Ultra-Short Fiber Bragg Grating Used for Spectral Analysis of Guided Light in Single-Mode Fibers

An ultra-short fiber Bragg grating with a grating length of 0.2 mm and constant grating period (uniform FBG) is proposed as an integrated dispersive element for spectral analysis in a single-mode glass fiber. This dispersive element is used to set up a fiber optical spectrometer that demonstrated an average spectral resolving power of 825, a pixel resolution of 0.02 nm/pixel, and a 40 nm bandwidth covering the 810–850 nm wavelength range. Furthermore, the dependence of this dispersive element on light polarization was examined, which induces a reduction in the wavelength accuracy up to ±0.05 nm. Finally, the investigated spectrometer to function well as a part of a readout device for fiber Bragg grating sensors is presented. Short uniform Bragg gratings as dispersive elements allow the manufacturing of miniaturized, compact spectrometers without additional focusing components between the FBG and detector.


I. INTRODUCTION
A FIBER Bragg grating (FBG) is one of the most striking integrated components in fiber optics. It has proved to be an asset in telecommunication and sensor technologies [1], [2], [3], [4], [5], [6], [7]. In the last five years, FBGs have become more popular as an integrated dispersive element for spectral analysis [8], [9], [10], thus opening up new research opportunities for grating applications.
The first demonstration of a spectrometer integrated within an optical fiber was reported by Russell and Ulrich [11] in 1985. Although this application is not based on a fiber Bragg grating, the way it monitors optical power diffracted out of the fiber is similar. The diffraction grating was formed on the surface of a side-polished single-mode fiber (grating-fiber coupler), allowing it to interact with the evanescent field of the light guided in the optical fiber diffracting the light of different wavelengths with Manuscript  different angles. The resulting out-coupled light was collected by a lens and focused on a photodiode which converted the light signal to an electrical signal. This fiber optical spectrometer showed a spectral resolution of ≈1 nm at a wavelength of around 633 nm. In 1997, Wagener et al. [12] demonstrated the first fiber optical spectrometer using a fiber Bragg grating as a dispersive element to focus the light onto a 256-element detector array. The grating was tilted to couple out the light of the fiber core, and the grating period was linearly varied (chirped FBG) to focus the light on the detector. This technique resulted in an excellent performance with a spectral resolution of 0.12 nm in 14 nm bandwidth ranging from 1546 nm to 1562 nm. Tilted fiber Bragg gratings (TFBG) without varied grating periods and TFBG with varied grating periods have been investigated in several studies [9], [13], [14], [15], [16], [17], [18] as dispersive elements for fiber optical spectrometers. The main weakness of TFBG is the polarization dependence of the tilt angle [13], [19]. Furthermore, the above-mentioned spectrometer approaches must be supported with optical components, such as prisms, lenses, and mirrors. Less polarization sensitivity may be achieved by using an untilted FBG. Waltermann et al. [8] showed the use of a chirped FBG (CFBG) as an integrated dispersive element for a fiber optical spectrometer to analyze FBG sensor signals in the 810-870 nm wavelength range. A similar design for the visible spectrum (450-635 nm) that used a filament geometry of the grating points instead of an ellipsoidal was investigated by Rahnama et al. [10]. The elongated form of the grating points enables azimuthally symmetric diffraction maxima to travel in opposite directions outside of the cladding. An additional chirping of such grating was introduced in order to focus the out-coupled light on the CCD array and thus provides a high-resolution of their fiber optical spectrometer without any focusing optics between fiber and detector, which is a significant advantage of the CFBG as a diffraction grating. The main disadvantage of their chirped FBG-based spectrometer is its low out-coupling efficiency of 1.5% [10], in contrast to the tilted FBG-based approach that can achieve an out-coupling efficiency of up to 54.8% [20]. Additionally, the production of chirped and tilted FBGs is much more complex compared to FBGs with constant grating periods (uniform FBGs).
However, tilting and chirping of a Bragg grating is not mandatory to couple out the light guided in the fiber and disperse it onto a linear CCD array. The standard grating with a constant grating period can also be applied to fiber spectrometers, which may simplify and reduce the cost of grating production. Froggatt and Erdogan, for example, demonstrated a fiber spectrometer based on the measurement of the light diffracted from two counterpropagating modes and Fourier transform analysis [21]. This device has a high resolution and is compact and simple in structure, but it needs a reflector at the fiber end. The main drawback is the polarization-dependent out-coupling efficiency.
In our research, we propose an ultra-short (grating length of 0.2 mm) fiber Bragg grating with a constant grating period as a dispersive element. The short grating is tuned so that the out-coupled light at the third diffraction order provides sufficient linear dispersion on the CCD array at the defined spectral range without overlapping with neighboring orders, as shown schematically in Fig. 1. This configuration allows for a high-resolution fiber optical spectrometer that does not need additional focusing optics and other components. The grating parameters were produced and optimized by point-by-point (PbP) inscription technology according to the spectral range between 810 nm and 850 nm.

