Optical fiber measuring device for use in a hot-spot method of tank measurement

By using fiber optic measurement devices with single-row cyclic arrangement and multi-row cross arrangement, the problem of excessively small bending radius caused by dense arrangement of fiber optic measurement points is solved, achieving higher measurement point density and larger arrangement space, which is suitable for measuring temperature gradients in storage tanks and planar structures.

CN123045547APending Publication Date: 2026-11-06BEIJING INST OF CONTROL ENG
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Patent Information

Application Number
CN202611103117.1
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2026-07-23
Publication Date
2026-11-06

AI Technical Summary

Technical Problem

In the prior art, the size of the fiber Bragg grating measurement points and the bending radius of the connecting optical fiber limit the density of the measurement points, making it difficult to lay out the optical fiber. In addition, the spacing between the grating measurement points is too small, making it difficult to accurately measure the temperature state inside the tank.

Method used

Design an optical fiber measurement device. Increase the bending radius of the connecting optical fiber by cyclically arranging a single column of measurement points, and increase the density of temperature measurement points by arranging multiple columns of measurement points in a cross pattern. Use a single or double optical fiber to connect multiple grating measurement points in series to form different types of measurement devices.

Benefits of technology

It increases the density of fiber Bragg grating measurement points in the measurement direction, provides more space for fiber optic splice arrangement, and is suitable for tank temperature gradient measurement and planar structure temperature gradient measurement, thus improving measurement accuracy and feasibility.

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Abstract

This invention relates to an optical fiber measurement device for measuring hot spots in a storage tank, belonging to the field of spacecraft propulsion system technology; it includes k optical fibers and j grating measurement points; k=1 or 2; j is an even number not less than 4; when k=1, one optical fiber connects j grating measurement points in series to form a single-fiber single-sided measurement device or a single-fiber double-sided measurement device; when k=2, each optical fiber connects a grating measurement point in series to form a double-fiber double-sided measurement device; this invention increases the bending radius of the connecting optical fibers by cyclically arranging measurement points in a single column, providing a larger space for the arrangement of optical fiber splice sections, and increases the density of temperature measurement points in the gradient measurement direction by crisscrossing multiple columns of measurement points.
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Description

Technical Field

[0001] This invention belongs to the field of spacecraft propulsion system technology and relates to an optical fiber measurement device for measuring hot spot in a propellant tank. Background Technology

[0002] In spacecraft propulsion systems, a hotspot method based on thermal response measurement is used to assess propellant remaining quantity for satellites employing plate-type propellant tanks. The hotspot method is a propellant remaining quantity estimation method based on thermal response. It involves densely arranging temperature measuring points along the direction of liquid level change in a specific area on the outer surface of the tank, and using a hotspot heater. The position of the liquid level inside the tank is primarily determined by the thermal response of each temperature measuring point, thereby indirectly judging the remaining liquid quantity in the tank.

[0003] The hot spot method requires densely arranged fiber optic grating temperature measurement points within a certain area of ​​the tank.

[0004] Furthermore, large spacecraft have extensive propellant tanks with significant temperature variations throughout. Traditional methods, using only a few thermistors at specific points within the tank, are insufficient to accurately represent the temperature state of the gas and liquid within. Therefore, it is necessary for large propellant tanks to utilize a dense array of optical grating measuring points to achieve accurate measurement and description of the internal temperature field, enabling monitoring of the tank's condition and providing measurement parameters.

[0005] The density of measurement points determines the resolution of the hotspot method and the temperature gradient resolution of the storage tank. However, factors such as the current size of the grating measurement points and the bending radius of the connecting optical fibers between the grating measurement points not only limit the density of measurement point arrangement but also cause difficulties in fiber routing. Dense arrangement of fiber measurement points leads to problems such as excessively small bending radii of the connecting optical fibers between measurement points, difficulty in arranging fiber fusion splices between measurement points, and excessively small spacing between grating measurement points. Summary of the Invention

[0006] The technical problem solved by this invention is to overcome the shortcomings of the prior art and propose an optical fiber measuring device for tank hot spot method measurement. By cyclically arranging a single column of measuring points, the bending radius of the connecting optical fiber is increased, providing a larger space for the arrangement of optical fiber fusion splices. By arranging multiple columns of measuring points in a cross pattern, the arrangement density of temperature measuring points in the gradient measurement direction is improved.

