Application of Gabor, Log-Gabor, and Adaptive Gabor Filters in Determining the Cut-Off Wavelength Shift of TFBG Sensors

Featured Application

TFBG sensors can be used to measure temperature, bending, rotation, twist and, in particular, refractive index. The value of the refractive index is useful in assessing the concentration of different substances in an aqueous solution. This finds applications in monitoring the environment and industrial processes, and in the field of medicine.

Abstract

Tilted fibre Bragg gratings are optical fibre structures used as sensors of various physical quantities. Their unique measurement capabilities result from the high complexity of the optical spectrum consisting of several dozen cladding mode resonances. TFBG spectra demodulation methods generate signal features that highlight changes in the spectrum due to changes in the interacting quantities. Such methods should enable the distinction between two slightly different values of the measured quantity. The paper presents an effective method of processing the TFBG spectrum for use in measuring the refractive index of liquids. The use of Gabor and log-Gabor filters and their adaptive version eliminates the problem of discontinuity in determining the SRI value related to the existence of the cladding mode comb. The Gabor filters used make visible the shifting and fading of spectral features related to the decrease in the intensity of leaking modes. Subsequent modifications of the proposed algorithm led to an increase in the quality factor of the processed spectrum.

1. Introduction

The development of fibre optic periodic structures is accompanied by an ever-increasing number of associated applications. Improvements and modifications to measurement systems are aimed at enabling the accurate determination of the value of analysed quantities. A fundamental issue in the case of sensors is the appropriate analysis of measurement signals. In the case of optical sensors, we analyse optical signals, and for periodic structures, the analysed signals are optical spectra. In order to fully use the information contained in optical spectra, they must be subjected to the appropriate mathematical operations. New demodulation algorithms are being developed for most sensors requiring optical spectrum measurement. Fibre optic Bragg gratings have a wide range of interesting applications. Despite the relatively simple spectrum, most often in the form of a single Gaussian shape, there are a significant number of algorithms for demodulating such structures. The spectra of tilted fibre Bragg gratings are much more complex. Information about the physical quantities affecting TFBG is encoded in the optical spectrum. The basic element of the TFBG spectrum is a single cladding mode resonance, which repeats to form a cladding mode comb. More advanced methods are used to demodulate such a complex spectrum. Most often, as a result of the demodulation algorithm, a parameter representing the spectrum and correlated with the analysed quantity is determined. The first parameter established on the basis of changes in the TFBG spectrum was the area occupied by cladding modes [1]. Its basic use was to determine the refractive index. The envelope area can be combined with various quantities acting on the sensor, such as bending or immersion level. Further algorithms using quantitative spectrum parameters include spectrum length and advanced methods for calculating the area of cladding modes. The second approach to determining parameters based on TFBG spectra is to identify the cut-off wavelength, i.e., a parameter in the wavelength domain. The cut-off wavelength shift can be preceded by filtering in the Fourier transform domain [2]. The filters used to analyse TFBG spectra have rectangular shapes in the frequency domain [3]. The third group are methods that use parameters in the domain of appropriate transformations, such as the Fourier transform mentioned earlier, for the purpose of filtering. An example would be the centroid under the curve of the fundamental frequency band calculated using fast Fourier transform [4]. There are also methods that use local features of the spectrum, such as small local shifts of individual cladding modes with a constant amplitude value [5,6].

Determining the value of the refractive index is used in many fields of science and technology. Refractometers can be used to determine the concentration of various substances such as paracetamol [7]. Finding the SRI (surrounding refractive index) value based on the TFBG optical spectrum requires establishing the appropriate spectrum parameter. Parameters that can be calculated in a simple and effective way are preferred. Unnecessary steps should be avoided, especially those that do not improve the quality of the results obtained. Frequency-selective, smoothing, and differentiation filters are commonly used in signal analysis. Differential filtering requires appropriate selection of the method due to the amplification of higher frequencies, which may result in the amplification of noise contained in the data. The selection of appropriate algorithms requires an in-depth analysis of the signals considered. Complex or sophisticated methods do not always bring the desired results. The TFBG spectrum is a specific signal in which the cut-off wavelength shifts with a change in the refractive index. It seems that detecting this place in the fundamental domain, i.e., wavelength, gives the best results in determining the SRI. This requires estimation of the local characteristics of the signal. The local signal amplitude can be determined using the so-called analytical signal, i.e., the Hilbert transformation. However, the algorithms should determine local features associated with a narrow frequency range. Band-pass quadrature filters have such properties [8]. Generally, signal enhancement methods are based on either fundamental domain or frequency domain signal processing. The use of appropriately selected filters allows us to obtain a more useful form of signal. As is well known, Gabor filters achieve optimal resolution in the time and frequency domains. Therefore, it seems that their application to TFBG spectra will allow the synthesis of an effective demodulation algorithm.