II. CALCULATION OF THE FIBER BRAGG GRATING OFF-AXIS LINEAR DIFFRACTION
A fiber Bragg grating (FBG) is a periodic perturbation of the refractive index along a fiber length [1], [22], conventionally inscribed in the core of an optical fiber with a constant grating period. The grating diffracts a light wave propagating along the fiber core [23], [24], [25], according to: where m is the diffraction order, θ i is the angle of incidence, n core is the refractive index of the fiber core, λ i is the central wavelength of the incident light, and θ m is the angle of the diffracted wave relative to the center of the FBG, as shown in Fig. 2.
Assuming θ i ≈ π/2 we find for the diffraction angle θ m : Considering the subsequent refraction of the dispersed light at the fiber-air interface (neglecting the refraction between fiber core and fiber cladding), we solve for the final angle of dispersion α with Snell's law: Hence, we find that for a given distance d between the optical fiber and the screen (CCD surface) the peak of the diffracted wave is shifted by l m relative to the center of the FBG, with: where the shift in the fiber cladding and coating is neglected. Fig. 3 shows the scheme of the experimental setup used to thoroughly analyze the diffracted light around the FBG (0°-360°).

III. EXPERIMENTAL SETUP
The setup design ensures that the out-coupled optical power will be analyzed three-dimensionally around and along the fiber length at different distances. A Toshiba TCD1304DG linear CCD image sensor consisting of 3648 elements with 8 μm pixel resolution was mounted on a motorized rotation stage, allowing a scan of the diffracted light profile around the fiber. As well as rotating around the fiber, the CCD array can be moved at distances ranging from 1.5 mm to 34 mm. A tunable diode laser (DL pro, TOPTICA Photonics AG) with a tuning range between 810 and 850 nm was applied as a light source. The light from the diode laser was coupled into a single-mode glass fiber (SM800, Fibercore) and split in half by an optical 3-dB coupler. The first coupler port was directed to an optical power meter (PM400, Thorlabs Inc.) and an optical spectrum analyzer (OSA: OSA202C, Thorlabs Inc.) to observe its characteristics. The second port guided the light to the tested Bragg grating. For all following experiments, the fiber Bragg gratings were fabricated with a point-by-point processing technology using a commercially available regenerative titan-sapphire laser amplifier (seed laser Tsunami, amplifier Spitfire Pro F-XP; SpectraPhysics) with 90-fs pulse duration and 5 kHz repetition rate. The laser pulses (approx. 50 nJ) were focused with a microscope objective (40×, NA = 0.6) into a commercially available single-mode glass fiber with acrylate coating (Fibercore SM800(5.6/125)). The produced fiber Bragg gratings had the following parameters: a centroid wavelength of 810 nm and a grating length of 0.2 mm. Additionally, a polarization rotator was employed to rotate the polarization state using quartz retardation plates (λ/2 or λ/4 plates) in order to measure its effect on the tested interrogation method.