[0007] The solution of the present invention is: An optical fiber measuring device for measuring hot spots in a storage tank includes k optical fibers and j grating measuring points; k = 1 or 2; j is an even number not less than 4. When k=1, one optical fiber connects j grating measurement points in series to form a single-sided or single-sided measuring device. When k=2, each fiber is connected in series. A grating measuring point is formed to create a measuring device with two roots and two sides.

[0008] In the aforementioned fiber optic measurement device for tank hotspot measurement, the single-strand, single-sided measurement device specifically takes the following form: Let j = 2n; n is a positive integer not less than 1; 2n grating measurement points are arranged horizontally in a straight line, and all 2n grating measurement points are located on the same side of the hot spot normal heater; each grating measurement point is set perpendicular to the hot spot normal heater; Number each grating measurement point as 1-2n; The 2n grating measurement points are arranged as follows: No. 1, No. 3, No. 5, ..., No. 2n-1, No. 2, No. 4, No. 6, ..., No. 2n; A single optical fiber is used to sequentially connect grating measurement points numbered 1, 2, ..., 2n in series to form a single-fiber, single-sided measuring device.

[0009] In the aforementioned fiber optic measuring device for measuring hot spots in a storage tank, 2n grating measuring points are evenly distributed along a straight line with an interval of s. Grating point 1 is set at the starting position of the measurement; odd-numbered grating points are set in sequence at intervals of s; grating point 2 is set at intervals of s after grating point 2n-1, and even-numbered grating points are set in sequence at intervals of s.

[0010] In the aforementioned fiber optic measuring device for measuring hot spots in a storage tank, the tail ends of all odd-numbered grating measuring points point towards the hot spot normal heater; the head ends of all even-numbered grating measuring points point towards the hot spot normal heater; the fiber is horizontally inserted into the head end of grating measuring point 1, and the tail end of grating measuring point 1 is connected to the head end of grating measuring point 2 by an arc bend, the tail end of grating measuring point 2 is connected to the head end of grating measuring point 3 by an arc bend, the tail end of grating measuring point 3 is connected to the head end of grating measuring point 4 by an arc bend, ..., the tail end of grating measuring point 2n-1 is connected to the head end of grating measuring point 2n by an arc bend, and the fiber exits from the tail end of grating measuring point 2n.

[0011] In the aforementioned fiber optic measuring device for tank hotspot measurement, the specific form of the dual-root, dual-sided measuring device is as follows: Set j=4n; n is a positive integer not less than 1; the two optical fibers are defined as fiber A and fiber B respectively; fiber A and 2n grating measurement points are combined to form the first single-fiber single-sided measurement device; fiber B and the other 2n grating measurement points are combined to form the second single-fiber single-sided measurement device. The hot spot normal heater is horizontally positioned in the middle; the first single-strand single-sided measuring device and the second single-strand single-sided measuring device are positioned on both sides of the hot spot normal heater.

[0012] In the aforementioned fiber optic measuring device for measuring hot spots in a storage tank, in two single-strand, single-sided measuring devices, the distance between the grating measuring point and the hot spot normal heater is h; h is 18-22 mm.

[0013] In the aforementioned fiber optic measuring device for measuring hot spots in a storage tank, two single-strand, single-sided measuring devices are staggered; the stagger interval is... .

[0014] In the aforementioned fiber optic measurement device for tank hotspot measurement, the specific form of the single-strand, double-sided measurement device is as follows: Let j = 4n; n is a positive integer not less than 1; The first column has 2n grating measurement points arranged horizontally and located on one side of the heater normal to the hot spot; the second column has another 2n grating measurement points arranged horizontally and located on the other side of the heater normal to the hot spot; each grating measurement point is set perpendicular to the heater normal to the hot spot. Number each grating measurement point as 1-4n; The serial numbers of the 2n grating measurement points in the first column are arranged as follows: 1, 5, 9, ..., 4n-3, 2, 6, 10, ..., 4n-2; The second column of 2n grating measurement points is ordered as follows: No. 3, No. 7, No. 11, ..., No. 4n-1, No. 4, No. 8, No. 12, ..., No. 4n; A single optical fiber is used to sequentially connect grating measurement points numbered 1, 2, ..., 4n in series to form a single-fiber, double-sided measuring device.

