Method for realizing CFBG multi-order dispersion dynamic tuning by using multiple temperature fields

Abstract

The invention discloses a method for achieving CFBG multi-order dispersion dynamic tuning through multiple temperature fields. According to the initial dispersion of each order of the ultrafast laser system at the central wavelength lambda and the initial dispersion of each order of the CFBG at the central wavelength lambda, the residual dispersion of each order of the CFBG at the central wavelength lambda is calculated, based on the thermo-optic effect and the thermal expansion effect, a multi-section discrete temperature field is designed to regulate and control the effective refractive index and period of the CFBG, and the multi-order dispersion parameters of the CFBG are finely regulated and controlled. The pulse compression quality is improved, and high-quality ultrafast laser output is achieved. According to the method provided by the invention, accurate registration can be simultaneously carried out on second-order, third-order and fourth-order dispersion by adding a multi-section discrete temperature field; in addition, the system has the advantages of large dispersion tuning range, high flexibility and the like.

Description

Technical Field

[0001] The present invention belongs to the field of ultrafast fiber lasers, and in particular relates to a method for realizing dynamic tuning of CFBG multi-order dispersion by utilizing multiple temperature fields. Background Art

[0002] In ultrafast fiber lasers, the dispersion of the CFBG itself, which acts as a pulse stretcher, is prone to mismatch with the intrinsic dispersion of the ultrafast fiber laser. When the second-order dispersion mismatch occurs, the output pulse waveform will be significantly broadened; when the third-order dispersion mismatch occurs, the output pulse waveform will produce a series of pulse sidelobes, namely relaxation oscillator pulses; and when the fourth-order dispersion mismatch occurs, a pedestal will be generated at the bottom of the output pulse. In summary, each order of dispersion mismatch will cause different distortions in the output pulse of the fiber laser, seriously degrading the pulse peak power and worsening the laser performance. Therefore, it is necessary to suppress multi-order dispersion mismatch to ensure that the ultrafast laser outputs high-quality pulses.

[0003] Typically, multi-order dispersion mismatch is suppressed by regulating the chirp rate of the CFBG. By changing the chirp rate from a constant to a linear function related to the grating position, the second-order dispersion of the CFBG is changed from linear to nonlinear, thereby achieving preliminary alignment of the various orders of dispersion. In the article "Research on Fiber Bragg Grating Technology for Laser Pulse Broadening and Compression," Zhao Weina analyzed this technical solution and pointed out that the chirp rate of the CFBG is determined by the mask, which is not very flexible and difficult to meet the dispersion alignment requirements in various situations. In addition, the dispersion generated by the amplifier itself and the residual dispersion caused by the Treacy grating due to calibration errors are not constant, and cannot meet the precise alignment of the various orders of dispersion. Therefore, a method with a large multi-order dispersion compensation range, high accuracy, and strong flexibility is needed to suppress multi-order dispersion mismatch and ensure the output of high-quality pulses from ultrafast lasers. Summary of the Invention

[0004] The present invention proposes a method for realizing dynamic tuning of CFBG multi-order dispersion using multiple temperature fields, which solves the problems of traditional dispersion matching methods such as small multi-order dispersion compensation range, inability to accurately match each order of dispersion, and low flexibility.

[0005] The technical solution to realize the present invention is: a method for realizing dynamic tuning of CFBG multi-order dispersion using multiple temperature fields, comprising the following steps:

[0006] Step 1: Perform preliminary alignment of the dispersion orders of the CFBG and Treacy grating pair in the ultrafast laser system. Based on the initial dispersion orders of the ultrafast laser system at the center wavelength λ and the initial dispersion orders of the CFBG at the center wavelength λ, calculate the preliminary residual dispersion orders of the CFBG at the center wavelength λ, and proceed to step 2.

[0007] Step 2: Assume the number of TECs is n. Based on the residual dispersion of each order of the CFBG at the center wavelength λ obtained in Step 1, design a multi-segment discrete temperature field. The parameters of the multi-segment discrete temperature field include the reference temperature T0, the position x of the TEC, the temperature difference ΔT between adjacent TECs, and the temperature field shape. Go to Step 3.

