Signal processing--Digital techniques.
Optical filter design and analysis : a signal processing approach. Madsen, Christi K.
Contributors: Zhao, Jian H. Wiley, c Madsen, Jian H. Series statement: Wiley series in microwave and optical engineering Series title: Wiley series in microwave and optical engineering. Contents note: Introduction -- Fundamentals of electromagnetic waves and waveguides -- Digital filter concepts for optical filters -- Multi-stage MA architectures -- Multi-stage AR architectures -- Multi-stage ARMA filters -- Optical measurements and filter analysis -- Future directions. Contributor: Zhao, Jian H.
Wiley series in microwave and optical engineering. Procedures for determining a transmission spectrum from such information are known and are described in more detail below. Referring to FIG.edge-jo.com/wp-includes/map2.php
Linear optical signal processing with optical filters: a tutorial
Thus, if the thicknesses of layers 1 and 2 are O and P, respectively, the error between the resulting transmission spectrum and the desired spectrum, as defined by the merit function, is the MF axis value of point B on surface 8. Once on surface 8 , a quasi-Newton method of nonlinear optimization may be used to find points on surface 8 that are successively closer to the surface's minimum point A, eventually finding point A or a point acceptably close to point A.
Point A corresponds to thicknesses M and N of layers 1 and 2, respectively. The merit function value at these thicknesses, using the assumed materials and substrate, is not zero. That is, it is impossible under the assumed conditions to achieve the desired transmission spectrum. If the error represented by this merit function value is unacceptable, a second guess is made that includes a third layer of a material with a refractive index different from that of the second layer. The third guess may begin with the thicknesses determined in the first iteration for the first two layers, or the user may choose three new thicknesses.
In either event, the algorithm re-optimizes for all three layer thicknesses, and the third guess therefore results in a four-dimensional space. The algorithm proceeds in the same manner outlined above, except with an additional thickness variable, and continues to add layers until the merit function value falls within the acceptable tolerance, thereby establishing the final filter design. A procedure for determining a transmission spectrum of an optical filter defined by an assumption of layer thicknesses and selection of layer materials, substrate material, substrate thickness and incident medium is based on a matrix method.
Linear optical signal processing with optical filters: a tutorial | SpringerLink
Each filter layer is associated with a matrix Y 1 as follows:. Light transmitted through the film layers includes an electric vector component, E, and a magnetic vector component, H. The characteristic matrix M relates the electric and magnetic fields at individual layer boundaries as follows:. The propagation of light through a single thin film coating is illustrated in FIG.
The total transmittance of the film and substrate at a given wavelength channel can be determined by summing the infinite series of the combined reflectance and transmittance terms, i. The method for calculating total reflectance is performed in a like manner. A full and enabling disclosure of the present invention, including the best mode thereof, directed to one of ordinary skill in the art, is set forth in the specification, which makes reference to the appended drawings, in which:.
Repeat use of reference characters in the present specification and drawings is intended to represent same or analogous features or elements of the invention. Reference will now be made in detail to presently preferred embodiments of the invention, one or more examples of which are illustrated in the accompanying drawings.
Each example is provided by way of explanation of the invention, not limitation of the invention. In fact, it will be apparent to those skilled in the art that modifications and variations can be made in the present invention without departing from the scope or spirit thereof. For instance, features illustrated or described as part of one embodiment may be used on another embodiment to yield a still further embodiment. Thus, it is intended that the present invention covers such modifications and variations as come within the scope of the appended claims and their equivalents.
In one embodiment of the present invention, optical filter design parameters are optimized to conform to sample spectroscopic measurements, as opposed solely to a predetermined regression vector. In known spectroscopic systems, several samples of a given material may be measured using conventional non-spectroscopic methods to determine a desired characteristic of the material for each sample. If a light spectrum is taken for each sample and compared to the measured characteristics, it is possible to determine a regression vector that can be used to find the value of the desired characteristic in future samples.
This procedure is described in the ' patent. Referring to equation 2, a sample's spectrum defines the wavelength channel values u n , and the optical filter's transmission spectrum is defined according to the regression vector constants b n. Thus, if light from the sample passes through the optical filter, the filter's output is the dot product of the sample's spectrum and the regression vector components in wavelength space. The addition of the offset value a 0 to the filter's output therefore provides the solution to the regression vector formula at equation 15, i.
Accordingly, it is possible to define the regression vector using conventional means and then develop an optical filter structure according to the iterative method described above in the Background section. In a preferred embodiment of the present invention, however, the filter's design parameters are optimized to the initial sample data itself, rather than solely to a predetermined regression vector derived from that data. As an example, assume it is desired to develop an optical filter to measure octane in gasoline.
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Initially, several gasoline samples are measured for octane by conventional means, and the spectrum for each sample is recorded. The number of samples may vary as desired, although it should be understood that measurement errors tend to decrease with an increasing number of samples. Each of the substrate and the layers has a thickness W and a refractive index R. Light, 12 , has an intensity I that varies across the relevant wavelength range.
