Photothermal effects in phase shifted FBG with varied light wavelength and intensity

Meng Ding; Dijun Chen; Zujie Fang; Di Wang; Xi Zhang; Fang Wei; Fei Yang; Kang Ying; Haiwen Cai

doi:10.1364/OE.24.025370

1.Introduction

The phase shifted fiber Bragg grating (PSFBG) has been used in many applications, such as narrow linewidth lasers [1–3], optical fiber sensing [4], and optical switching [5–7], where its transmission peak is utilized as an ultra-narrow filter. The transmission of PSFBG is usually measured by optical spectrum analyzer (OSA) with a broadband source, such as the amplified spontaneous emission (ASE) of erbium doped fiber, with low spectral intensity. It was found if the power of incident light was increased to a certain degree, PSFBG showed wavelength shifts varied with increasing incident power [8, 9]. The phenomenon was attributed to the fact that distribution of light intensity along PSFBG is not uniform, but with a maximum at the position of phase shift [7]. The effect is called intensity enhancement. Similar effect was found in active devices, such as DFB semiconductor lasers [10] and DFB fiber lasers [11]. The photothermal effect in ordinary FBG was investigated by [12]. The dynamic photothermal behavior of a fiber Bragg grating F-P resonator was examined in [13] by using a tunable laser and its push–pull effect was analyzed. The dynamic thermal behavior of resonators is investigated in [14], and generalized discussions are given.

However, the dependence of intensity enhancement on light wavelength near the transmission peak of PSFBG has not been reported in detail. In this paper we give a systematic theoretical analysis of the spectral characteristics of PSFBG varied both with the incident light wavelength and its intensity, and present results of experimental investigation. The temperature rising near the phase shift point was obtained from the transmission peak shift with the light wavelength and intensity. The dynamic transmission of PSFBG transmission was measured by a wavelength modulated laser, showing asymmetric behaviors in red/blue forward tunings. By the measured data, the loss coefficient of grating fiber was evaluated. The effect of different media surrounding the grating on temperature rising was also measured. We believe that these results are helpful for applications with PSFBG used.

2. Theoretical model and analyses

2.1 Intensity enhancement

The intensity distribution of PSFBG can be obtained by a model of two cascaded uniform FBGs with a phase shift in between. The intensity enhancement factor is defined as the ratio of optical power along grating to optical power of incident light P_in. Its amplitude is dependent on the wavelength of incident light. When the wavelength of incident light (pump light, here-in-below) is at Bragg wavelength the intensity enhancement reaches its maximum. By continuity of field amplitude and phase the enhancement of a symmetric π - phase shifted FBG with the phase shift at the middle point (z = 0 for grating length of 2l), the factor is deduced to be

H (z, δ) \approx \frac{σ^{2} {κ^{2} \cos h [2 σ (l - | z |)] - δ^{2}}}{κ^{2} {[κ^{2} + (δ / 2)}^{2} e^{4 σ l}]}

where

κ = π δ n / λ

is the coupling coefficient of fiber grating with index modulation depth δn,

δ = β - β_{B}

is the difference of wave vectors, and

σ = \sqrt{κ^{2} - δ^{2}}

. Details of the deduction of Eq. (1) are given in Appendix. The factor shows an exponentially decay function of position z, and a Lorenzian-like relation to the detuning from the Bragg wavelength. At Bragg wavelength

H (z, 0) = \cosh [2 κ (l - | z |)]

, it is the same as that given in [7]. For the PSFBG we used in the experiment with parameter of κ = 0.3mm⁻¹ and l = 10mm, the intensity enhancement at the phase shift point is calculated to be 201.

2.2 Nonuniform index distribution induced by photothermal effect

The intensity enhancement will causes non-uniform temperature distribution along the grating fiber due to the photo-thermal effect caused by optical loss [12]. The index changed by photothermal effect is determined by the temperature rising inside the core through the thermo-optic effect [4]. In the stationary state the temperature rising is determined by equilibrium between the generated heat and the dissipated heat. The later is the heat flow from fiber core to its surrounding media by thermal conduction, which should obey Fourier heat conduction equation. Since the fiber core is very thin, it can be regarded as a heat source with a uniform temperature in radius, the heat conduction equation in the stationary state is simplified as $d^{2} T / d r^{2} + r^{- 1} d T / d r = 0$ . Its solution gives the temperature distribution in cladding layer $T_{c l} (r)$ and in a heat dissipation layer of surrounding medium $T_{e x} (r)$ [15]:

\begin{matrix} T_{c l} (r) = T_{c l} - \frac{Q}{2 k} \ln (\frac{r}{r_{c l}}) \\ T_{e x} (r) = T_{m} - \frac{Q}{2 k_{m}} \ln (\frac{r}{r_{m}}) \end{matrix}

where

r_{c l}

and

r_{m}

are the outer radiuses of fiber and of the heat dissipation layer, respectively; the later dependent on the thermal conductivities of media is approximated as a ring surrounding the fiber whose surface temperature is constant at

T_{m}

, i.e.

T_{e x} (r_{m}) = T_{m}

.

k

and

k_{m}

are the thermal conductivity of fiber and the surrounding medium, respectively. The temperatures at the cladding surface

T_{c l} = T_{e x} (r_{c l})

is determined by the thermal flow continuity. The heat generated in fiber core per unit length is

Q = α P_{i n} H (z, δ)

where α is the fiber loss coefficient. The temperature of fiber core in the stationary state is then obtained to be:

Δ T = T_{c o} - T_{m} = \frac{Q}{2 π} [\frac{\ln (r_{m} / r_{c l})}{k_{m}} + \frac{\ln (r_{c l} / r_{c o})}{k}]

Then the index increment caused by photothermal effect is given as

Δ n_{h} = ξ Δ T = ξ α R_{H} P_{i n} H (z, δ) \propto H (z, δ)

where

ξ

is the thermo-optic coefficient and

R_{H} = \frac{1}{2 π} [\frac{1}{k} l n (\frac{r_{c l}}{r_{c o}}) + \frac{1}{k_{m}} l n (\frac{r_{m}}{r_{c l}})]

is the thermal resistance.

2.3 The effect of Δn on characteristics of PSFBG

The index change will affect spectrum characteristics of PSFBG by two factors. Firstly, the original grating with a uniform period becomes a chirped grating with opposite signs of chirping rates in the left and right due to nonuniform index distribution. Due to temperature rising the local Bragg period changes as well, expressed as $Δ Λ (z) = Λ_{0} α_{T E} Δ T (z)$ , where $α_{T E}$ is the thermal expansion coefficient.

Secondly, the value of phase shift is changed too. Although the index increases inside the whole fiber grating, the phase shift point gets the maximum Δn(0) so that the value of phase shift increases relatively. If the original phase shift is π, corresponding to quarter wavelength, it is now increased by Δϕ = ϕ −π ∼π δn/n, where δn is the difference of index increment of the phase shift point and the average value $\bar{Δ n}$ . It is known that the position of transmission peak of PSFBG will move towards the long wavelength from Bragg wavelength when the phase shift deviates from π [16]. The peak movement with phase deviation is deduced for a strong coupling FBG with 2κl >>1 as [17]

Δ λ_{ϕ} \approx \frac{κ λ^{2}}{4 π n} \sin Δ ϕ .

It is meant that the transmission peak will not appear at the central position of the reflection band, not symmetric as in the case of weak incident light, and shift faster than the reflection band.

2.4. Transmission peak shift with light wavelength and intensity

Taking the photothermal effect into account, the wavevector difference $δ = β - β_{B}$ in Eq. (1) has to be replaced by $δ = 2 π n (λ_{p u m p}^{- 1} - λ_{p e a k}^{- 1})$ . When the pump light wavelength increases towards the transmission peak, the intensity enhancement will increase and the transmission peak shifts from the original position. The process can be described by equation of

λ_{p e a k} - λ_{B 0} = Δ λ_{B} + Δ λ_{ϕ}

where Δλ_Β and Δλ_ϕ are the reflection band shift and the transmission peak shift over the band, respectively. They can be expressed as

Δ λ_{B} = a \bar{Δ T}

and

Δ λ_{ϕ} = b [Δ T (0) - \bar{Δ T}]

, where a and b are the coefficients related to the grating under test and the experimental condition. The reflection band shift is roughly proportional to averaged temperature rising, at the resonance wavelength it is expressed as

\bar{Δ T} \propto P_{i n} \bar{H} = P_{i n} \frac{1}{l} \int_{0}^{l} \cos h [2 κ (l - z)] d z = P_{i n} \frac{\sin h 2 κ l}{2 κ l} .