Later in this article, the applications of Gabor filters and their mathematical description will be explained. Following this, the TFBG spectra will be analysed using the Gabor and log-Gabor filters, and the adaptive algorithms will be applied to the previously differentiated TFBG spectrum. The results will be compared based on the shapes of the envelope derivatives determined for each of the algorithms. The proposed algorithm produces accurate results and allows us to predict its practical usefulness. It should be added that the TFBG optical spectrum has a multi-scale nature, which has not been used so far. To analyse the optical spectrum, we use band-pass Gabor filters, and their selection is based on Fourier analysis.

2. Gabor Filters and Their Properties

A characteristic feature of Gabor filters is optimal localisation in the time and frequency domains. Individual filters can be adjusted to the characteristics of the processed signal. Gabor filter banks are used in time-frequency analysis. The features determined for the set of filters can then be combined using data fusion [9]. Multiresolution analysis may also be useful in the process of algorithm development and optimisation [10]. The disadvantage of Gabor filters is that their bandwidth is limited to one octave of frequencies. Log-Gabor filters do not have this limitation, as they also better attenuate lower frequencies.

Gabor filters are widely used in various image analysis problems. This applies to both individual filters tailored to a specific application and a set of filters. Most applications of Gabor filters concern various classification problems. Very often, they are used to detect various types of defects [11] or identify features [12]. Gabor filters are also used to determine a shift based on recorded images [13]. Improving the quality of images, as well as achieving resistance to interference, can be achieved by using a log-Gabor filter [14]. Initial quality improvement in the case of images is most often understood as increasing the contrast between the details of interest or between the object of analysis and the background. The Gabor transform is also used in the analysis of one-dimensional signals, such as in nuclear magnetic resonance spectroscopy [15]. The transform is also useful for investigating various signals in mechanics [16], including vibration analysis [17].

The Gabor function in the one-dimensional case is a continuous complex function with two parameters related to the centre frequency ${\omega }_t$ and the bandwidth ${\sigma }_t$. The Gabor function can be written as follows:

$$g\left(t\right)={\displaystyle \frac{1}{{\sigma }_t·\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{t^2}{2{\sigma }_t}}}·e^{i{\omega }_{t}t}.$$

(1)

This function is therefore the product of the Gaussian envelope with a complex sine wave. The Gabor function consists of a real part and an imaginary part, which can be written as:

$$g\left(t\right)={\displaystyle \frac{1}{{\sigma }_t·\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{t^2}{2{\sigma }_t^2}}}·\left(cos\left({\omega }_{t}t\right)+i·sin\left({\omega }_{t}t\right)\right),$$

(2)

where ${\omega }_t=2\pi f_t$ is the central pulsation of the filter, and the parameter ${\sigma }_t$ is responsible for the location of the filter in the time domain. The real part of the filter has even symmetry, and the imaginary part has odd symmetry. There is a phase shift of pi/2 between the real and imaginary parts. The optimal representation in the fundamental domain and in the frequency domain results from the shape of the Gabor function itself. A sine wave has the best localization in the frequency domain without the possibility of localisation in time. In turn, a single signal value (a sample) represents the exact position, but extends over the entire frequency range. Superimposing a Gaussian shape on sine waves introduces localisation in the time domain while limiting it in the frequency domain. To process and analyse digital, i.e., sampled, signals, we use the sampled version of the Gabor function. This sampled function is treated as the impulse response of a finite impulse response filter. Signal processing therefore involves convolving signal $x\left(t\right)$ with such an impulse response:

$$y\left(t\right)=x\left(t\right)\ast g\left(t\right).$$

(3)

The output signal consists of a real and imaginary part:

$$y\left(t\right)=y_r\left(t\right)+y_i\left(t\right).$$

(4)

