**1. Introduction**

24 Will-be-set-by-IN-TECH

224 Photonic Crystals – Introduction, Applications and Theory

Tada, K. & Karasawa, N. (2010). Broadband coherent anti-Stokes Raman scattering

Tada, K. & Karasawa, N. (2011). Single-beam coherent anti-Stokes Raman scattering

Takabatake, Y., Tada, K. & Karasawa, N. (2010). Optical coherence tomography using a soliton

Zolla, F., Renversez, G., Nicolet, A., Kuhlmey, B., Guenneau, S. & Felbacq, D. (2005). *Foundations of Photonic Crystal Fibres*, Imperial College Press, London.

*Electro-Optics*, OSA Technical Digest, paper JTuD72.

fiber, *Appl. Phys. Express* Vol. 4 pp. 092701-1–092701-3.

*Technologies*, PWC Publishing, Chitose, Japan, pp. 116–119.

spectroscopy using a quasi-supercontinuum light source, *in Conference on Lasers and*

spectroscopy using both pump and soliton Stokes pulses from a photonic crystal

pulse train, *in* Kawabe, Y. & Kawase, M. eds., *Polymer Photonics, and Novel Optical*

The photonic crystal fiber (PCF) is a special class of components incorporating photonic crystals with a two-dimensional (2D) periodic variation in the plane perpendicular to the fiber axis and an invariant structure along it [1-3]. Typically these bers incorporate a number of air holes that form a so-called photonic crystal cladding and run along the length of the bers, and the shape, size, and distribution of the holes can be designed to achieve various novel wave-guiding properties that may not be achieved readily in conventional bers [2-19], so that they have attracted significant attention in recent ten years.

A long-period fiber grating (LPG) is a one dimension (1D) periodic structure, and is formed by introducing periodic modulation of the refractive index along an optical ber. Since its period is about 100 to several hundreds μm and longer than that of fiber Bragg grating (FBG), LPG resonantly couples light from the fundamental core mode to some copropagating cladding modes and leads to dips in the transmission spectrum. LPGs have been widely used in optical ber communications and sensors. Examples of LPG-based devices include all-ber band-rejection lters [20, 21], gain atteners in erbium-doped ber ampliers [22], and sensors for strain, temperature, and external refractive index measurement [23-25]. When a LPG is formed on a PCF, a 2-D periodic structure is combined with a 1-D periodic structure. LPGs based on PCFs (PCF-LPGs) have been fabricated recently [26-36] and shown many unique properties compared with a conventional LPG (1-D periodic structure) [27, 31-34, 37, 38], which provide wide and novel applications [38-47].

In this chapter, we will first introduce the basic operation principle of LPGs, secondly, will demonstrate in detail the strain and temperature characteristics of a LPG based on an endlessly-single-mode (ESM) solid silica core PCF theoretically. To account for the effect of dispersive characteristics of the PCF, we identify a dispersion factor , which offers a deeper understanding into the behavior of LPGs in PCF. Following, we will move on to the fabrication of a PCF-LPG by using a CO2 laser and demonstrate the experimental observations on the strain and temperature characteristics, which agree with the theoretical predictions very well. Finally, we will demonstrate their applications in optical sensors, including a temperature-insensitive strain sensor, and demodulation technologies for fiber Bragg grating and fiber loop mirror sensors.

Long-Period Gratings Based on Photonics Crystal Fibers and Their Applications 227

*d*

**3. Theoretical properties of an LPG based on PCF [38]** 

where *nn n e co cl* and *<sup>g</sup> <sup>g</sup>*

an LPG based on the PCF.

strain is applied.

**3.1 Properties of the ESM-PCF** 

PCF, showing the quarter used in calculation

*co cl <sup>e</sup> <sup>g</sup> <sup>e</sup>*

differential group index between the core mode and the cladding mode. As will be discussed in Section 3, plays a signicant role on the strain and temperature sensitivity of

In this section, we investigate in detail the strain and temperature characteristics of an LPG based on an endlessly single-mode (ESM) solid silica core PCF theoretically. To account for the effect of dispersive characteristics of the PCF, we identify a dispersion factor, which offers a deeper understanding into the behavior of PCF-LPGs. Theoretical results show that is always negative, and this causes blue-shifting of the resonant wavelength when an axial