[0015] In the aforementioned fiber optic measuring device for tank hotspot measurement, the 2n grating measuring points in the first column are equally spaced along a straight line with an interval of s; the 2n grating measuring points in the second column are equally spaced along a straight line with an interval of s; the first and second columns are staggered with an alternation interval of [missing information]. ; The distance between the grating measuring points in the first and second columns and the heater normal to the hot spot is h; h is 18-22 mm.

[0016] In the aforementioned fiber optic measuring device for measuring hot spot in a storage tank, the tail ends of all grating measuring points in the first column point towards the hot spot normal heater; the tail ends of all grating measuring points in the second column point towards the hot spot normal heater. The optical fiber is horizontally connected to the head end of grating point 1. The tail end of grating point 1 is connected to the head end of grating point 2 using an arc bend. The tail end of grating point 2 is connected to the head end of grating point 3 using an arc bend. The tail end of grating point 3 is connected to the head end of grating point 4 using an arc bend. ... The tail end of grating point 4n-1 is connected to the head end of grating point 4n using an arc bend. The fiber is then exited from the tail end of grating point 4n.

[0017] The advantages of this invention compared to the prior art are: (1) The arrangement configuration and method of the grating measurement points of the measuring fiber designed in this invention can improve the arrangement density of the fiber grating measurement points in the measurement direction by the hot spot method. (2) The grating measurement point arrangement configuration and arrangement method of the measuring fiber designed in this invention provide more arrangement space for the connecting fiber and fiber fusion splice between the grating measurement points; (3) The arrangement configuration and method of the grating measurement points of the present invention are also applicable to the measurement of temperature gradient of storage tank based on fiber optic grating measurement points, temperature gradient of planar structure, strain field measurement of planar structure, etc., and have wide application value. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the single-root, single-sided measuring device of the present invention; Figure 2 This is a schematic diagram of the measuring device of the present invention in the form of a double root and double sides; Figure 3 This is a schematic diagram of the single-root, double-sided measuring device of the present invention. Detailed Implementation

[0019] The present invention will be further described below with reference to the embodiments.

[0020] This invention provides an optical fiber measuring device for tank hot spot method measurement. By cyclically arranging a single column of measuring points, the bending radius of the connecting optical fiber is increased, providing a larger space for the arrangement of optical fiber fusion splices. By arranging multiple columns of measuring points in a cross pattern, the arrangement density of temperature measuring points in the gradient measurement direction is increased.

[0021] The fiber optic measurement device used for tank hot spot method measurement mainly consists of optical fibers and grating measurement points, including k optical fibers and j grating measurement points; k=1 or 2; j is an even number not less than 4.

[0022] When k=1, one optical fiber connects j grating measurement points in series to form a single-sided or single-sided measuring device. When k=2, each fiber is connected in series. A grating measuring point is formed to create a measuring device with two roots and two sides.

[0023] like Figure 1 As shown, the specific form of the single-root, single-sided measuring device is as follows: Let j = 2n; n is a positive integer not less than 1.

[0024] 2n grating measurement points are arranged horizontally in a straight line, and all 2n grating measurement points are located on the same side of the hot spot normal heater; each grating measurement point is set perpendicular to the hot spot normal heater.

[0025] Number each grating measurement point as 1-2n.

[0026] The 2n grating measurement points are ordered as follows: No. 1, No. 3, No. 5, ..., No. 2n-1, No. 2, No. 4, No. 6, ..., No. 2n.

[0027] A single optical fiber is used to sequentially connect grating measurement points numbered 1, 2, ..., 2n in series to form a single-fiber, single-sided measuring device.

[0028] 2n grating measurement points are distributed at equal intervals along a straight line, with an interval of s; Grating point 1 is set at the starting position of the measurement; odd-numbered grating points are set in sequence at intervals of s; grating point 2 is set at intervals of s after grating point 2n-1, and even-numbered grating points are set in sequence at intervals of s.