A filter that uses only the real part of the Gabor function is called group detection. The imaginary part is called edge detection [18]. Determining the modulus from the resulting Gabor filter response shows the amplitude (envelope) and phase of the signal at the frequency of this filter:

$$y_a\left(t\right)=\sqrt{y_r^2\left(t\right)+y_i^2\left(t\right)},$$

(5)

$$\varphi \left(t\right)=arctan\left({\displaystyle \frac{y_i\left(t\right)}{y_r\left(t\right)}}\right).$$

(6)

For theoretical signals, such an envelope waveform should be smooth. Due to the presence of noise and distortions in real signals, the received envelope signal may require additional smoothing. The next step in such a case may be a smoothing Gabor filter optimised for the low frequency value of the processed signal [19]. In the frequency domain, the Gabor filter can be described by the following equation:

$$G\left(f\right)=e^{-2{\pi }^2{\sigma }_t^2·{\left(f-f_t\right)}^2}.$$

(7)

The half-peak bandwidth of a Gabor filter can be defined by the following equation [20]:

$$B=lo{g}_2{\displaystyle \frac{{\sigma }_t{f}_t\pi +\sqrt{\frac{ln2}{2}}}{{\sigma }_t{f}_t\pi -\sqrt{\frac{ln2}{2}}}} .$$

(8)

The average value of the impulse response of a Gabor filter is not zero; therefore, such a filter does not suppress the DC component. In general, the basic version of the Gabor filter over-represents low-frequency components. A modification of the Gabor filter is the log-Gabor filter [21], which suppresses the DC component and reduces low-frequency components. It has been proven that the log-Gabor filter is insensitive to changes in the illumination level of the analysed images [22]. The log-Gabor function additionally has a wider bandwidth for higher frequencies. The frequency response of a log-Gabor filter has a Gaussian shape if the frequency scale is logarithmic. For the basic Gabor filter, the Gaussian shape of the filter frequency response occurs for a linear frequency scale. The frequency spectrum of the log-Gabor filter is described by the following function [23]:

$$G\left(f\right)=e^{-0.5·log{\left({\displaystyle \frac{f}{f_t}}\right)}^2/log{\left(\sigma /f_t\right)}^2}$$

(9)

The bandwidth of the log-Gabor filter depends on the parameter σ. Larger values of σ increase the filter bandwidth. The frequency band of the log-Gabor filter can be determined using the following formula [24]:

$$B=2\sqrt{2/log2}·\left|log\left(\sigma /f_t\right)\right|$$

(10)

Log-Gabor filters have their representation only in the frequency domain. To implement such a filter, the multiplication operation of the filter transfer function and signal frequency spectrum must be performed. The frequency spectrum of the signal processed in this way is recovered from the frequency to the original domain by inverse Fourier transform. The reproduced signal is a complex signal. Its real part corresponds to the symmetric part of the filter, and its imaginary part to the odd part. Therefore, it is the equivalent of a quadrature filter, as in the case of the classic Gabor filter. The optical spectrum signal can also be considered as an amplitude modulation signal in telecommunications. Then, the derivative of the signal envelope can be used to estimate the cut-off wavelength. To determine the envelope, a band-pass filter can be used, after which we calculate the absolute value of the signal, and in the last step, we apply a smoothing filter. Such a filter should be a low-pass filter that specifically attenuates twice the frequency of the band-pass filter. A doubling of the fundamental frequency of the signal occurs after rectification of the signal. The signal envelope can also be calculated using the analytical signal. It is a complex signal in which the real part is the original signal, and the imaginary part is the Hilbert transformation of this signal:

$$x_A\left(t\right)=x\left(t\right)+x_H\left(t\right)$$

(11)

The Hilbert transformation of the signal $x\left(t\right)$ can be written as its convolution with the kernel $1/\pi x$:

$$x_H\left(t\right)=x\left(t\right)\ast {\displaystyle \frac{1}{\pi x}}.$$

(12)

This operation can also be performed in the Fourier domain by changing the sign to the opposite for negative frequencies of a signal. However, the analytical signal does not have the property of simultaneous localisation in time and frequency. Gabor filters calculate the equivalent of an analytical signal with simultaneous band-pass pre-filtering.