The PCF used in the work is an endlessly single-mode PCF fabricated by Crystal Fiber A/S. The fiber has a standard triangular air/silica cladding structure, as shown in Fig. 1 (a). The mode field diameter is ~6.4 μm, the center-to-center distance between the air holes (L) is ~7.78 μm, and the diameter of the air holes is ~3.55 μm. The diameter of the entire holey region is ~ 60 μm, and the outer cladding diameter of the PCF is 125 μm. A full-vector finiteelement method (FEM) was used to calculate the effective index of modes of PCF. Because of the symmetric nature of the PCF, only a quarter of the cross-section as shown in Fig.1 (b) is used during calculation. A perfect electric or perfect magnetic conductor (PEC or PMC) was

applied at boundaries [7]. The refractive index of pure silica was taken as 1.444.

Fig. 1. (a) Micrograph of the PCF used in the experiment; (b) Schematic cross-section of a

*d n n n n d n <sup>n</sup> <sup>n</sup>*

*d*

*e e*

*g co cl nnn* are respectively the differential effective index and

(6)

## **2. Basic operation principle of LPGs [23, 38, 48]**

A LPG is formed usually by a periodic modulation of the refractive index in a fiber core, which allows coupling from the fundamental core mode to some resonant cladding modes and leads to some dips in the transmission spectrum at wavelengths that satisfy the resonant condition. The phase matching condition of a LPG can be expressed as [23]:

$$
\lambda = (n\_{co} - n\_{cl})\Lambda \tag{1}
$$

where λ is the resonant wavelength, Λ is the index modulation period of the LPG, and *nco* and *ncl* are the effective indices of the fundamental core mode, and the forward-propagating cladding mode, respectively.

When an axial strain is applied on the LPG, the resonant wavelength of the LPG will shift because the Λ of the LPG will increase with stretching axially and at the same time the effective refractive index of both core and cladding modes will decrease due to the photoelastic effect of the fiber [31]. Meanwhile, if the ambient temperature changes, the wavelength of the LPG may also be changed by linear expansion or contraction and the thermo-optic effect. From equation (1), the sensitivity of the LPG to strain or temperature is a function of the differential effective index between the core and cladding modes (or the differential propagation constant). Thus, from equation (1), the strain and temperature sensitivity can be written as [48]:

$$\frac{d\lambda}{d\varepsilon} = \lambda \cdot \gamma \cdot \left(1 + \frac{\eta\_{co}n\_{co} - \eta\_{cl}n\_{cl}}{n\_{co} - n\_{cl}}\right) \tag{2}$$

$$\frac{d\mathcal{N}}{dT} = \lambda \cdot \gamma \cdot \left(\alpha + \frac{\mathsf{\dot{\xi}}\_{\rm co} \mathsf{n}\_{\rm co} - \mathsf{\dot{\xi}}\_{\rm cl} \mathsf{n}\_{\rm cl}}{n\_{\rm co} - n\_{\rm cl}}\right) \tag{3}$$

where ε is the axial strain, *T* is the ambient temperature, *co* and *cl* are strain-optic coefficients of the core and cladding, ξ *co* and ξ *cl* are the thermo-optic coefficient of the core and cladding, respectively, and α is the linear expansion coefficient. and ξ are defined as [23]:

$$
\eta = \frac{1}{n} \frac{dn}{d\varepsilon} \tag{4}
$$

$$
\xi = \frac{1}{n} \frac{dn}{dT} \tag{5}
$$

Different materials have different and ξ. and ξ may also have some difference due to the different effective index of waveguides made by the same material [49].

Since the effective index (or propagation constant) both of the fundamental mode in the fiber core and cladding modes in the fiber cladding will be affected by the waveguide change which is caused by the applied axial strain on the LPG, the dispersion factor is used to describe the effect of waveguide dispersion and is expressed as [48]:

$$\gamma = \frac{d\lambda\_{\text{el}}}{n\_{\text{co}} - n\_{\text{cl}}} = \frac{\Delta n\_{\text{e}}}{\Delta n\_{\text{e}} - \lambda} \frac{d\Delta n\_{\text{e}}}{d\lambda} = \frac{\Delta n\_{\text{e}}}{\Delta n\_{\text{g}}} \tag{6}$$

where *nn n e co cl* and *<sup>g</sup> <sup>g</sup> g co cl nnn* are respectively the differential effective index and differential group index between the core mode and the cladding mode. As will be discussed in Section 3, plays a signicant role on the strain and temperature sensitivity of an LPG based on the PCF.

