**4.1 The model of an effective planar waveguide**

Taking advantage of the already computed PBG diagrams of the photonic crystal membranes with one and two waveguide channels (see Section 3), the phase constant of the supermodes may be easily determined. That knowledge is essential to build an equivalent effective waveguide model, which enables an approximate analytical representation of a field distribution of the guided modes in passive structures (Lesniewska-Matys, 2011). As it can be seen in Fig.17, in the proposed model, a photonic crystal waveguide is replaced with a two dimensional planar one with the same membrane's thickness but the channel's width adjusted so as to obtain the same phase constant.

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 111

*E x y az E x y*

(,) (,) *N p*

1

where: *Et*

structure.

consecutive waveguide channels.

defined as follows:

The matrices *S* and *M* are defined as follows:

, *Ht*

components, respectively.

, ( *Ez*

, *Hz*

*z pz p*

> 1 (,) (,) *N*

Fig. 18. A top view of a dual-channel defect waveguide processed in a membrane with a photonic crystal square lattice. Shaded regions indicate an equivalent effective waveguide

*<sup>d</sup> C a z iSa z*

According to (Kogelnik & Shank, 1972; Schmidt et al., 1974; Chen & Wang, 1984), the amplitudes of the modes guided in the coupled waveguides satisfy the following conditions:

*dz* or *<sup>d</sup> a z iMa z*

where an *N*-element vector *a z*( ) consists of *ap(z)* amplitudes of the modes propagating in

where *C* is an *N*x*N* square matrix, where each element of the overlapping integrals *Cpq* is

*pq qp*

*C C*

2 *pq qp*

*C C*

*dz* (7)

*S CB K* or *M* 1 1 *CS BCK* (8)

(9)

*z pz p H xy a z H xy* 

*p*

) denote transverse (longitudinal) electric and magnetic field

*p*

(5)

(6)

Refractive indices in the planar equivalent structure are chosen in the following way: the refractive index of a waveguide core (*n1* in Fig.17b) is the same as in the photonic crystal structure, while *n4* and *n5* are equal to the value of the material filling the holes in the photonic crystal structure (air in the examples considered in this Chapter).

Fig. 17. A perspective view of a) a single-channel in a square-PhC membrane and b) an effective planar waveguide equivalent.

Next, to evaluate the field distribution in the already defined effective planar waveguide, a method proposed in (Marcuse, 1974) is applied. Those analytically derived waveguide modes are used, afterwards, to describe the operation of laser modes in such effective planar waveguides. Subsequently, the field distribution in a multichannel structure with the propagating supermodes is obtained using a non-orthogonal (strongly) coupled mode theory as proposed in (Chuang, 1987a; Chuang, 1987b; Chuang, 1987c). The overlapping integrals between the modes and their coupling coefficients in planar dual-channel waveguides are derived using formulae describing an EM field distribution in the singlechannel planar structure. Eventually, the obtained field distributions for the single- and dual-channel structures may be used to estimate approximate operation conditions of laser structures above a generation threshold. (see Section 4.3).

#### **4.2 Mode propagation in an effective N-waveguide structure**

Fig.18 shows a top view of a dual-channel defect waveguide, where shaded regions indicate the equivalent effective planar structure. The channels' widths in the equivalent model are adjusted so as to provide the same phase constant of the fundamental supermode as in the corresponding PhC channels.

A total EM field distribution in the coupled planar waveguides may be represented as a weighted sum of the modes propagating in each of the waveguides separately:

$$\vec{E}\_l(\mathbf{x}, \mathbf{y}) = \sum\_{p=1}^{N} \vec{a}\_p(z) \,\vec{E}\_l^{(p)}(\mathbf{x}, \mathbf{y}) \tag{3}$$

$$\vec{H}\_l(\mathbf{x}, \mathbf{y}) = \sum\_{p=1}^{N} \vec{a}\_p(z) \, \vec{H}\_l^{(p)}(\mathbf{x}, \mathbf{y}) \tag{4}$$

Refractive indices in the planar equivalent structure are chosen in the following way: the refractive index of a waveguide core (*n1* in Fig.17b) is the same as in the photonic crystal structure, while *n4* and *n5* are equal to the value of the material filling the holes in the

Fig. 17. A perspective view of a) a single-channel in a square-PhC membrane and b) an

Next, to evaluate the field distribution in the already defined effective planar waveguide, a method proposed in (Marcuse, 1974) is applied. Those analytically derived waveguide modes are used, afterwards, to describe the operation of laser modes in such effective planar waveguides. Subsequently, the field distribution in a multichannel structure with the propagating supermodes is obtained using a non-orthogonal (strongly) coupled mode theory as proposed in (Chuang, 1987a; Chuang, 1987b; Chuang, 1987c). The overlapping integrals between the modes and their coupling coefficients in planar dual-channel waveguides are derived using formulae describing an EM field distribution in the singlechannel planar structure. Eventually, the obtained field distributions for the single- and dual-channel structures may be used to estimate approximate operation conditions of laser

Fig.18 shows a top view of a dual-channel defect waveguide, where shaded regions indicate the equivalent effective planar structure. The channels' widths in the equivalent model are adjusted so as to provide the same phase constant of the fundamental supermode as in the

A total EM field distribution in the coupled planar waveguides may be represented as a

1 (,) , *N*

1 (,) (,) *N*

*t p t p H x y a zH x y* 

*t p t p E x y a zE x y* 

*p*

(3)

(4)

*p*

weighted sum of the modes propagating in each of the waveguides separately:

effective planar waveguide equivalent.

corresponding PhC channels.

structures above a generation threshold. (see Section 4.3).