3. Selection of Gabor Filters for the Analysis of TFBG Spectra

The spectrum of a tilted Bragg grating is represented by many cladding resonances that are only visible when measuring in transmission mode. TFBG gratings used for experiments were inscribed in fibre using the phase mask method. During recording, the phase mask was rotated by 6 degrees. The length of the grating was 1 cm. The recording was made on a typical single-mode optical fibre (SMF). Figure 1 shows a schematic view of the experimental setup. Optical spectra were measured by an AQ6370D optical spectrum analyser (Yokogawa).

The effect of changing the SRI coefficient on the TFBG spectrum is shown in Figure 2. The SRI changes the amplitudes of the modes and their location. However, the changes are different for each individual mode. Changes in the wavelength position of a single mode are relatively small. The position of the cut-off wavelength has the greatest influence on the spectrum, as its shift causes the modes to disappear. As the SRI increases, the cut-off wavelength shifts towards longer wavelengths, which is visible as a decrease in the cladding resonance amplitudes. When the TFBG grating used in the research was immersed in distilled water, the first significant cladding modes could be observed at a wavelength of 1490 nm. Immersing the grating in a water-glucose solution with SRI = 1.3479 reduced the mode amplitudes below 1502 nm. Subsequently, for SRI = 1.3672, the amplitudes of modes below 1514 nm decreased. For SRI = 1.3883, the disappearance of modes below 1524 nm could be observed. Modes for wavelengths above the cut-off wavelength shifted slightly towards longer wavelengths, but they did not experience a significant reduction in amplitude.

The selection of filters was preceded by the analysis of spectra using the Fourier transform (Figure 3). The goal of optical spectrum processing is to obtain a smooth function representing the decay of cladding modes. Spectrum processing should eliminate the mode-hopping problem when determining the cut-off wavelength. In order to analyse the algorithms, TFBG spectrum measurements were performed for 15 SRI values from 1.333 to 1.4056 refractive index values. The glucose content was up to 43%. For each SRI value, the measurement was repeated 20 times. The figures later in the article show a single optical spectrum for each of the 15 SRI values.

The first step was filtration using filters with impulse response, shown in Figure 4. The filter’s frequency response was matched to the fundamental frequency associated with the occurrence of cladding modes, and covered the band shown in Figure 3.

The real part of the optical spectrum obtained as a result of filtration is shown in Figure 5. The spectrum is symmetrical about the wavelength axis. A place can be noticed on the wavelength axis where the mode envelope is bent. Figure 6 shows the spectra of both the real and imaginary parts of the spectrum along with the envelope they create. The spectra after filtration are noise-free and smooth. Thanks to this, the envelope determined on their basis will also be smooth. Additionally, the envelope calculated with the real and imaginary parts of the filtered spectrum uses all points of the spectrum. Therefore, this is a much better method than envelopes calculated from the spectrum peaks alone. The envelope determined in this way represents the signal amplitude at the frequency characteristic of the filter operation.

Figure 7 shows the envelopes determined by the Gabor filtration method for individual SRI values. The envelope shift is directly related to the change in cut-off wavelength. The disappearance of subsequent modes with increasing SRI values causes the envelope to shift towards longer wavelengths. Estimating the value of the cut-off wavelength may involve finding the place where the envelope function has the fastest change with wavelength. This will correspond to the maximum value of the derivative, as shown in Figure 8.

The shift in the maximum value of the Gabor envelopes derivatives depending on SRI is shown in Figure 9. The shape of the derivatives is similar to the resonance spectra of a surface plasmon sensor. To determine the measurement capabilities of these sensors, the Q-F parameter was developed, which is the ratio of the sensor sensitivity to the FWHM width [25]. In order to improve it, special methods of producing surface plasmon sensors are used, taking into account the use of appropriate materials [26,27]. The basic algorithm for determining surface plasmon resonance (SPR) is centroid. Additional modifications to this algorithm have been developed, such as fixed-boundary centroid [28] or enhanced centroid [29], which are characterised by a significant improvement in the resolution of resonance determination. It should be noted, however, that modifications to the algorithm are optimised precisely to the characteristics of the data obtained from SPR measurements. In the case of TFBG, the sensitivity is dependent on the shift of the maximum value of the envelope derivative under the influence of the SRI, which is constant regardless of the spectrum transformation method. To increase the value of the Q-F parameter, the width of the FWHM of the envelope derivative shape should be reduced. Improving the quality (Q-factor) of TFBG envelope derivatives can only be achieved by selecting appropriate numerical methods.