### **3. Theoretical properties of an LPG based on PCF [38]**

In this section, we investigate in detail the strain and temperature characteristics of an LPG based on an endlessly single-mode (ESM) solid silica core PCF theoretically. To account for the effect of dispersive characteristics of the PCF, we identify a dispersion factor, which offers a deeper understanding into the behavior of PCF-LPGs. Theoretical results show that is always negative, and this causes blue-shifting of the resonant wavelength when an axial strain is applied.

#### **3.1 Properties of the ESM-PCF**

226 Photonic Crystals – Introduction, Applications and Theory

A LPG is formed usually by a periodic modulation of the refractive index in a fiber core, which allows coupling from the fundamental core mode to some resonant cladding modes and leads to some dips in the transmission spectrum at wavelengths that satisfy the

where λ is the resonant wavelength, Λ is the index modulation period of the LPG, and *nco* and *ncl* are the effective indices of the fundamental core mode, and the forward-propagating

When an axial strain is applied on the LPG, the resonant wavelength of the LPG will shift because the Λ of the LPG will increase with stretching axially and at the same time the effective refractive index of both core and cladding modes will decrease due to the photoelastic effect of the fiber [31]. Meanwhile, if the ambient temperature changes, the wavelength of the LPG may also be changed by linear expansion or contraction and the thermo-optic effect. From equation (1), the sensitivity of the LPG to strain or temperature is a function of the differential effective index between the core and cladding modes (or the differential propagation constant). Thus, from equation (1), the strain and temperature

> (1 ) *co co cl cl co cl*

> ( ) *co co cl cl co cl*

*d n n d n n* 

*d n n dT n n*

where ε is the axial strain, *T* is the ambient temperature, *co* and *cl* are strain-optic coefficients of the core and cladding, ξ *co* and ξ *cl* are the thermo-optic coefficient of the core and cladding, respectively, and α is the linear expansion coefficient. and ξ are

> 1 *dn n d*

> > 1 *dn*

Different materials have different and ξ. and ξ may also have some difference due to

Since the effective index (or propagation constant) both of the fundamental mode in the fiber core and cladding modes in the fiber cladding will be affected by the waveguide change which is caused by the applied axial strain on the LPG, the dispersion factor is

the different effective index of waveguides made by the same material [49].

used to describe the effect of waveguide dispersion and is expressed as [48]:

( ) *n n co cl* (1)

(2)

(3)

(4)

*n dT* (5)

resonant condition. The phase matching condition of a LPG can be expressed as [23]:

**2. Basic operation principle of LPGs [23, 38, 48]** 

cladding mode, respectively.

sensitivity can be written as [48]:

defined as [23]:

The PCF used in the work is an endlessly single-mode PCF fabricated by Crystal Fiber A/S. The fiber has a standard triangular air/silica cladding structure, as shown in Fig. 1 (a). The mode field diameter is ~6.4 μm, the center-to-center distance between the air holes (L) is ~7.78 μm, and the diameter of the air holes is ~3.55 μm. The diameter of the entire holey region is ~ 60 μm, and the outer cladding diameter of the PCF is 125 μm. A full-vector finiteelement method (FEM) was used to calculate the effective index of modes of PCF. Because of the symmetric nature of the PCF, only a quarter of the cross-section as shown in Fig.1 (b) is used during calculation. A perfect electric or perfect magnetic conductor (PEC or PMC) was applied at boundaries [7]. The refractive index of pure silica was taken as 1.444.