A. Calculations of Off-Axis Diffraction Maxima Positions on the CCD Line
In this Section, we presented a set of calculated parameters (grating period, order of diffraction, distance to the CCD, and overlapping conditions of spectral imaging of different orders) in order to adapt the diffraction order to the best distance between the CCD and the FBG, as well as to avoid overlapping of spectral images between different orders of refraction on the CCD surface. The light dispersion for three different gratings was calculated for a defined spectral range to determine a maximum pixel resolution that was carried out if the diffracted spectrum covers most of the CCD area. The calculation was performed for the spectral range of 810 nm to 850 nm at a 1.5 mm distance between the grating and the CCD array. The spectral range was identical to the tuning range of the tunable diode laser (810 nm-850 nm). The grating periods provide a back reflection within the fiber core for wavelengths at 810 nm (according to the third-, fourth-and fifth-order at 810 nm of a FBG signal for a single mode fiber with an effective refractive index of n ef f = 1.4574). We selected these grating periods to monitor the grating quality while manufacturing by observing the Bragg reflection using a light source at 810 nm. The numerical simulation results for all three gratings are presented in Fig. 4. Here, the x-axis corresponds to the direction of light propagation, from positive to negative values. The y-axis is the distance from the grating to the detector. Zero is the position of the grating. The dotted and solid lines indicate the direction and position of the diffraction maxima of two different wavelengths. The third-order maximum in (a) is not present since it represents the Bragg reflection within the fiber core of this third-order FBG. The same applies to the fourth-order in (b) and the fifth-order in (c), as they are the Bragg reflections within the fiber core of the fourth-and fifth-order FBGs, respectively. Fig. 4(a) shows the simulation of the diffraction order maxima m = 1 and m = 2 at a fiber grating with a period of 834 nm. Both out-coupled orders are incompatible with the tested wavelength range because of their low angular separation (angle difference). As a result, they cover only 0.1 mm and 0.3 mm of the detector surface, corresponding to 13 and 38 pixels, respectively, which does not provide sufficient resolution. The third diffraction order of the grating with a period of 1112 nm shows better coverage of 87 pixels (Fig. 4(b)) that can be further enlarged by increasing the distance between the optical fiber and detector surface. An increased distance of 34 mm (setup limit) resulted in 2000 pixels, providing a suitable pixel resolution of 0.02 nm/pixel. A higher resolution can be achieved by increasing the number of diffraction orders, as demonstrated by the grating with a period of 1389 nm in Fig. 4(c), where m = 4 covers 952 pixels, even at the original distance of 1.5 mm. However, the higher the diffraction order, the lower the out-coupled optical power. Thus, the fourth-order fiber Bragg grating with a distance of 34 mm between the fiber and detector was the optimum configuration for the experiments.

B. Grating Investigation
According to the results of our calculations, we selected the FBG with a grating period of Λ 2 = 1112 nm to be best suited within our preferred configuration (CCD grating distance of 34 mm). This fiber Bragg grating was produced and placed into the experimental setup. For our investigations, monochromatic light at 820 nm emitted from a tunable diode laser was coupled into the fiber resulting in six out-coupled beams, as depicted in Fig. 5(a). The six beams are three pairs that represent azimuthally symmetric diffraction maxima of diffraction orders m = 1, m = 2, and m = 3. They were measured by rotating the CCD array around the fiber at a fixed distance of 1.5 mm and an azimuth resolution of 3°.
The azimuthally symmetric diffraction is associated with the geometrical form of the individual grating points: elongated ellipsoids. The highest intensity diffracts perpendicular to the long axis of the created ellipsoidal grating points [8], [10], [26], [27], [28]. The azimuthal mapping of the out-coupled radiation around the fiber axis is shown in Fig. 5(b), indicating an azimuthal spread of ≈20°in both directions.
Next, we fixed the CCD array at an azimuth position (at 104°) with the highest intensity of the out-coupled light to test its resultant orders at low and high spectral limits. Fig. 6(a)   indicates overlapping of the first and second orders, thus not ensuring an adequate resolution in the predetermined spectral range. On the other hand, the third order of both wavelengths is completely distinguishable, with a distance of 88 pixels between peak maxima. These measurements corroborate our calculations in Section IV.A, providing an optimum resolution by increasing the distance between the optical fiber and the detector surface to 34 mm.
The grating length is the second significant factor affecting the resolution of this spectrometer (aside from diffraction order). The longer the Bragg grating, the broader the peak width recorded by the CCD array. To exemplify it, we contrasted the proposed 0.2 mm-long grating with 180 points to the more commonly used 1.0 mm-long grating with 900 points (Fig. 6). The 1.0 mm-long grating provides a peak width of 1.0 mm, leading to heavy overlap between the selected wavelengths, even at the third order (see m = 3 Fig. 6(b)). The 0.2 mm-long grating, in turn, represents no overlapping at m = 3 (see Fig. 6(a)), thus providing the required spectrometer resolution for the spectrometer application demonstrated in Section V. Therefore, it was inferred that the grating length should not be too long leading to peak overlapping, and not too short decreasing the out-coupling efficiency. Thus, the grating length and, accordingly, the peak width (resolution) should be set based on the required resolution of the spectrometer, the output power of the light source, and the detector sensitivity: The choice of the grating length and the distance between the grating and CCD array can be used to control the resulting peak width considering the intensity of the diffracted light illuminating the CCD pixels.