[0008] Step 3: Place the CFBG in a temperature-controlled array consisting of n TECs. By controlling the current, the TEC temperature at different positions is precisely adjusted so that the grating field temperature reaches the target value. This changes the effective refractive index and period at different positions of the grating, thereby regulating the multi-order dispersion parameters of the CFBG, changing its dispersion curve from linear to nonlinear. Ultimately, strict alignment of the various orders of dispersion between the CFBG and the Treacy grating pair is achieved in the ultrafast laser system, completing dynamic tuning of the CFBG's multi-order dispersion.

[0009] Compared with the prior art, the present invention has the following significant advantages:

[0010] (1) Existing methods typically suppress multi-order dispersion mismatch by regulating the CFBG chirp rate. However, the CFBG chirp rate is determined by the mask, which cannot be changed after design and production. This results in limited flexibility and makes it difficult to achieve dispersion alignment requirements in various situations. The proposed method only performs preliminary alignment of each order of dispersion by regulating the chirp rate, and then uses an additional discrete temperature field to achieve high-precision multi-order dispersion matching. This method is highly flexible, reduces mask design requirements, and saves usage costs.

[0011] (2) Existing methods (such as suppressing multi-order dispersion mismatch by regulating the chirp rate) are difficult to achieve simultaneous matching of multi-order dispersions. However, the method described uses three different additional discrete temperature fields to match the second-order, third-order, and fourth-order dispersions respectively, which can effectively ensure the simultaneous matching of multi-order dispersions.

[0012] (3) Existing methods (such as suppressing multi-order dispersion mismatch by regulating the chirp rate) are difficult to achieve precise matching of each order of dispersion. The tuning accuracy of each order of dispersion in the above method depends on the temperature accuracy of the temperature control device (TEC). Currently, the temperature accuracy of the temperature control device can usually reach ±0.1°C, which can achieve precise matching of each order of dispersion.

[0013] (4) Existing methods (such as suppressing multi-order dispersion mismatch by controlling the chirp rate) have a small tuning range for each order of dispersion. However, the tuning range for each order of dispersion in the proposed method depends on the temperature range of the temperature control device (TEC) and the tolerable temperature range of the CFBG, which is usually 40-60°C. The tuning range for each order of dispersion is relatively large. Among them, the tuning range for the second-order dispersion can reach 10%, the tuning range for the third-order dispersion can reach 15%, and the tuning range for the fourth-order dispersion can reach 20%, which can meet the requirements for compensating for residual dispersion of each order. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 This is a schematic diagram of the secondary temperature field described in the present invention and its effect on regulating the second-order dispersion GDD.

[0015] Figure 2 This is a schematic diagram of the three-dimensional temperature field and its effect on the control of third-order dispersion TOD described in the present invention.

[0016] Figure 3 This is a schematic diagram of the W-shaped temperature field described in the present invention and its effect on regulating fourth-order dispersion FOD.

[0017] Figure 4 It is a schematic diagram of the process of the present invention. DETAILED DESCRIPTION

[0018] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0019] The technical solutions between the various embodiments of the present invention can be combined with each other, but they must be based on the fact that ordinary technicians in this field can implement them. When the combination of technical solutions is mutually contradictory or cannot be implemented, it should be deemed that such a combination of technical solutions does not exist and is not within the scope of protection required by the present invention.

[0020] The following will further introduce the specific implementation methods, as well as the technical difficulties and inventive points of this invention in combination with this design example.

[0021] Combine Figure 4 The method of realizing dynamic tuning of CFBG multi-order dispersion by using multiple temperature fields according to the present invention comprises the following steps:

[0022] Step 1: Perform a preliminary alignment of the dispersion orders of the CFBG and Treacy grating pair in the ultrafast laser system. Based on the initial dispersion orders of the ultrafast laser system at the center wavelength λ and the initial dispersion orders of the CFBG at the center wavelength λ, calculate the residual dispersion orders of the CFBG at the center wavelength λ as follows:

[0023] When the temperature field T(z) is not added, according to the second-order initial dispersion GDD(λ), third-order initial dispersion TOD(λ) and fourth-order initial dispersion FOD(λ) of the ultrafast laser system at the central wavelength λ, the CFBG and Treacy grating in the ultrafast laser system can be preliminarily aligned for each order of dispersion by adjusting the CFBG chirp rate α(λ). At this time, the second-order initial dispersion GDD of the CFBG at the central wavelength λ isCFBG (λ) is shown in the following formula (1):

[0024]

[0025] Where c is the speed of light, and α(λ) is the chirp rate of the CFBG at the center wavelength λ.

