### Notch Filters

 Notch filters (= minus filters, narrow band pass filters) reflect a narrow band from a wide specified spectral range and transmit on both short wavelength and long wavelength sides of this band. These filters used in Raman spectroscopy, laser-based fluorescence instrumentation, and multiphoton microscopy. Typical target reflectance is as high as possible in the high reflectance range and as small as possible in the rest of the specified spectral range. Design approaches: rugate type coatings and conventional multilayer stacks. The advantage of the rugate and quasi-rugate coatings is that they suppress ripples in the transmission spectral ranges.   The disadvantages of conventional multilayer solutions is that they might contain layers of four, five, and more materials. Notch filters with narrow and extremely narrow high-reflection zones can be obtained using efficient numerical design algorithms. Two-component solutions typically contain thin layers which may be a problem in the course of the practical realization of the obtained designs. However, it has been demonstrated by several companies that production of multilayer notch filters containing thin layers is realistic. See for example, these references [1], [2], [3], and [4]. There are two periodic multilayer structures that can be used as good starting designs for notch filters. Using these structures you may achieve the target width of the high reflection zone and target values of reflectance in this zone. If we denote layer materials as A and B and their refractive indices as $$n_1$$ and $$n_s$$, then two structures S1 and S2 provide good starting design for notch filters: S1:  $$n_s | \left((2-p)A\; pB \right)^n |n_a$$ S2:  $$n_s | \left((4-p)A\; pB \right)^n |n_a$$ $$p$$ is a fraction quarter wave thickness and $$n$$ is the number of layer pairs. Analytical estimations of the reference wavelength reflectance $$R(\lambda_0)$$ and the width of the high-reflection zone $$\Delta\lambda_0$$ are obtained allow you to find the number of layer pair $$n$$ and fraction $$p$$ that you need to achieve target specifications. Reference wavelength reflectance are calculated as follows:    $$R(\lambda_0)=\frac bc,$$          (Equation 1)            where $$n_a, n_s$$ are refractive indices of incident medium and substrate                                                                               $$b=a_{11}^2 (n_1-n_s)^2+a^2 (n_a n_s\alpha - 1)^2 a_{12}^2$$ $$c=a_{11}^2 (n_1+n_s)^2+a^2 (n_a n_s\alpha + 1)^2 a_{12}^2$$ $$a=\pi p (n_1=n_2)/2, \; \alpha=-1/(n_1 n_2)$$ $$a_{11}=0.5\left[ (1+\sqrt{-\alpha} |a|)^{-n}+(1-\sqrt{-\alpha} |a|)^{-n}\right]$$ $$a_{12}=0.5\left[ (1-\sqrt{-\alpha} |a|)^{-n}-(1+\sqrt{-\alpha} |a|)^{-n}\right] /(\sqrt{-\alpha} |a|)$$ Example 1: $$n_1=2.35, n_2=1.45, n_s=1.52, n_a=1$$, number of layer pairs $$n=64, \lambda_0=532$$ nm, $$p=0.12$$. R(λ0)=99.9975% Example 2: $$n_1=2.0, n_2=1.5, n_s=1.52, n_a=1$$, number of layer pairs $$n=32, \lambda_0=532$$ nm, $$p=0.2$$. R(λ0)=98.9856% The width of the high reflection zone can be estimated as: Structure S1: $$\Delta\lambda_0=\displaystyle\frac {4\lambda_0}{\pi} \frac{\sqrt{C^2-1}\;\sin(\pi p/2)}{\left[C+1-(1-p)^2(C-1)\cos(\pi p)\right]}$$    (Equation 2) ghgh Structure S2: $$\Delta\lambda_0=\displaystyle\frac {2\lambda_0}{\pi} \frac{\sqrt{C^2-1}\;\sin(\pi p/2)}{\left[C+1-(1-2p)^2(C-1)\cos(\pi p)\right]}$$    (Equation 2) where $$C=\displaystyle \frac 12\left(\frac{n_1}{n_2}+\frac{n_2}{n_1}\right)$$ If you have a specified R(lam0) value and width of high reflection zone then using Equation 1 and Equation 2 you can estimate fraction p and number of layer pairs n as follows: $$\Delta \lambda_0$$  --> p from Equation 2 --> n from Equation 1 See the details on the next page. Here you can NOTCH CALCULATOR presented above. Simply press Calculate button. In our video example we illustrate how to work with NOTCH CALCULATOR.
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