[0029] The tail ends of all odd-numbered grating measurement points point towards the hot spot normal heater; the head ends of all even-numbered grating measurement points point towards the hot spot normal heater; the optical fiber is horizontally inserted into the head end of grating measurement point 1, and is connected to the head end of grating measurement point 2 by an arc bend, the tail end of grating measurement point 2 is connected to the head end of grating measurement point 3 by an arc bend, the tail end of grating measurement point 3 is connected to the head end of grating measurement point 4 by an arc bend, ..., the tail end of grating measurement point 2n-1 is connected to the head end of grating measurement point 2n by an arc bend, and it exits from the tail end of grating measurement point 2n.

[0030] like Figure 1For a single-strand, single-sided measuring device, the grating measurement points are numbered 1 to 2n. In the case of an odd number of grating measurement points, the last number, 2n, can be left vacant. The grating measurement points are arranged sequentially according to their numbers. Grating measurement point 1 is placed at the starting position of the measurement. Grating measurement point 2 is placed at a distance s×n from grating measurement point 1, with its ends facing opposite directions, ensuring that the optical fiber connecting grating measurement points 1 and 2 has only one large-circle bend. Grating measurement point 3 is placed at a distance s from grating measurement point 1, with its ends facing opposite directions, ensuring that the optical fiber connecting grating measurement points 3 and 2 has only one large-circle bend. Grating measurement point 4 is placed at a distance s×n from grating measurement point 2, with its ends facing opposite directions, ensuring that the optical fiber connecting grating measurement points 4 and 3 has only one large-circle bend. Similarly, the grating measurement points of the entire measurement fiber are arranged in the order of grating measurement point 1, grating measurement point 3, ..., grating measurement point 2n-1, grating measurement point 2, grating measurement point 4, ..., grating measurement point 2n, with an interval of s between each other.

[0031] like Figure 2 As shown, the specific form of the double-root, double-sided measuring device is as follows: Set j=4n; n is a positive integer not less than 1; the two optical fibers are defined as fiber A and fiber B respectively; fiber A and 2n grating measurement points therein form the first single-fiber single-sided measurement device; fiber B and the other 2n grating measurement points form the second single-fiber single-sided measurement device.

[0032] The hot spot normal heater is horizontally positioned in the middle; the first single-strand single-sided measuring device and the second single-strand single-sided measuring device are positioned on both sides of the hot spot normal heater.

[0033] In the two single-root, single-sided measuring devices, the distance between the grating measuring point and the hot spot normal heater is h; h is 18-22 mm.

[0034] Two single-strand, single-sided measuring devices are staggered; the stagger interval is... .

[0035] In this invention, measuring optical fibers A and B are positioned on either side of the hotspot heater, within its longitudinal coverage area. The edges of the grating measurement points on both measuring fibers A and B are each h away from the hotspot heater; the value of h is determined by the hotspot design and is typically 20 mm. The grating measurement points on measuring fiber B and measuring fiber A are offset by a distance s / 2 in the parallel direction, where s is the spacing between adjacent grating measurement points on a single measuring fiber.

[0036] like Figure 3 As shown, the specific form of the single-root double-sided measuring device is as follows: Let j = 4n; n is a positive integer not less than 1.

[0037] The first column has 2n grating measurement points arranged horizontally and located on one side of the hot spot normal heater; the second column has another 2n grating measurement points arranged horizontally and located on the other side of the hot spot normal heater; each grating measurement point is set perpendicular to the hot spot normal heater.

[0038] Number each grating measurement point as 1-4n.

[0039] The first column of 2n grating measurement points is numbered as follows: 1, 5, 9, ..., 4n-3, 2, 6, 10, ..., 4n-2. The second column of 2n grating measurement points is numbered as follows: 3, 7, 11, ..., 4n-1, 4, 8, 12, ..., 4n.

[0040] A single optical fiber is used to sequentially connect grating measurement points numbered 1, 2, ..., 4n in series to form a single-fiber, double-sided measuring device.

[0041] The first column has 2n grating measurement points evenly spaced along a straight line with an interval of s; the second column also has 2n grating measurement points evenly spaced along a straight line with an interval of s; the first and second columns are staggered, with an alternation interval of [missing value]. The distance between the grating measuring points in the first and second columns and the heater normal to the hot spot is h; h is 18-22 mm.