[0026] The first-order and second-order derivatives of formula (1) are obtained respectively, which corresponds to the third-order initial dispersion expression TOD of CFBG at the central wavelength λ when no temperature field T(z) is added. CFBG (λ) and the fourth-order initial dispersion expression FOD of CFBG at the central wavelength λ CFBG (λ).

[0027] By adjusting the CFBG chirp rate, the CFBG and Treacy grating are preliminarily aligned for each order of dispersion. However, at this time, the residual dispersion of each order of CFBG at the central wavelength λ is not zero (ΔGDD CFBG (λ)≠0, ΔTOD CFBG (λ)≠0, ΔFOD CFBG (λ)≠0), this is because the dispersion generated by the amplifier itself and the residual dispersion caused by the calibration error of the Treacy grating are not aligned. According to the initial dispersion GDD(λ), TOD(λ), FOD(λ) of each order of the ultrafast laser system at the central wavelength λ and the initial dispersion GDD of each order of the CFBG at the central wavelength λ CFBG (λ), TOD CFBG (λ), FOD CFBG (λ), calculate the residual dispersion of each order of CFBG at the central wavelength λ, and satisfy the following formula:

[0028]

[0029] Among them, ΔGDD CFBG (λ) represents the second-order residual dispersion of the CFBG at the central wavelength λ, ΔTOD CFBG (λ) represents the third-order residual dispersion of the CFBG at the central wavelength λ, ΔFOD CFBG (λ) represents the fourth-order residual dispersion of the CFBG at the central wavelength λ.

[0030] Go to step 2.

[0031] Step 2: Assume that the number of TECs is n. Based on the residual dispersion of each order of the CFBG at the central wavelength λ obtained in step 1, a multi-segment discrete temperature field is designed. The parameters of the multi-segment discrete temperature field include the reference temperature T0, the position x of the TEC, the temperature difference ΔT between adjacent TECs, and the temperature field shape. The specific parameters are as follows:

[0032] The multi-segment discrete temperature field uses n discrete TECs for temperature control, with a linear temperature field between adjacent TECs. By adjusting the reference temperature T0, the position x of the TEC, and the temperature difference ΔT between adjacent TECs, the dispersive CFBG additional temperature field T(z) is generated, thus determining the shape of the temperature field.

[0033] Go to step 3.

[0034] Step 3: Place the CFBG in a temperature-controlled array consisting of n TECs. By controlling the current, the TEC temperature at different positions is precisely adjusted to achieve the target grating field temperature. This changes the effective refractive index and period at different positions of the grating, thereby regulating the CFBG's multi-order dispersion parameters, transforming its dispersion curve from linear to nonlinear. This achieves dynamic tuning of the CFBG's multi-order dispersion, compensating for the CFBG's residual dispersion of each order at the central wavelength λ, and ultimately achieving strict alignment of the dispersion orders of the CFBG and Treacy grating pair in the ultrafast laser system. The details are as follows:

[0035] The corresponding relationship between the temperature field shape and the residual dispersion of each order of CFBG at the central wavelength λ is established. The precise compensation of each order of residual dispersion includes the following three methods:

[0036] First, the second-order residual dispersion is compensated using a quadratic temperature field;

[0037] Second, the third-order residual dispersion is compensated using a cubic temperature field;

[0038] The third method is to compensate the fourth-order residual dispersion using a W-shaped temperature field.

[0039] The dispersion parameters of each order of CFBG are usually determined by the chirp rate, effective refractive index and period. After the CFBG is written, its chirp rate cannot be changed, so the dispersion parameters of each order can only be tuned by changing its effective refractive index and period. When the temperature field T(z) is added, due to the thermo-optic effect and thermal expansion effect, the initial effective refractive index n of the CFBG changes. eff The initial period Λ(z) will change, and the effective refractive index N after tuning eff (z) and the tuning period Λ T (z) are shown in formulas (3) and (4) respectively:

[0040] N eff (z) = n eff [1+ξ(T(z)-T0)] (3)

[0041] Λ T (z)=Λ(z)[1+ε(T(z)-T0)] (4)

[0042] Among them, the reference temperature T0 is 20℃, neff is the initial effective refractive index of CFBG, ξ is the thermo-optical coefficient, which reflects the effect of temperature on the effective refractive index N after tuning. eff (z); Λ(z) is the initial period distribution, ε is the thermal expansion coefficient, reflecting the effect of temperature on the period Λ after tuning T (z) modulation.