C. Wavelength Calibration
To calibrate the investigated grating for spectrometer application, a wavelength calibration procedure was applied by correlating the illuminated CCD pixels excited by the diffracted laser light from the FBG with the spectrum of the OSA (spectral reference) and the power meter (grating efficiency). The tunable diode laser was utilized to tune nine characteristic lines in the 810-850 nm range at intervals of approximately 5 nm. Each line had a bandwidth of 0.05 nm and the same optical power irrespective of the wavelength (controlled by the power meter). A set of calibration lines distributed over the specified spectral range assign the characteristic wavelengths to individual pixels on the detector by defining the calibration function. Fig. 7(a) shows the output calibration lines measured by the OSA. The peaks detected on the CCD array are presented in Fig. 7(b), along with the assigned wavelengths. The measurements were done at a fixed room temperature of 19°C and the detector integration time of 10 ms. Next, the detected peaks were fitted using a standard Gaussian function to assign the pixel number on the detector to characteristic wavelengths, as shown in Fig. 7(c).
The relationship between the pixel numbers and their corresponding characteristic wavelengths leads to the calibration function, which is a second-order polynomial function with a coefficient of determination R 2 of 0.99993, as shown in Fig. 7(c). Here, y is the characteristic wavelength, x is the pixel number, and B1, B2 are coefficients.

D. Resolution of the Spectrometer
The resolution of a spectrometer is the ability to resolve the maximum number of spectral peaks (wavelengths) in a defined spectral range. There are two concepts to describe the spectrometer resolution: (1) the optical concept that is the spectral resolving power R, and (2) the digital one that is the pixel resolution (Δλ) pixel .
The resolving power describes the ability of a diffraction grating to separate the diffraction maxima of two closely spaced wavelength lines. It is defined as R = λ/(Δλ) min where λ is the given wavelength and (Δλ) min is the minimum resolvable wavelength difference (spectral resolution), following the Rayleigh criterion: two spectral lines of different wavelengths are separate when the maximum of one peak matches the minimum of the next peak. Considering the operating wavelength of 826 nm and the Rayleigh criterion of 1.09 nm ( Fig. 8(a)), the spectral resolving power of 758 was determined. It should be noted that according to the dispersion angle α (3) and the constant, parallel position of the grating to the detector, the resolving power increases with decreasing the dispersion angle. Accordingly, the lowest resolving power of 650 is at the lower wavelength limit (810 nm), the highest of 1000 is at the upper wavelength limit (850 nm), and the average is 825, respectively. The relationship between the dispersion angle as a function of the resolving power is presented in Fig. 8(b).
The second concept to describe the spectrometer resolution is the pixel resolution (Δλ) pixel that is the spectral bandwidth detected by one pixel. Based on the measurements, the pixel resolution constitutes 0.02 nm/pixel, corroborating the calculation in Section IV.A. Therefore, the pixel-to-spectral resolution ratio is greater than 43, providing efficient peak detection in the predetermined spectral range.