[0042] In the first column, the tail ends of all grating measurement points point towards the heater normal to the hot spot; in the second column, the tail ends of all grating measurement points point towards the heater normal to the hot spot.

[0043] The optical fiber is horizontally connected to the head end of grating point 1. The tail end of grating point 1 is connected to the head end of grating point 2 using an arc bend. The tail end of grating point 2 is connected to the head end of grating point 3 using an arc bend. The tail end of grating point 3 is connected to the head end of grating point 4 using an arc bend. ... The tail end of grating point 4n-1 is connected to the head end of grating point 4n using an arc bend. The fiber is then exited from the tail end of grating point 4n.

[0044] In this invention, the grating measurement points are numbered from 1 to 4n. If the number of grating measurement points exceeds 4n but is less than 4(n+1), the extra grating measurement points are flexibly arranged and not included in the arrangement. The grating measurement points are arranged sequentially according to their numbers. Grating measurement point 1 is placed at the starting position of the measurement. Grating measurement point 2 is placed on the same side of the hot spot heater as grating measurement point 1, at a distance of s×n from grating measurement point 1. The head and tail of grating measurement point 2 and grating measurement point 1 face opposite directions, so that the optical fiber connecting grating measurement point 1 and grating measurement point 2 has only one large arc bend. Grating measurement point 3 is on the opposite side of grating measurement point 2, at a distance of s / 2 from grating measurement point 1. The head and tail of grating measurement point 3 and grating measurement point 2 face opposite directions, so that the optical fiber connecting grating measurement point 3 and grating measurement point 2 has only one large arc bend. Grating measurement point 4 is placed on the same side as grating measurement point 3, at a distance of s×n from grating measurement point 3. Grating measurement point 4 and grating measurement point 3 have opposite orientations, ensuring that the connecting fiber between grating measurement point 4 and grating measurement point 3 has only one large-circle bend. Similarly, the grating measurement points along the entire measurement fiber are arranged in the following order on the left side of the hot-spot heater: grating measurement point 1, grating measurement point 5, ..., grating measurement point 4n-3, grating measurement point 2, grating measurement point 6, ..., grating measurement point 4n-2, with intervals of s between them. On the right side of the hot-spot heater, the grating measurement points are arranged in the following order: grating measurement point 3, grating measurement point 7, ..., grating measurement point 4n-1, grating measurement point 4, grating measurement point 8, ..., grating measurement point 4n, with intervals of s between them.

[0045] The fiber optic measurement point arrangement configuration and method designed in this invention can, on the one hand, improve the arrangement density of fiber optic measurement points in the measurement direction using the hot spot method, and on the other hand, provide more space for the connecting fibers and fiber optic splices between the measurement points. Furthermore, this fiber optic measurement point arrangement configuration and method are also applicable to tank temperature gradient measurement, planar structure temperature gradient measurement, and planar structure strain field measurement based on fiber optic measurement points, and have broad application value.

[0046] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention by utilizing the methods and techniques disclosed above without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

Claims

1. An optical fiber measuring device for tank hotspot method measurement, characterized in that: It includes k optical fibers and j grating measurement points; k = 1 or 2; j is an even number not less than 4; When k=1, one optical fiber connects j grating measurement points in series to form a single-sided or single-sided measuring device. When k=2, each fiber is connected in series. A grating measuring point is formed to create a measuring device with two roots and two sides.

2. The fiber optic measuring device for tank hotspot method measurement according to claim 1, characterized in that: The specific form of the single-root, single-sided measuring device is as follows: Let j = 2n; n is a positive integer not less than 1; 2n grating measurement points are arranged horizontally in a straight line, and all 2n grating measurement points are located on the same side of the hot spot normal heater; each grating measurement point is set perpendicular to the hot spot normal heater; Number each grating measurement point as 1-2n; The 2n grating measurement points are arranged as follows: No. 1, No. 3, No. 5, ..., No. 2n-1, No. 2, No. 4, No. 6, ..., No. 2n; A single optical fiber is used to sequentially connect grating measurement points numbered 1, 2, ..., 2n in series to form a single-fiber, single-sided measuring device.