Fig. 1. (a) Micrograph of the PCF used in the experiment; (b) Schematic cross-section of a PCF, showing the quarter used in calculation

Long-Period Gratings Based on Photonics Crystal Fibers and Their Applications 229

Fig. 4 shows the calculated resonance wavelength as a function of the period of PCF-LPG. It is clear that the resonance wavelength of PCF-LPG decreases with increasing LPG period, which is consistent with other experimental observations [27, 33, 34]. This is in contrast to LPGs written in conventional SMFs and is because of the highly dispersive property of the cladding mode due to the existence of the air-holes. In other words, ( ) *n n co cl* as shown in

> 400 450 500 550 600 **Period of LPG (μm )**

 is a special factor to describe the effect of waveguide dispersion, and may be positive or negative. Because *ne* is always positive, the sign of is determined by *ng* . When *ne* equals to *ng* , the factor is 1. This means dispersive properties of the core and the cladding mode is similar and this is the case normally for SMFs. When the group indices of the core mode is less than that of the cladding mode, will be negative. This has been observed in the case of the coupling from core mode to higher-order cladding modes in B-Ge co-doped fiber [48]. In Fig. 5, we show the relationship of with the period of a LPG based on the theory of ESM-PCF, and is in the range of -1.15~ -1.35 for LPG period of

> 400 450 500 550 600 **Period of LPG (μm)**

Fig. 5. Dispersion factor at the resonance wavelengths vs. grating period of the PCF-LPG

**3.2 Properties of an LPG based on the PCF in theory** 

equation (1), varies significantly with wavelength.

**Wavelength (**

from 420 to 570μm.

**μm)**

1.2 1.3 1.4 1.5 1.6 1.7 1.8




**Dispersion factor**


0

Fig. 4. Calculated resonance wavelength as a function of grating period

Fig. 2. Calculated intensity distribution of the PCF with L=7.78 μm and d=3.55 μm. (a) The fundamental core mode; (b) The cladding mode

Fig. 2 (a) and (b) shows the intensity distribution of the core and the cladding mode, which are considered as the two coupling modes in our PCF-LPGs. Fig. 3 shows the effective indices of the fundamental and cladding modes as functions of wavelength in the ESM-PCF. The group indices of these two modes, which were calculated by using *<sup>e</sup> g e dn n n d* , are also shown in Fig. 3. The curves of ng are not so smooth because of the limited data available for the calculation of *<sup>e</sup> dn d* but the trend is clear. The curve ng-cl shows the highly dispersive characteristics of the cladding mode. For any wavelength in the range of 1.2 ~ 1.8 μm, the group index of the cladding mode is higher than that of the core mode, which is in contrast to a conventional SMF.

Fig. 3. Calculated dispersion curves for core and cladding modes as shown in Fig. 2.

#### **3.2 Properties of an LPG based on the PCF in theory**

228 Photonic Crystals – Introduction, Applications and Theory

Fig. 2. Calculated intensity distribution of the PCF with L=7.78 μm and d=3.55 μm. (a) The

Fig. 2 (a) and (b) shows the intensity distribution of the core and the cladding mode, which are considered as the two coupling modes in our PCF-LPGs. Fig. 3 shows the effective indices of the fundamental and cladding modes as functions of wavelength in the ESM-PCF.

also shown in Fig. 3. The curves of ng are not so smooth because of the limited data available

characteristics of the cladding mode. For any wavelength in the range of 1.2 ~ 1.8 μm, the group index of the cladding mode is higher than that of the core mode, which is in contrast

but the trend is clear. The curve ng-cl shows the highly dispersive

*dn*

*d*

, are

*n n*

The group indices of these two modes, which were calculated by using *<sup>e</sup> g e*

Fig. 3. Calculated dispersion curves for core and cladding modes as shown in Fig. 2.

fundamental core mode; (b) The cladding mode

*d*

for the calculation of *<sup>e</sup> dn*

to a conventional SMF.

Fig. 4 shows the calculated resonance wavelength as a function of the period of PCF-LPG. It is clear that the resonance wavelength of PCF-LPG decreases with increasing LPG period, which is consistent with other experimental observations [27, 33, 34]. This is in contrast to LPGs written in conventional SMFs and is because of the highly dispersive property of the cladding mode due to the existence of the air-holes. In other words, ( ) *n n co cl* as shown in equation (1), varies significantly with wavelength.

Fig. 4. Calculated resonance wavelength as a function of grating period

 is a special factor to describe the effect of waveguide dispersion, and may be positive or negative. Because *ne* is always positive, the sign of is determined by *ng* . When *ne* equals to *ng* , the factor is 1. This means dispersive properties of the core and the cladding mode is similar and this is the case normally for SMFs. When the group indices of the core mode is less than that of the cladding mode, will be negative. This has been observed in the case of the coupling from core mode to higher-order cladding modes in B-Ge co-doped fiber [48]. In Fig. 5, we show the relationship of with the period of a LPG based on the theory of ESM-PCF, and is in the range of -1.15~ -1.35 for LPG period of from 420 to 570μm.