[0043] Bragg reflection wavelength λ corresponding to different positions within the CFBG bandwidth after tuning T (z) is shown in formula (5):

[0044] λ T (z)=2N eff (z)Λ T (z) (5)

[0045] At this time, the second-order dispersion GDD of CFBG under the additional temperature field T(z) is CFBG-T (z) is shown in formula (6):

[0046]

[0047] Where c represents the speed of light; z represents the position coordinate within the length range of the CFBG; λ(z) represents the central wavelength of the CFBG at position z; and λ(0) represents the central wavelength at the starting position of the CFBG.

[0048] Solving formula (4) yields λ T The inverse function z of (z) T (λ), and put it into formula (6) to obtain the second-order dispersion GDD of CFBG at the central wavelength λ under the additional temperature field T(z): CFBG-T (λ); for GDD CFBG-T (λ) respectively calculate the first-order and second-order derivatives, that is, the third-order dispersion expression TOD of CFBG at the central wavelength λ under the additional temperature field T(z) is obtained CFBG-T (λ) and the fourth-order dispersion expression FOD CFBG-T (λ), and subtracting it from the initial dispersion of each order of CFBG at the center wavelength λ, we can get the residual dispersion of each order of CFBG at the center wavelength λ.

[0049] At this point, the mathematical relationship between the additional temperature field T(z) and the tuning of each order of residual dispersion of the CFBG at the center wavelength λ is obtained. By setting the second-order residual dispersion constant, the third-order residual dispersion constant, and the fourth-order residual dispersion constant, the corresponding additional temperature field T(z) is calculated, thereby establishing a correspondence between the temperature field shape and each order of residual dispersion of the CFBG at the center wavelength λ. Based on this correspondence, three different additional discrete temperature fields can be used to match the second-order, third-order, and fourth-order dispersions, respectively, effectively ensuring simultaneous matching of multiple orders of dispersion. The temperature accuracy of current temperature control devices typically reaches ±0.1°C, enabling precise alignment of each order of dispersion. Furthermore, by setting the temperature range of the three different additional temperature fields T(z) to 40-60°C, the tuning range of each order of dispersion can be calculated. The tuning range for second-order dispersion reaches 10%, for third-order dispersion reaches 15%, and for fourth-order dispersion reaches 20%, meeting the requirements for compensating for each order of residual dispersion.

[0050] The method.

[0051] Example 1

[0052] The simulation results show that when the second-order residual dispersion ΔGDD of the CFBG at the central wavelength λ is a non-zero constant value, and the third-order residual dispersion ΔTOD and the fourth-order residual dispersion ΔFOD are both zero, the required additional temperature field curve is the quadratic temperature field, as shown in Figure 1 , meaning the quadratic temperature field can effectively compensate for the CFBG's second-order residual dispersion, ΔGDD, at the central wavelength λ, while maintaining a neutral effect on third-order dispersion (TOD) and fourth-order dispersion (FOD). Simulations show that using the quadratic temperature field allows for a 10% tuning range for second-order dispersion within the TEC temperature range of 40-60°C. The second-order residual dispersion, ΔGDD, is typically only 2% of the second-order dispersion GDD within the ultrafast laser system, effectively meeting the requirement for compensating for second-order residual dispersion.

[0053] The simulation results show that when the second-order residual dispersion ΔGDD of the CFBG at the central wavelength λ is not zero, the third-order residual dispersion ΔTOD is a non-zero constant value, and the fourth-order residual dispersion ΔFOD is zero, the required additional temperature field curve is the cubic temperature field, as shown in Figure 2 , indicating that the cubic temperature field can effectively compensate for the CFBG's third-order residual dispersion ΔTOD at the central wavelength λ without affecting the fourth-order dispersion FOD. Simulations show that using the cubic temperature field can achieve a 15% tuning range for third-order dispersion when the TEC temperature range is 40-60°C. The third-order residual dispersion ΔTOD is typically 10% of the third-order dispersion TOD in an ultrafast laser system, meeting the requirement for compensating third-order residual dispersion.