3. The fiber optic measuring device for tank hotspot measurement according to claim 2, characterized in that: 2n grating measurement points are distributed at equal intervals along a straight line, with an interval of s; Grating point 1 is set at the starting position of the measurement; odd-numbered grating points are set in sequence at intervals of s; grating point 2 is set at intervals of s after grating point 2n-1, and even-numbered grating points are set in sequence at intervals of s.

4. The fiber optic measuring device for tank hotspot measurement according to claim 3, characterized in that: The tail ends of all odd-numbered grating measurement points point towards the hot spot normal heater; the head ends of all even-numbered grating measurement points point towards the hot spot normal heater; the optical fiber is horizontally inserted into the head end of grating measurement point 1, and is connected to the head end of grating measurement point 2 by an arc bend, the tail end of grating measurement point 2 is connected to the head end of grating measurement point 3 by an arc bend, the tail end of grating measurement point 3 is connected to the head end of grating measurement point 4 by an arc bend, ..., the tail end of grating measurement point 2n-1 is connected to the head end of grating measurement point 2n by an arc bend, and it exits from the tail end of grating measurement point 2n.

5. The fiber optic measuring device for tank hotspot method measurement according to claim 4, characterized in that: The specific form of the double-root, double-sided measuring device is as follows: Set j=4n; n is a positive integer not less than 1; the two optical fibers are defined as fiber A and fiber B respectively; fiber A and 2n grating measurement points are combined to form the first single-fiber single-sided measurement device; fiber B and the other 2n grating measurement points are combined to form the second single-fiber single-sided measurement device. The hot spot normal heater is horizontally positioned in the middle; the first single-strand single-sided measuring device and the second single-strand single-sided measuring device are positioned on both sides of the hot spot normal heater.

6. The fiber optic measuring device for tank hotspot method measurement according to claim 5, characterized in that: In the two single-root, single-sided measuring devices, the distance between the grating measuring point and the hot spot normal heater is h; h is 18-22 mm.

7. The fiber optic measuring device for tank hotspot measurement according to claim 5, characterized in that: Two single-strand, single-sided measuring devices are staggered; the stagger interval is... .

8. The fiber optic measuring device for tank hotspot method measurement according to claim 4, characterized in that: The specific form of the single-root double-sided measuring device is as follows: Let j = 4n; n is a positive integer not less than 1; The first column has 2n grating measurement points arranged horizontally and located on one side of the heater normal to the hot spot; the second column has another 2n grating measurement points arranged horizontally and located on the other side of the heater normal to the hot spot; each grating measurement point is set perpendicular to the heater normal to the hot spot. Number each grating measurement point as 1-4n; The serial numbers of the 2n grating measurement points in the first column are arranged as follows: 1, 5, 9, ..., 4n-3, 2, 6, 10, ..., 4n-2; The second column of 2n grating measurement points is ordered as follows: No. 3, No. 7, No. 11, ..., No. 4n-1, No. 4, No. 8, No. 12, ..., No. 4n; A single optical fiber is used to sequentially connect grating measurement points numbered 1, 2, ..., 4n in series to form a single-fiber, double-sided measuring device.

9. The fiber optic measuring device for tank hotspot measurement according to claim 8, characterized in that: The first column has 2n grating measurement points evenly spaced along a straight line with an interval of s; the second column also has 2n grating measurement points evenly spaced along a straight line with an interval of s; the first and second columns are staggered, with an alternation interval of [missing value]. ; The distance between the grating measuring points in the first and second columns and the heater normal to the hot spot is h; h is 18-22 mm.

10. The fiber optic measuring device for tank hotspot measurement according to claim 9, characterized in that: In the first column, the tail ends of all grating measurement points point towards the heater normal to the hot spot; in the second column, the tail ends of all grating measurement points point towards the heater normal to the hot spot. The optical fiber is horizontally connected to the head end of grating point 1. The tail end of grating point 1 is connected to the head end of grating point 2 using an arc bend. The tail end of grating point 2 is connected to the head end of grating point 3 using an arc bend. The tail end of grating point 3 is connected to the head end of grating point 4 using an arc bend. ... The tail end of grating point 4n-1 is connected to the head end of grating point 4n using an arc bend. The fiber is then exited from the tail end of grating point 4n.