The temperature at the phase shift position is obtained from Eq. (1) and Eq. (3). The transmission shift from the position of the case with weak incident light is given by

λ_{p e a k} - λ_{B 0} \propto Δ T (0) \propto P_{i n} \frac{σ^{2} [κ^{2} \cos h (2 σ l) - δ^{2}]}{κ^{2} {[κ^{2} + (δ / 2)}^{2} e^{4 σ l}]}

where

d = δ / κ \approx 2 π (λ_{p e a k} - λ_{p u m p}) / (κ {\bar{λ}}^{2})

.

The process of peak shifting is that more heat is generated as the pump wavelength approaches the peak due to the intensity enhancement, and the red shift of transmission peak occurs in pace. The transmission peak red-shift has a limit, determined by the pump power and corresponding to the case of λ_pump towards λ_peak, i.e. i.e. d = 0. The maximum of Δλ_peak can then be deduced to be,

λ_{p e a k}^{[\max]} - λ_{B 0} \propto α R_{H} P_{i n} \cos h (2 κ l)

At the point, the transmission peak red-shift stops, and the pump wavelength exceeds; then the intensity enhancement effect declines. It is an accelerated process; the peak goes back quickly to the original Bragg wavelength λ_B0.

When the pump wavelength scans from the longer wavelength side down to λ_B0, the heat generation occurs also. However, the shifts of pump wavelength and peak wavelength are in opposite direction, so that their difference diminishes very quickly, a jump up occurs at the point of λ_pump = λ_peak, and a small red shift of peak wavelength occurs, showing a bistable behavior. The simulations of characteristics are shown in Fig. 1, where the curve is calculated by using Eq. (10). It is seen that the red-scanning of pump pushes the transmission peak co-directionally, whereas the blue-scanning pulls the peak counter-directionally. Such a pull-push process is shown clearer in the experimental results below.

Fig. 1 Calculated curve of λ_peak v.s. λ_pump.

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3. Experiment method and results

3.1 Experiment setup and static measurement

The effect of light wavelength and intensity on PSFBG’s transmission spectrum is investigated experimentally by using a setup shown in Fig. 2. Two different sources are used as pump and probe to avoid unwanted coupling between them. The DFB-LD is used as a pump laser, which can be tuned by current and temperature. The probe is provided by the OSA (APEX Technologies, AP-2041B)’s affiliated tunable laser with linewidth of 500kHz. The PSFBG under test was written in a polarization maintaining fiber (PMF) with grating length of 20mm, and the phase shift was made at the middle by the mask for UV exposure. The rate of wavelength variation with temperature of the PSFBG is measured to be Δλ/ΔT = 13.3 pm/K with a low power source. Two transmission peaks at 1548.71nm and 1549.07nm are measured for the fast and slow axes, respectively; the FWHM linewidth of transmission line is ~100MHz, i.e. ~0.8pm. The pump laser is tuned around the peak of slow axis. Its photothermal effect on the fast axis light is measured by OSA with resolution of 0.04pm. The two polarizations are separated by the polarization beam splitter (PBS); and the polarization directions are selected by the angles of connection, as shown in the figure. The power of probe light is 0.1mW, which is low enough thought below the level for high intensity enhancement. Two isolators (ISO1 and ISO2) are used to avoid disturbances. The pump power is measured by the optical power meter (OPM) through a 50/50 coupler (OC).

Fig. 2 Schematic diagram of experiment setup.

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The transmission spectra were measured for pump light with varied power and wavelength. Figure 3(a) shows the variation of transmission peak with the pump wavelength increasing from blue side of Bragg wavelength for two different powers; and also with the pump wavelength decreasing from red side of Bragg wavelength. The measured results are agree well with the analysis. It is also shown that the maximal wavelength shift increases with the pump power; and the red shift starts earlier for the higher pump power. For example, for 11mW pump power the peak starts shifting at the pump wavelength ~10 pm below Bragg wavelength, quite far away from the transmission peak, which is with FWHM width of ~0.8pm. .

Fig. 3 (a) The variation of transmission peak with the pump wavelength up (the left axis) and down (the right axis); (b) Transmission spectra of PSFBG for different pump powers.