In the next stage of the research, Gabor filters were used (Figure 10), adjusted to the second harmonic, resulting from the distance between cladding modes in the transmission spectrum (Figure 11). The frequency spectrum for the second harmonic of the cladding modes is between the frequencies of 1.5 and 2.5 nm⁻¹. The value of the central wavelength is therefore 2 nm⁻¹, and is twice as large as the central value of the first frequency band. This means that the filtered optical spectrum will be twice as dense as the spectrum considered in the first part of the analysis. This can be seen in Figure 12. More densely located maxima allow for more accurate mapping of the envelope.

Figure 13 shows the derivatives of the spectral envelopes after the operation of a Gabor filter adjusted for the second harmonic of the cladding modes. Figure 14 shows the dependence of the shift on the maximum value of the envelope’s derivatives depending on the SRI value.

The calculations clearly show that the proposed algorithms allow for effective processing of TFBG spectra and feature extraction, including the indirect estimation of the cut-off wavelength. However, when analysing Figure 8 and Figure 14, one can notice the characteristic shape of the baseline of the curves presented there. Such a background has a maximum value of 1518 nm, and a minimum of 1530 nm. This is a slowly changing component that results from the poor attenuation of lower frequencies by the basic version of the Gabor filter. The next step is to use log-Gabor filters. The first filter adjusted to the fundamental harmonic of the cladding modes is shown in Figure 15. Unfortunately, the derivative of the cladding mode envelope did not change its shape significantly. In Figure 16, as in Figure 13, the shapes similarly contain a slow-changing component. Additionally, for extreme SRI values, the envelope derivative does not have the clear Gaussian shape present at the cut-off wavelength. Especially for the highest SRI value, the shape of the envelope derivative is wide without a clear maximum. It can therefore be concluded that the envelope derivative curves have a low value equivalent to the quality factor known from SPR resonances.

The problem with selecting the bandwidth of the log-Gabor filter is that a filter that is too narrow will not highlight the derivative maxima for the entire SRI range, while a filter band that is too wide causes a broadening of the shape and even the creation of shapes with two maxima lying close to each other. Similarly, the centre frequency of the filter cannot be optimally selected. It is possible to improve the quality of the envelope derivative for either smaller or larger SRI values. It seems that in such a case, a flexible approach should be used, consisting of modifying the filter parameters depending on the SRI scope. When calculating the envelope using the log-Gabor filter, we perform mathematical operations in the frequency domain. In the first step, the frequency spectrum is calculated based on a given optical spectrum. The centre frequency of the filter can therefore be adjusted based on this spectrum. The parameter may be the frequency at which the maximum intensity of the first harmonic occurs. The frequency of the maximum intensity of the spectrum can be calculated directly or using the centroid method. Figure 17 shows the shift of the central frequency of the log-Gabor filter for successively increasing SRI values. Figure 18 shows the advantages of this solution. The improvement in peak quality for high SRI values is particularly visible. There is also a noticeable reduction in the slowly changing background component.

In a similar way to the first harmonic, the adaptation method can be applied to the second harmonic. Changes in the frequency of the adaptive filter can be observed in Figure 19. The derivatives of the envelopes shown in Figure 20 do not differ significantly from those in Figure 18. This means that the log-Gabor filter for both the first and second harmonics is able to eliminate ripples associated with the occurrence of cladding modes. The problem of jumps between modes when determining the cut-off wavelength is completely eliminated. A smooth derivative indicates a uniform approximation of the envelope without significant high-frequency ripples. The shape of the envelope derivative is equivalent to the resonance curve located at the cut-off wavelength. The final filtering operation that smooths the envelope only allows for giving the final shape to the derivative. However, in order to improve the quality of the final resonance shapes of the envelope’s derivatives for the entire SRI range, additional actions must be performed. The goal is to initially eliminate slowly changing components of the transmission spectrum. This is especially true for the portion of the optical spectrum remaining after mode leakage. Therefore, it is necessary to eliminate the low-frequency components while strengthening the frequencies associated with the first and second harmonics of the cladding modes. Such properties are characteristic of algorithms that determine the derivative of a signal. For discrete signals, the derivative can be approximated by first difference. Taking the derivative causes significant amplification of the higher frequencies above the second harmonic, which can be identified as noise. However, a Gabor filter will significantly reduce these frequencies. In the frequency domain, the first and second harmonics of the modes lie in the frequency range up to 3 nm⁻¹. Higher frequencies can be relatively easily removed by using, for example, a smoothing filter.