It passes through an incident medium typically air that has its own refractive index, R o. Exemplary filter layer materials include Nb 2 O 5 , which has a high refractive index, and SiO 2 , which has a low refractive index, although it should be understood that various materials could be used. These materials alternatingly form the successive filter layers.
In general, it is desirable that the materials have minimal internal stress and tend not to absorb light or form non-uniform films. The substrate material and thickness may also vary as desired. In one embodiment, a 1 mm thick BK-7 glass substrate is used. It should be understood, however, that one or more of these variables may also be included in an optimization algorithm. An initial guess is made at 16 for the number of layers and the thickness of each layer.
Although these variables may be selected at random, the method for manufacturing the filter is preferably considered. For example, if a reactive magnetron sputtering method is used, it is generally undesirable to have layers with a thickness less than 20 nm or greater than nm, and the initial thickness guesses are preferably within this range. Additional initial guess procedures are described in detail below. For example, it can be desirable to make multiple guesses to account for the possibility of different optimization solutions.
The transmission spectrum is converted to a regression vector at It is likely that the best regression vector solution will have negative values. Thus, a filter arrangement is used that translates an all-positive transmission spectrum to a regression vector with both positive and negative components. Prior to reaching filter 10 , light 12 passes through a pin-hole 11 , a collimating lens 13 , a band pass filter 15 , and the sample substance itself, Light 20 and 24 is focused on the respective detectors by lenses 19 and Each of detectors 22 and 26 outputs a signal corresponding to the intensity of its incident light to a processor 28 , which may be any suitable device or arrangement such as a computer or logic circuitry capable of performing the appropriate mathematical functions.
Processor 28 subtracts the output of detector 26 from the output of detector 22 and outputs a signal equal to this value at S is the spectrum of incident light beam 12 , in terms of percentage values over the wavelength range. Thus, the intensity of light beam 12 at any wavelength channel is the percentage value of spectrum S at that wavelength multiplied by intensity I.
For ease of explanation, the present discussion assumes that detector sensitivity Q is the same for both detectors 22 and 26 and that no light is absorbed by filter Where filter 10 absorbs light, reflectance R is calculated as in equation 14b. This does not, however, fundamentally change the characteristics of the method, as should be understood by those skilled in this art.
Thus, by using the difference between the filter's transmitted and reflected light, the system translates an all-positive transmission pattern into a full-range regression vector. Accordingly, the user provides a guess for these regression formula variables at Since the gain and offset values may be determined arbitrarily through processor 28 , these variables may be chosen at random. Given these assumptions, equation 21 provides a regression formula into which the spectra of the measured gasoline samples may be applied.
Assuming there are ten samples, equation 21 produces ten octane values. Given that the values are the result of the design parameter guesses at 16 and 32 , it is expected that the ten values calculated at 34 will not equal the actual octane values measured for the ten samples.
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Accordingly, the system executes a merit function that compares the calculated and actual values, and optimizes the merit function optimizing for the calculated values , at It should be understood that various suitable functions may be used to compare the calculated and actual values. A plot of all possible RMS values as a function of the thicknesses of each filter layer and the regression formula's gain and offset values i.
The RMS value resulting from the initial guesses at 16 and 32 corresponds to a point on this surface, and a quasi-Newton method optimizes this point to a local minima on the merit function surface, as should be understood in this art.
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That is, the quasi-Newton method iteratively changes the thickness values of each layer, and also changes the regression formula gain and offset, while minimizing the RMS value. If, at 38 , the optimized RMS value is within an acceptable range, the filter design parameters produced by the optimization correspond to an acceptable filter. If not, the algorithm assumes an additional filter layer at 40 and begins a new optimization sequence with an initial guess of the thicknesses of the filter layers at The new guesses at 16 and 32 may include the optimized values from the previous optimization sequence, although it should be understood that completely new guesses may be made.
The optimization converges to a solution corresponding to the lowest RMS value when the partial derivatives of z go to zero, i. As noted above, layers of large thicknesses may become difficult or impractical to manufacture. In addition, very thin layers do not contribute appreciably to the filter's transmission curve. Thus, during optimization at 36 , if a layer thickness falls below a specified threshold value e.
The optimization resumes with the design parameter values at the point where the optimization stopped, less the thickness value and dimension associated with the removed layer. In another embodiment, the procedure also removes any layer exceeding a predetermined threshold thickness, for example nm.
As described above, the merit function used in the optimization at 36 may have multiple local minima. The optimization result therefore depends on the starting point defined by the guesses at 16 and In a preferred embodiment, therefore, several solutions are derived through the procedure illustrated in FIG. Again, it is possible to randomly select the design parameter guesses, but it is preferred that they be spaced relatively evenly over a desired range of values to assure that the initial guesses are spaced far enough apart on the merit function surface that an effective number of solutions are achieved.
In this embodiment, the loop indicated at 40 in FIG.