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Figure 3(b) shows transmission spectra measured near the wavelength shift maxima for different powers. It is also seen that the reflection band is shifted with the pump power, indicating variation of the averaged Bragg wavelength; and the transmission peak shifts larger than the band, showing its asymmetric position in the band. These phenomena are coincident with the theoretical analyses. The peak’s shift rate is read to be 2.0 pm/mW, and the shift rate of reflection band is read to be 0.58 pm/mW. The transmission peak moved from the middle point (50%) to the red side at 63%. It can be seen in Fig. 3(b) that the ratio of peak shift rate to reflection band shift is Δλ_peak /Δλ_B = 3.4.

3.2 Dynamic characteristics

From the mechanism analyzed above, PSFBG will show different characteristics when the pump laser is tuned in opposite direction. When λ_pump decreases from blue side toward λ_B0, λ_peak shifts towards red side from λ_B0 and the detuning decreases quickly. This behavior was observed in a dynamic experiment with the setup shown in Fig. 4. The wavelength of pump laser is now modulated in triangle waveforms; the transmitted power of PSFBG is detected by PD1. To remove the residue amplitude modulation of the current tuning, the modulated power is detected by PD2, and their ratio is displayed in the oscilloscope.

Fig. 4 Setup for measurement of PSFBG dynamic characteristics.

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Figure 5 shows the detected signals for different modulation frequencies: (a) 0.1Hz, (b) 1Hz, (c) 10Hz, (d) 1kHz; corresponding to wavelength tuning rate of 12pm/sec, 120pm/sec, 1.2nm/sec, and 12nm/sec. The pump power is P_in = 13mW for all the 4 curves. The abscissa scales are normalized to be one modulation period for comparison. It is seen that the detected signals show completely different waveforms in ascend slope and in descend slope; and the retardation behavior in the ascend slope is weakened as the tuning rate increases. Similar phenomena were presented in transmissions from a Fabry-Perot cavity composed of two FBGs, and was explained by the push-pull effect induced by heat aggregate in the cavity [15]. The observed phenomena for PSFBG here can be understood based on the analysis in section 2.4 of this paper.

Fig. 5 Transmitted signals with pump tuning rates of (a) 12pm/s; (b) 120pm/s; (c) 1.2nm/s; (d) 12nm/s(Abscissa normalized to one scanning period).

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3.3 Effect of the heat dissipation condition

From the analysis of thermal conduction in 2.2 of this paper, the temperature rising is dependent on the thermal dissipation condition of the fiber. The experiment verifies this property. The transmission spectra were measured by the same setup as in Fig. 2 for PSFBG placed on aluminum clippers and immersed in water with aluminum clippers, respectively. The maximal transmission peak shifts were measured by tuning the pump, as did for data of Fig. 3. Its variations with the input power are shown in Fig. 6. It is seen that the wavelength shift is proportional to the incident power. The shift rate for air surrounded FBG is read to be 2.3 pm/mW, while it is 0.8pm/mW for water surrounded FBG.

Fig. 6 Transmission peak wavelength shift vs. pump power in different heat dissipations. The straight line is fitting curve using Eq. (11).

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4. Discussions

From the above experimental results and the theoretical model, some conclusions may be obtained, and some points are left to be discussed.

The temperature rising can be evaluated from the model and the measure data. The transmission peak wavelength shift rate has been measured to be 2.3 pm/mW from the data of Fig. 6, which corresponds to temperature rising rate of 0.17K/mW, by using Δλ/ΔT = 13.3 pm/K measured with a low power source.