[0054] The simulation results show that when the second-order residual dispersion ΔGDD and the third-order residual dispersion ΔTOD of the CFBG at the central wavelength λ are not zero and the fourth-order residual dispersion ΔFOD is a non-zero constant value, the required additional temperature field curve is a W-shaped temperature field, as shown in Figure 3 , indicating that a W-shaped temperature field can effectively compensate for the CFBG's fourth-order residual dispersion, ΔFOD, at the central wavelength λ. Simulations show that using a W-shaped temperature field can achieve a 20% tuning range for fourth-order dispersion when the TEC temperature ranges from 40°C to 60°C. While the fourth-order residual dispersion, ΔFOD, is typically 15% of the fourth-order FOD in an ultrafast laser system, it meets the requirements for compensating for fourth-order residual dispersion.

Claims

1. A method for realizing dynamic tuning of CFBG multi-order dispersion using multiple temperature fields, characterized in that: The following steps are involved: Step 1: Perform preliminary alignment of the various order dispersions of the CFBG and Treacy grating pair in the ultrafast laser system. Based on the various order initial dispersions of the ultrafast laser system at the center wavelength λ and the various order initial dispersions of the CFBG at the center wavelength λ, calculate the preliminary residual dispersions of the various orders of the CFBG at the center wavelength λ, and proceed to step 2. Step 2: Assume the number of TECs is n. Based on the residual dispersion of each order of the CFBG at the center wavelength λ obtained in Step 1, design a multi-segment discrete temperature field. The parameters of the multi-segment discrete temperature field include the reference temperature T0, the position x of the TEC, the temperature difference ΔT between adjacent TECs, and the temperature field shape. Go to Step 3. Step 3: Place the CFBG in a temperature-controlled array consisting of n TECs. By controlling the current, the TEC temperature at different positions is precisely adjusted so that the grating field temperature reaches the target value. This changes the effective refractive index and period at different positions of the grating, thereby regulating the multi-order dispersion parameters of the CFBG, changing its dispersion curve from linear to nonlinear. Ultimately, strict alignment of the various orders of dispersion between the CFBG and the Treacy grating pair is achieved in the ultrafast laser system, completing dynamic tuning of the CFBG's multi-order dispersion.

2. The method for realizing dynamic tuning of CFBG multi-order dispersion using multiple temperature fields according to claim 1, characterized in that: In step 1, the dispersion orders of the CFBG and Treacy grating pair in the ultrafast laser system are preliminarily aligned. Based on the initial dispersion orders of the ultrafast laser system at the center wavelength λ and the initial dispersion orders of the CFBG at the center wavelength λ, the residual dispersion orders of the CFBG at the center wavelength λ are calculated as follows: When the temperature field T(z) is not added, according to the second-order initial dispersion GDD(λ), third-order initial dispersion TOD(λ) and fourth-order initial dispersion FOD(λ) of the ultrafast laser system at the central wavelength λ, the CFBG and Treacy grating in the ultrafast laser system are preliminarily aligned for each order of dispersion by adjusting the CFBG chirp rate α(λ). At this time, the second-order initial dispersion GDD of the CFBG at the central wavelength λ is CFBG (λ) is shown in the following formula (1): Where c represents the speed of light, and α(λ) represents the chirp rate of the CFBG at the central wavelength λ; The first-order and second-order derivatives of formula (1) are obtained respectively, which corresponds to the third-order initial dispersion expression TOD of CFBG at the central wavelength λ when no temperature field T(z) is added. CFBG (λ) and the fourth-order initial dispersion expression FOD of CFBG at the central wavelength λ CFBG (λ); Based on the initial dispersion GDD(λ), TOD(λ), FOD(λ) of each order of the ultrafast laser system at the central wavelength λ and the initial dispersion GDD of each order of the CFBG at the central wavelength λ CFBG (λ), TOD CFBG (λ), FOD CFBG (λ), calculate the residual dispersion of each order of CFBG at the central wavelength λ, and satisfy the following formula: Among them, ΔGDD CFBG (λ) represents the second-order residual dispersion of the CFBG at the central wavelength λ, ΔTOD CFBG (λ) represents the third-order residual dispersion of the CFBG at the central wavelength λ, ΔFOD CFBG (λ) represents the fourth-order residual dispersion of the CFBG at the central wavelength λ.