Figure 21 shows the envelopes determined by the log-Gabor filter based on the derivative of the optical spectrum. It is noticeable that a clear shape of the envelope derivative has been achieved, which can be defined as the edge on the side of smaller wavelengths (edge on the left). In most cases, these edges for individual SRI values are parallel to each other. The second feature of these edges is their uniform growth. As a result, the envelope shaped in this way will, as a result of the differentiation operation, be transformed into the curves shown in Figure 22. In the entire SRI range, the derivatives of the envelopes are close to the Gaussian shape and can be assigned a high value of quality factor (Q-factor). The slope of the calibration curve in Figure 23 is greater than that in Figure 14.

In subsequent calculations, the envelopes for the second harmonic derivatives of the optical spectra were determined. The derivatives of the envelopes are shown in Figure 24. The shapes of the peaks in the entire range of the measured SRI differ significantly from the background level. Additionally, the half width of these shapes can be determined over the entire range. Thus, methods such as centroid can be used here to determine the cut-off wavelength shift.

The curves in Figure 24 are largely symmetrical. Symmetry deformations occur only in the lower part of individual curves. Based on 20 repetitions for each of the 15 SRI values, the measurement resolution was determined to be 2.3 × 10⁻⁵ RIU. The SRI value versus cut-off shift curve was approximated by a linear function. The maximum error in determining the SRI value was 1.4 × 10⁻⁴ RIU. For comparison, calculations were made for the method of area between envelopes determined on the basis of peaks. When calculated directly, the resolution (standard deviation from the mean value for 20 repeated measurements of a single SRI value) was 10⁻⁴ RIU. Using additional smoothing filters and interpolating the envelope between peaks using spline functions, the best resolution was 5 × 10⁻⁵. Moreover, the function between the value of the envelope area and SRI was non-linear, and its approximation using a 5th degree polynomial turned out to be optimal. The maximum error in determining the SRI was 3.3 × 10⁻⁴ RIU. It seems that the proposed methods using Gabor filters are characterized by better quality of metrological parameters. Their main advantage is the linear relationship between the parameter determined on the basis of the spectrum, i.e., the shift in the cut-off wavelength (Figure 25), and the measured SRI value of the substance.

4. Conclusions

We have proposed a new method for processing the spectra of tilted TFBG gratings to determine the SRI value. The use of Gabor filters processes the optical spectra in a very convenient way, enabling the use of simple methods in subsequent stages of signal processing algorithms. Such simple filters eliminate the need for operations such as denoising or baseline correction of the measured spectra. This is a big practical advantage. The log-Gabor filter has a better enhancement and separation effect of optical spectra, and consequently, it better highlights the wavelengths between leaked and non-leaked modes.

The basic concept of filter selection was to limit the spectrum to a single frequency associated with the occurrence of cladding modes. If the cladding modes were equidistant from each other, such a signal would have a spectrum characteristic of a periodic signal. There would be the first and subsequent harmonics. Four such harmonics can be seen in the spectrum of the analysed TFBG grating. However, since the distances between the modes decrease with increasing wavelength, the first harmonic of the cladding modes does not create a single frequency, but a chirp. Therefore, in the frequency spectrum, individual harmonics are broadened and partially overlap. Therefore, simply adjusting the filter to the first or subsequent harmonics is not sufficient. A filter that is too narrow does not cover the frequency of the entire chirp. Broadening the filter results in the signal transfer not only of the selected harmonic, but also the partial transfer of the next harmonic. As a result, the calculated derivative of the envelope will not have a clear maximum. The filter band should therefore be selected so that it does not cover another harmonic. The main frequency of the filter can be taken as the frequency of the maximum amplitude of the frequency spectrum for a given harmonic.

The proposed spectrum analysis method improves the quality of determining the SRI coefficient value by estimating the change in the cut-off wavelength position. This is the result of precisely and adaptively extracting the frequency range in which the optical spectra are modified by the cut-off wavelength value.