Fig. 5. Dispersion factor at the resonance wavelengths vs. grating period of the PCF-LPG

Long-Period Gratings Based on Photonics Crystal Fibers and Their Applications 231

Furthermore, the dependence of temperature sensitivity on the grating period is approximately linear for =1 while it is non-monotonic for the other case. Similar to the discussion for the strain coefficient, for the ESM-PCF, since the effective index *nc*o is larger than *n*cl, from eq. (5), we expect that *co* is slightly smaller than *cl* , which has also been

Fig. 7. Theoretical strain sensitivity at resonance wavelength vs. LPG period for various

400 450 500 550 600 **Period of LPG (μm)**





*cl* =




Fig. 8. Theoretical temperature sensitivity at resonance wavelength vs. LPG period for

7.85×10-6

400 450 500 550 600 **Period of LPG (μm)**

7.827×10-6

7.8×10-6

ξcl =7.75×10-6

verified by the experiment in Section 4.

values of *cl* and with taken from Fig. 5.



0

0.02

**Temperature sensitivity (nm/ oC)**

0.04

0.06




0

**Strain sensitivity (nm/1000με)**

2

4

6

various values of ξcl and = 1.

The strain sensitivity *dλ/dε* of a LPG based on the ESM-PCF is determined by four parameters: the elasto-optic coefficients of the core and cladding materials, waveguide properties ( ), the period of the LPG, and the mode order. Now, we choose the same coupling modes, and focus on the effect of the first three parameters on the strain sensitivity of a PCF-LPG. Fig. 6 shows the calculated strain sensitivity as a function of LPG period with different *cl* when we assume =1. In the calculation, *co* is assumed to be constant at a value of -0.22 for the pure silica core. For LPGs with period ranging from 400 to 600μm, the strain sensitivity is positive and relatively independent of the grating period when *cl* is larger than 0.22. The strain sensitivity becomes negative and decreases with grating period when *cl* is smaller than 0.218. On the other hand, when the value of is taken the value as shown in Fig. 5, the strain sensitivity as a function of LPG period is as shown in Fig. 7. The strain sensitivity is negative when *cl* is larger than 0.22. This is the opposite of what is shown in Fig. 6. In ref. [49], A. Bertholds et. al. showed that the strain-optic coefficients of a bulk silica and a silica fiber are different. It's believed that, owing to the different geometry of solid core and micro-structured air-silica cladding of the ESM-PCF, *co* will be slightly different from *cl* .

Fig. 6. Theoretical strain sensitivity at resonance wavelength vs. LPG period for various values of *cl* and = 1.

Similarly, the temperature sensitivity *dλ/dT* of a LPG is determined by the thermo-optic coefficients of the core and cladding materials, waveguide properties ( ), period of LPG, and the mode order. We calculated the temperature sensitivity as a function of LPG period by assuming that the thermo-optic coefficient of the pure silica core is ξco =7.8×10-6 / oC and thermal expansion coefficient is α = 4.1×10-7/ oC. Figs. 8 and 9 show respectively the results for the cases of =1 and taking from Fig.5. The temperature characteristics are quite different for the two cases. With ξcl less than ξco =7.8×10-6, the LPG has positive temperature sensitivity for =1 but negative temperature sensitivity for the case of taking from Fig.5.

The strain sensitivity *dλ/dε* of a LPG based on the ESM-PCF is determined by four parameters: the elasto-optic coefficients of the core and cladding materials, waveguide properties ( ), the period of the LPG, and the mode order. Now, we choose the same coupling modes, and focus on the effect of the first three parameters on the strain sensitivity of a PCF-LPG. Fig. 6 shows the calculated strain sensitivity as a function of LPG period with different *cl* when we assume =1. In the calculation, *co* is assumed to be constant at a value of -0.22 for the pure silica core. For LPGs with period ranging from 400 to 600μm, the strain sensitivity is positive and relatively independent of the grating period when *cl* is larger than 0.22. The strain sensitivity becomes negative and decreases with grating period when *cl* is smaller than 0.218. On the other hand, when the value of is taken the value as shown in Fig. 5, the strain sensitivity as a function of LPG period is as shown in Fig. 7. The strain sensitivity is negative when *cl* is larger than 0.22. This is the opposite of what is shown in Fig. 6. In ref. [49], A. Bertholds et. al. showed that the strain-optic coefficients of a bulk silica and a silica fiber are different. It's believed that, owing to the different geometry of solid core and micro-structured air-silica cladding of the ESM-PCF, *co* will be slightly