3. The method for realizing dynamic tuning of CFBG multi-order dispersion using multiple temperature fields according to claim 1, characterized in that: Its characteristics are: In step 2, let the number of TECs be n. Combined with the residual dispersion of each order of CFBG at the central wavelength λ obtained in step 1, a multi-segment discrete temperature field is designed as follows: The multi-segment discrete temperature field uses n discrete TECs for temperature control, with a linear temperature field between adjacent TECs. By adjusting the reference temperature T0, the position x of the TEC, and the temperature difference ΔT between adjacent TECs, the dispersive CFBG additional temperature field T(z) is generated, thus determining the shape of the temperature field.

4. The method for realizing dynamic tuning of CFBG multi-order dispersion using multiple temperature fields according to claim 3, characterized in that: Its characteristics are: In step 3, the CFBG is placed in a temperature-controlled array consisting of n TECs. The TEC temperature at different locations is precisely adjusted by controlling the current, so that the grating field temperature reaches the target value. This changes the effective refractive index and period at different locations of the grating, thereby regulating the CFBG's multi-order dispersion parameters, transforming its dispersion curve from linear to nonlinear. This completes dynamic tuning of the CFBG's multi-order dispersion, compensates for the CFBG's residual dispersion of each order at the central wavelength λ, and ultimately achieves strict alignment of the dispersion orders of the CFBG and Treacy grating pair in the ultrafast laser system, as follows: The corresponding relationship between the temperature field shape and the residual dispersion of each order of CFBG at the central wavelength λ is established. The precise compensation of each order of residual dispersion includes the following three methods: First, the second-order residual dispersion is compensated using a quadratic temperature field; Second, the third-order residual dispersion is compensated using a cubic temperature field; The third method is to compensate the fourth-order residual dispersion using a W-shaped temperature field.

5. The method for realizing dynamic tuning of CFBG multi-order dispersion using multiple temperature fields according to claim 4, characterized in that: The corresponding relationship between the temperature field shape and the residual dispersion of each order at the central wavelength λ of the CFBG is established to accurately compensate for the residual dispersion of each order, as follows: When the temperature field T(z) is added, due to the thermo-optic effect and thermal expansion effect, the initial effective refractive index n of CFBG eff The initial period Λ(z) will change, and the effective refractive index N after tuning eff (z) and the period after tuning Λ T (z) are shown in formulas (3) and (4) respectively: N eff (z)=n eff [1+ξ(T(z)-T0)] (3) Λ T (z)=Λ(z)[1+ε(T(z)-T0)] (4) Among them, the reference temperature T0 is 20℃, n eff is the initial effective refractive index of CFBG, ξ is the thermo-optical coefficient, which reflects the effect of temperature on the effective refractive index N after tuning. eff (z); Λ(z) is the initial period distribution, ε is the thermal expansion coefficient, reflecting the effect of temperature on the period Λ after tuning T (z) modulation; Bragg reflection wavelength λ corresponding to different positions within the CFBG bandwidth after tuning T (z) is shown in formula (5): λ T (z)=2N eff (of)Λ T (of) (5) At this time, the second-order dispersion GDD of CFBG under the additional temperature field T(z) is CFBG-T (z) is shown in formula (6): Where c represents the speed of light; z represents the position coordinate within the length range of the CFBG; λ(z) represents the central wavelength of the CFBG at position z; λ(0) represents the central wavelength at the starting position of the CFBG; Solving formula (4) yields λ T The inverse function z of (z) T (λ), and put it into formula (6) to obtain the second-order dispersion GDD of CFBG at the central wavelength λ under the additional temperature field T(z): CFBG-T (λ); for GDD CFBG-T (λ) respectively calculate the first-order and second-order derivatives, that is, the third-order dispersion expression TOD of CFBG at the central wavelength λ under the additional temperature field T(z) is obtained CFBG-T (λ) and the fourth-order dispersion expression FOD CFBG-T (λ), and subtracting it from the initial dispersion of each order of CFBG at the center wavelength λ, we can get the residual dispersion of each order of CFBG at the center wavelength λ.

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