure) and the mechanical compliance of the membrane supporting the moving mirror. Since mechanical compliance frequently can be adjusted independently of the thicknesses of the layers used to fabricated the mirror, we will consider t a freely adjustable design variable. If we wish to achieve the best accuracy [12] for a given initial gap gi and travel t we must choose a fitting function so that the error

(6)

is minimized.

In addition to the simple errors due to response fitting, we should also evaluate a particular design's tolerance to manufacturing errors; i.e., how errors propagate to g via process-induced uncertainties in the zi's. In this case there exists an uncertainty [[Delta]]g in the gap at a given reflected light intensity

(7)

where the functional dependencies of

. (8)

Combining eqs. 7 and 8 then gives [[Delta]]g; the maximum uncertainty [[Delta]]gproc in gap due to process-induced thickness variations is

. (9)

In terms of yield, if for each mirror layer i (i != k) the fraction of devices with thickness between and is Pi, then for a fixed value of

. (10)

To select a particular design, it is critical to remember that manufacturing uncertainty in layer thicknesses also includes the process that determines gi. Even if we assume a single calibration measurement is made to determine the specific value of gi for a given manufactured sensor, the overall design should still allow gi to vary over the range gi

. (11)

The best design (i.e., the best values of gi and t) is the one that minimizes . In terms of yield, if the fraction of sensors with actual gap in the range gi

. (12)

If both fitting and process-induced errors are included, to find the design that would give the best accuracy we must find the combination of initial gap gi and travel t so that the total error

(13)

is minimized.

To actually evaluate sensor errors the measurand

. (14)

This response curve will be periodic in g, with period [[lambda]]1/2. Unfortunately, if there are any optical system losses, it is usually not possible to measure the absolute value of reflectance, and hence g cannot be determined. An alternative is the dual wavelength technique, in which relative reflected intensities at two different wavelengths ([[lambda]]1 and [[lambda]]2) are separately measured, and then a ratio is calculated using

(15)

where accounts for any losses induced by the optical system. If over the wavelength range used these losses are wavelength independent (such as bending loss or temperature-induced variations) the ratio given by eq. 10 becomes equal to the corresponding ratio of absolute reflectances, and so

. (16)

This technique eliminates errors resulting from wavelength-independent changes in the fiber interconnect to the sensor [1, 2, 4] . The response curve is still a periodic function with respect to gap, but with a period equal to the lowest common multiple of and .

Figure 2 shows the calculated single wavelength absolute reflectance curve (from eqs. 1-4), assuming [[lambda]]1 = 700 nm. The periodicity of the curve suggests two basic operating branches, one between 1000 Å and 2750 Å, and the other between 2750 Å and 4500 Å. To illustrate a design process, we first assume it is desirable to find a design that produces the best accuracy when using a linear response approximation. Figure 3 shows the accuracy contour plot ( from eq. 6) over the design space of gi and t. This shows, for instance, that 1 % accuracy could be obtained from two designs, gi = 2625 Å and t = 375 Å, or gi = 3450 Å and t = 375 Å, if there were no manufacturing variations, and the only source of error was the linear fitting approximation.

For this device and wavelength choice, we have also calculated the process-induced errors using eq. 9 (Fig. 2). For comparison, extensive random combinations of Au mirror thicknesses from 67 Å to 73 Å have been tested to verify that the maximum change in g is produced by the perturbed layer thicknesses used in eq. 9. Agreement between the two approaches indicates that the first order Taylor series approximation used in eq. 8 is sufficient for this device. Figure 4 shows the error contours including both the manufacturing layer thickness variations and the linear response fitting error (from eq. 13). In this case, the sensitivity to process variations is not the same on the two branches of the response curve, and is much larger than . The optimum design is now gi = 2600 Å and t = 325 Å, which will yield better than 5 % accuracy for at least 97 % of the sensors manufactured.

Figure 5 shows the response curve and associated manufacturing-induced errors for dual wavelength detection, assuming detection wavelengths of 560 nm and 700 nm. There are now nine distinct operating branches. Calculation of the contours of linearity error only (using eq. 6) indicates that an accuracy not worse than 1 % could be achieved using gi = 6250 Å and t = 1050 Å. Note that when using dual wavelength detection linearity is maintained over a much longer travel than for single wavelength detection, as has been noted in [4]. Consideration of process-induced errors only (from eq. 11, using the full non-linear response curve as a reference) suggests an optimum design for gi = 9325 Å and t = 825 Å, with at least 97 % of the sensors producing an accuracy of not worse than 1 %. Figure 6 shows the contour plot of the error induced by both linearity and layer thickness variation for the branch with best performance. This shows that when both errors are considered the optimum design is gi = 6050 Å and t = 650 Å, with at least 97 % of the manufactured sensors producing an accuracy of not worse than 3.5 %.

Acknowledgments: This work was sponsored by the Advanced Research Projects Agency (ARPA) Embedded Microsystems Program under contract # DABT63-92-C-0027 AMD P00002 - DOD-ARPA.

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Optically interrogated Fabry-Perot pressure sensor.

Figure 1: Schematic diagram of multilayer Fabry-Perot sensor. The device consists of q layers, where layers 1 through k - 1 make up the "moving" mirror, layer k is the gap to be measured, and layers k + 1 through q make up the "fixed" mirror. Each mirror layer i is zi thick, with manufacturing-induced thickness error [[Delta]]zi.

Figure 2: Solid line: absolute reflectance (single wavelength response curve, eq. 14, [[lambda]]1 = 700 nm) versus gap for Fabry-Perot cavity described in the text. Dashed line: process-induced response variations (from eq. 9).

Figure 3: Accuracy contour map for single wavelength detection as a function of initial gap and mechanical travel, assuming the only errors are those produced by using a linear response approximation.

Figure 4: Accuracy contour map for single wavelength detection, assuming both process-induced variations and linear response errors. Optimum performance occurs for gi = 2600 Å and t = 325 Å, which will yield better than 5 % accuracy for at least 97 % of the sensors manufactured.

Figure 5: Solid line: dual wavelength response curve (from eq. 16, [[lambda]]1 = 560 nm and [[lambda]]2 = 700 nm) versus gap for Fabry-Perot cavity described in the text. Dashed line: process-induced response variations (from eq. 9). The period of the response function is 14,000 Å, containing nine distinct branches.

Figure 6: Accuracy contour map for due wavelength detection, assuming both process-induced variations and linear response errors. Optimum performance occurs for gi = 6050 Å and t = 650 Å, with at least 97 % of the manufactured sensors producing an accuracy of not worse than 3.5 %.