The fiber loss coefficient can then be estimated by the measured temperature rising from relation of $Δ T = R_{H} Q = R_{H} α P_{i n} H$ . The thermal resistance is related to the heat dissipation condition of the fiber. The model giving Eq. (5) is over-simplified for quantitative calculation. The viscosities of air and water are different; the contribution of their convection to heat dissipation is unknown yet. Besides, the PSFBG under test is packaged in clippers; it is hard to estimate their influence. However, when the PSFBG attached to the aluminum block is immersed into the water, it may be assumed that the temperature at fiber outer radius can be approximated as the room temperature. The dissipation of fiber cladding layer makes the main contribution to heat resistance. With r_cl / r_co = 12.5 and k = 0.014W/(cm·K), the minimum heat resistance $R_{H}^{[\min]} = \ln (r_{c l} / r_{c o}) / (2 π k)$ can be calculated to be ~29cm·K/W. By using the data of Fig. 6, the minimum heat resist for air surrounding fiber is estimated as ~83cm·K/W. With the intensity enhancement of 201 at the phase shift point, the upmost fiber loss is estimated to be 0.001 ${mm}^{- 1}$ . The estimation can be compared with the data reported previously. A relation between the loss and the coupling coefficient κ was investigated; it is read that κ of 0.3 mm-1 corresponds to α of 0.0005 mm-1. The estimated loss here agrees well in the order with that in [12]. On the other hand, fiber loss will cause the transmission at peak decreasing from unity. According to the model and data of [18], 0.001 mm-1 loss will cause transmission peak decrease of 2.52dB, which agrees also with the measured 2.84dB loss from the data of Fig. 3(b). It is worthy of noticing that the fiber loss in FBG depends on the techniques in its fabrication. Different hydrogen-loading process, different UV exposure condition, and different annealing process may affect the loss greatly.

The thermal relaxation time τ of single mode silica fiber may be estimated from the waveforms given in Fig. 5. It is seen that similar waveforms in ascend slope and in descend slope are observed in the tuning rate of 12pm/sec. The relaxation time may be estimated in the order of 20μs, much shorter than that given by [15] for a FBG F-P cavity. Generally, the thermal relaxation is related to the heat capacity C_H and the heat resist R_H, τ~C_H·R_H. Since the fiber core is very thin, a short relaxation time is reasonable.

It is noticed that some factors are omitted in the theoretical model proposed in this paper just for simplicity. The intensity enhancement given in Eq. (1) is for π-phase shift FBG. It has to be revised for phase shift deviated from π; this revision may change the phenomena in some degree. The chirping induced by the intensity enhancement is another problem to be investigated. It may be necessary to take the heat conduction in axial direction into consideration for a better model. Nevertheless, the observation presented in this paper can be explained satisfactorily by the first order model.

5. Conclusion

The effect of intensity enhancement of PSFBG on its transmission spectrum is presented. A systematic theoretical model is proposed. It is found experimentally that the transmission peak is red-shifted when the wavelength and power of pump increase, and the peak shifts more than the reflection band of FBG. The push-pull effect and the asymmetric behaviors in red/blue-forward tuning are observed. The peak shift rate is measured with different heat dissipation conditions. The fiber loss of ~0.001mm⁻¹ is estimated from the measured data.

The push-pull effect and the asymmetry observed in this paper is an important feature of PSFBG. Attentions have to be paid to such characteristics when PSFBG is used as a narrow band filter or as a frequency discriminator for applications, such as laser mode selection, frequency stabilization and in sensors. It will cause different temporal responses for positive and negative error signals, and thus degrade the performances of e.g. frequency stabilization. Further researches are undertaken in our group.

6 Appendix: Detailed analysis of the intensity enhancement factor

Consider a PSFBG who has two equal, uniform, periodic regions separated by a phase shift of π at its center z=0. When calculating the intensity enhancement factor, the length of phase shift region is ignored. Thus the PSFBG has an index distribution as follows:

n = {\begin{matrix} n_{0} + n_{1} \cos (2 β_{B} z - \frac{π}{2}), - l \leq z < 0 \\ n_{0} + n_{1} \cos (2 β_{B} z + \frac{π}{2}), 0 \leq z < l \end{matrix}

The electric field in each region can be solved by a set of coupled-mode equations [17]. And it can be written as the sum of forward- and backward-traveling waves:

E_{i} (z) = E_{+}^{i} (z) e x p (j β z) + E_{-}^{i} (z) e x p (- j β z)

where the subscript i = 1,2 means region

z \in (- l, 0)

and

z \in (0, l)

respectively. The coupled-mode equations that correspond to each region are given by

\begin{matrix} \frac{d E_{+}^{i}}{d z} = - j κ E_{-}^{i} e^{j [2 δ z - {(- 1)}^{i} \frac{π}{2}]} \\ \frac{d E_{-}^{i}}{d z} = j κ E_{+}^{i} e^{- j [2 δ z - {(- 1)}^{i} \frac{π}{2}]} \end{matrix}