Fig. 6. Theoretical strain sensitivity at resonance wavelength vs. LPG period for various


**Strain sensitivity (nm/1000με)**

Similarly, the temperature sensitivity *dλ/dT* of a LPG is determined by the thermo-optic coefficients of the core and cladding materials, waveguide properties ( ), period of LPG, and the mode order. We calculated the temperature sensitivity as a function of LPG period by assuming that the thermo-optic coefficient of the pure silica core is ξco =7.8×10-6 / oC and thermal expansion coefficient is α = 4.1×10-7/ oC. Figs. 8 and 9 show respectively the results for the cases of =1 and taking from Fig.5. The temperature characteristics are quite different for the two cases. With ξcl less than ξco =7.8×10-6, the LPG has positive temperature sensitivity for =1 but negative temperature sensitivity for the case of taking from Fig.5.

400 450 500 550 600 **Period of LPG (μm)**








*cl* =

different from *cl* .

values of *cl* and = 1.

Furthermore, the dependence of temperature sensitivity on the grating period is approximately linear for =1 while it is non-monotonic for the other case. Similar to the discussion for the strain coefficient, for the ESM-PCF, since the effective index *nc*o is larger than *n*cl, from eq. (5), we expect that *co* is slightly smaller than *cl* , which has also been verified by the experiment in Section 4.

Fig. 7. Theoretical strain sensitivity at resonance wavelength vs. LPG period for various values of *cl* and with taken from Fig. 5.

Fig. 8. Theoretical temperature sensitivity at resonance wavelength vs. LPG period for various values of ξcl and = 1.

Long-Period Gratings Based on Photonics Crystal Fibers and Their Applications 233

Fig. 10 shows the experimental setup of the PCF-LPG fabrication. The CO2 laser operates at a frequency of 10 kHz and has a maximum power of 10 W. The laser power is controlled by the width of the laser pulses. In the experiment, the pulse-width of the CO2 laser was chosen to be 3.8 μs. The laser beam was focused to a spot with a diameter of ~60 μm and scanned across the ESM-PCF transversely and longitudinally along the fiber by use of a two-dimensional optical scanner attached to the laser head. The scanning step of the focused beam was 1 μm and the delay time of each step was 350 μs. The LPG inscribed has a period of about 467 μm and a period number of 40. The process of the CO2 laser scanning is repeated 9 times, which results in a LPG with a deep transmission dip and no observable deformity in the fiber structure. The spectrum measurements were performed using a broadband light source (a light-emitting diode, LED, with the wavelength range of 1200 ~1700 nm) in combination with an optical

Fig. 11 shows the growing process of a PCF-LPG as a function of the number of scanning procedures. The resonant wavelengths of the PCF-LPG are about 1552.45 nm and 1363.3 nm which are due to coupling of the fundamental core mode to two different cladding modes. The dip at the wavelength 1552.45 nm is nearly 20 dB. The insertion loss of the LPG is about 1.5 dB. The resonance at 1552.45 nm is due to coupling of the core mode to the cladding mode shown in Fig. 2 (b) and is in good agreement with the theoretical result (1552.45 nm

Fig. 12 shows the experimental setup for measuring the characteristics of the PCF-LPG. The spectrum measurements were performed by using a broadband LED and an OSA with a

spectrum analyzer (OSA, ADVANTEST Q8384) with a resolution of 0.5 nm.

CO2 laser

2-D scanning

light source OSA

Fig. 10. Schematic of the PCF-LPG fabrication setup

Computer

Broadband

X

Y

resonance wavelength corresponds 467.2 μm LPG period) in Fig. 4.

**4.2 Properties of a LPG based on PCF in experiment** 

Fig. 9. Theoretical temperature sensitivity at resonance wavelength vs. LPG period for various values of ξcl and with taken from Fig. 5.