E. Polarization Dependence
Polarization state changes of light in fiber optics can be caused by birefringence in the fiber core due to stress, strain, or temperature [29], [30]. It is essential that the polarization state of the guided light should not decrease the accuracy of measurements if a spectrometer is used for the spectral characterization of sensor signals. Therefore, we evaluated the polarization effect of our spectrometer with the test setup demonstrated in Fig. 3. The measurement procedure requires a linear polarizer and two types of quartz retardation plates: a quarter-wave plate and a half-wave plate. Each component is mounted on a separate rotation stage where the azimuth angles can be controlled. Then, a polarizer is inserted into the beam to define the initial linear polarization state. Behind the polarizer, the quarter-wave and half-wave plates are applied to create different polarization states by rotating either λ/2or λ/4-wave plate at 10°-intervals, respectively, beginning with 0°and ending with 360°. The analyzed peak shifts and amplitude changes are shown in Fig. 9. Fig. 9(a) shows a characteristic polarization dependency by the quarter-wave plate, where the wavelength shift totaled up to ±1.0 pixel (±0.02 nm) and the intensity loss up to ±25%. On the other hand, the spectral shifts by the half-wave plate were less than ±2.5 pixel (±0.05 nm), and the maximum variation of intensity was ±40%, as shown in Fig. 9(b).

F. Relative Diffraction Efficiency
The relative diffraction efficiency of the tested grating was determined using the setup shown in Fig. 3. The wavelength signal was first set to 810 nm, that is, the lower wavelength limit on the tested spectrometer, and then stepped away to the upper limit of 850 nm; thus, the relative diffraction efficiency at the specified spectral range was characterized. The central wavelength and optical power of the laser peaks were controlled by applying the OSA and the power meter. The relative diffraction efficiency as a function of the wavelength is presented in Fig. 10. The results show that the relative, wavelength-dependent intensity shifts were found within ±1.5% in the specified spectral range.

V. APPLICATION OF THE PROPOSED SPECTROMETER FOR READING OUT FBG SENSORS
The proposed spectrometer was connected to the FBG sensor array and a light source via a 3 dB coupler to create a readout device. Additionally, a commercial spectrometer (FLAME, Ocean Insight) with a spectral resolution of 1.34 nm was paired with the test spectrometer via another 3 dB coupler as a reference spectrometer, as shown in Fig. 11(a). The light source is a superluminescent diode (SLED) (EXS210037-01, Exalos). The signals of the Bragg gratings used as sensors for experiments were placed in the spectral range of 815-850 nm and had the following parameters: a reflectivity of >30%, an FWHM of about 0.35 nm, and a grating length of 1 mm. Fig. 11 Fig. 11(c) indicates a Pearson's correlation coefficient of r = 0.99.

VI. DISCUSSION AND CONCLUSION
The reported numerical and experimental results indicate that a short, uniform fiber Bragg grating can be used as a dispersive element for fiber spectrometer. This element shows a relative, wavelength-dependent intensity shift within ±1.5% in the used spectral range and depends on light polarization, reducing the wavelength accuracy up to ±0.05 nm. In addition, the proposed element was tuned so that the diffracted light was focused on the CCD array with the required optical resolution and without overlap between the diffraction orders, thereby creating a fiber optical spectrometer. The resultant spectrometer was characterized, calibrated, and applied to the readout of the FBG sensor array. The performance of this spectrometer resulted in an average resolving power of 825, a pixel resolution of 0.02 nm/pixel, and 40 nm bandwidth, covering a 810-850 nm wavelength range.
The advantage of our spectrometer is its excellent structural operating stability due to the lack of additional optical components between fiber and detector that reduces the cost and size of the device. In addition, the uniform Bragg grating as an integrated dispersive element may simplify the production of the diffraction gratings for specific applications.
The significant limitation of this method is the low outcoupling diffraction efficiency of <1.0%. For the 0.2 mm-long FBG, a detailed quantification of the out-coupling efficiency using a monochromatic light source at 835 nm shows an out-coupling diffraction efficiency of 0.66% summed over all diffraction orders, whereby the third order accounts for 0.16%, the second for 0.35% by, and the first for 0.15%. In principle, tilting the ultra-short grating can increase the out-coupling efficiency but complicates the grating production and increases the polarization dependence. The relatively narrow spectral range of 40 nm is also a drawback, but it can be varied by adapting the grating parameters and the length of the CCD array with minor technological effort.
In conclusion, the proposed fiber optical spectrometer has potential applications for analyzing fiber-guided light in a specific wavelength range; for instance, it can be used as a readout system for quasi-distributed FBG sensors. Furthermore, using ultra-short uniform Bragg grating as a diffraction grating allows for miniaturized, cost-effective spectrometers, which are critical to meet growing demand in scientific and industrial research [31], [32], [33], [34].