The general solutions are obtained to be

\begin{array}{l} E_{+}^{i} (z) = [A_{_{1}}^{i} e^{σ z} + A_{_{2}}^{i} e^{- σ z}] e^{j δ z} \\ E_{-}^{i} (z) = [B_{_{1}}^{i} e^{σ z} + B_{_{2}}^{i} e^{- σ z}] e^{-j δ z} \end{array}

Considering a unit wave propagates in + z direction and inputs to the PSFBG, the constants should satisfy the boundary conditions at z = -l and z = l:

\begin{array}{l} A_{1}^{1} e^{- σ l} + A_{2}^{1} e^{σ l} = E_{0} \\ B_{1}^{2} e^{σ l} + B_{2}^{2} e^{- σ l} = 0 \end{array}

and at z = 0, it should satisfy the electric field continuity:

\begin{matrix} A_{1}^{1} + A_{2}^{1} = A_{1}^{2} + A_{2}^{2} \\ B_{1}^{1} + B_{2}^{1} = B_{1}^{2} + B_{2}^{2} \end{matrix}

Then the field in each region can be deduced to be

\begin{array}{l} E_{+}^{1} = E_{0} \frac{κ^{2} \cos h [σ (z + l)] - δ^{2} \cos h [σ (z - l)] - j σ δ \sin h [σ (z - l)]}{D_{M}} \\ E_{-}^{1} = E_{0} \frac{σ κ s h [σ (z + l)] + j δ κ c h [σ (z + l)] - j δ κ c h [σ (z - l)]}{D_{M}} \\ E_{+}^{2} = E_{0} \frac{σ^{2} c h [σ (z - l)] - j σ δ s h [σ (z - l)]}{D_{M}} \\ E_{-}^{2} = E_{0} \frac{- σ κ s h [σ (z - l)]}{D_{M}} \end{array}

where

D_{M} = κ^{2} - δ^{2} \cosh (2 σ l) + j δ σ \sinh (2 σ l)

. Hence the field enhancement is given by

H (z, δ) = \frac{{| E_{_{1}}^{i} |}^{2} + {| E_{_{2}}^{i} |}^{2}}{{| E_{0} |}^{2}} = {\begin{matrix} \begin{array}{l} {κ^{4} ch[2 σ (z + l)] + κ^{2} δ^{2} c h [2 σ (z - l)] - \\ 4 κ^{2} δ^{2} c h [σ (z + l)] c h [σ (z - l)] + δ^{2} κ^{2} + δ^{4}} / {| D_{M} |}^{2}, - l \leq z < 0 \end{array} \\ \frac{σ^{2} κ^{2} c h [2 σ (z - l)] - σ^{2} δ^{2}}{{| D_{M} |}^{2}}, 0 \leq z < l \end{matrix}

and in the case of δ << κ and 2κl >>1, it can be approximated as

H (z, δ) \approx \frac{σ^{2} κ^{2} \cos h [2 σ (l - | z |)]}{κ^{2} [κ^{2} + δ^{2} \sin h^{2} (2 σ l)]} \approx \frac{σ^{2} κ^{2} \cos h [2 σ (l - | z |)]}{κ^{2} {[κ^{2} + (δ / 2)}^{2} e^{4 σ l}]}

Funding

National Natural Science Foundation of China (NSFC) (61405218, 61535014, 61475165); Shanghai Natural Science Foundation (14ZR1445100); Excellent Academic Leaders of Shanghai (15XD1524500); and Youth Innovation Promotion Association of Chinese Academy of Sciences (CAS).

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Photothermal effects in phase shifted FBG with varied light wavelength and intensity

Abstract

1.Introduction

2. Theoretical model and analyses

2.1 Intensity enhancement

2.2 Nonuniform index distribution induced by photothermal effect

2.3 The effect of Δn on characteristics of PSFBG

2.4. Transmission peak shift with light wavelength and intensity

3. Experiment method and results

3.1 Experiment setup and static measurement

3.2 Dynamic characteristics

3.3 Effect of the heat dissipation condition

4. Discussions

5. Conclusion

6 Appendix: Detailed analysis of the intensity enhancement factor

Funding

References and links

Cited By

Figures (6)

Equations (20)

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