**4.2 Mode propagation in an effective N-waveguide structure**

photonic crystal structure (air in the examples considered in this Chapter).

$$\vec{E}\_z(\mathbf{x}, y) = \sum\_{p=1}^{N} \vec{a}\_p(z) \frac{\mathbf{c}^{(p)}}{\mathbf{c}} \vec{E}\_z^{(p)}(\mathbf{x}, y) \tag{5}$$

$$
\vec{H}\_z(\mathbf{x}, \mathbf{y}) = \sum\_{p=1}^N \vec{a}\_p(z) \, \vec{H}\_z^{(p)}(\mathbf{x}, \mathbf{y}) \tag{6}
$$

where: *Et* , *Ht* , ( *Ez* , *Hz* ) denote transverse (longitudinal) electric and magnetic field components, respectively.

Fig. 18. A top view of a dual-channel defect waveguide processed in a membrane with a photonic crystal square lattice. Shaded regions indicate an equivalent effective waveguide structure.

According to (Kogelnik & Shank, 1972; Schmidt et al., 1974; Chen & Wang, 1984), the amplitudes of the modes guided in the coupled waveguides satisfy the following conditions:

$$i\overline{\mathcal{C}}\,\frac{d}{dz}\overline{a}\,(z) = i\,\,S\,\,\overline{a}\,\,(z) \quad\text{or}\quad \frac{d}{dz}\overline{a}\,(z) = i\,\,M\,\,\overline{a}\,\,(z)\tag{7}$$

where an *N*-element vector *a z*( ) consists of *ap(z)* amplitudes of the modes propagating in consecutive waveguide channels.

The matrices *S* and *M* are defined as follows:

$$S = \overline{C}B + \tilde{K} \quad \text{or} \quad M = \overline{C}^{-1}S = B + \overline{C}^{-1}\tilde{K} \tag{8}$$

where *C* is an *N*x*N* square matrix, where each element of the overlapping integrals *Cpq* is defined as follows:

$$
\overline{\mathbf{C}}\_{pq} = \overline{\mathbf{C}}\_{qp} = \frac{\mathbf{C}\_{pq} + \mathbf{C}\_{qp}}{2} \tag{9}
$$

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 113

structures, energy theorem is used (Szczepanski et al., 1989; Szczepanski, 1988). It allows us to represent a normalised small-signal gain saturation of the laser as a function of a saturation power, a distributed losses coefficient and a laser's geometry. The field distribution of the modes generated in the membrane was obtained in two ways: applying the FDTD method (Taflove & Hagness, 2005; QWED) and using analytical formulas derived for the effective planar waveguides. As it is shown in the next Section, discrepancy between

Fig. 19. A schematic view of phased array photonic crystal lasers processed in a membrane with a square photonic crystal lattice with mirrors made of a) a 1D photonic crystal and b) a

*E xyz R xyz S xyz f z E xyz e*

,, ,, ,, , ,

*i z*

is a phase constant of a laser mode, *fR(z)* and *fS(z)* denote complex amplitudes of

 

forward and backward propagating waves, whereas *ER(x,y,z)* and *ES(x,y,z)* represent a

The coupled-mode equations for the considered laser structures may be written as

*L R o RR R*

*df z <sup>i</sup> <sup>f</sup> <sup>z</sup> <sup>g</sup> D d <sup>f</sup> zE x <sup>y</sup> z dxdy dz <sup>N</sup>*

\*

<sup>1</sup> , ,

<sup>1</sup> , ,

*df z <sup>i</sup> <sup>f</sup> <sup>z</sup> <sup>g</sup> D d <sup>f</sup> zE x <sup>y</sup> z dxdy dz <sup>N</sup>*

*L S o SS S*

\* \*

<sup>1</sup> ,, ,, , <sup>2</sup>

*g D d f z E x y z E x y z dxdy <sup>N</sup>*

*o RR S*

<sup>1</sup> ,, ,, , <sup>2</sup>

*g D d f z E x y z E x y z dxdy <sup>N</sup>*

*o SS R*

*R R*

*i z*

2

2

(14)

(15)

*f z E xyz e*

, ,

lasers characteristics achieved with both methods remains below ca. 10 %.

The field distribution in the laser is written in the following way:

*S S*

transverse distribution of the laser mode.

*R*

 

 

 

*R*

*S*

*S*

2D photonic crystal.

(Szczepanski, 1994):

where 

where

$$C\_{pq} = \frac{1}{2} \int\_{-\infty}^{\infty} \int\_{0}^{\infty} \left( \vec{E}\_t^{(q)} \times \vec{H}\_t^{(p)} \right) \hat{z} \, dx \, dy \tag{10}$$

It should be emphasized that the electromagnetic field was normalised so as to:

$$\mathbb{C}\_{pp} = \mathbb{C}\_{qq} = 1 \tag{11}$$

Another *N*x*N* matrix *K* , applied in Eq.8, consists of coupling coefficients between all the *N* waveguides, which are defined in the following way:

$$\tilde{\mathcal{K}}\_{pq} = \frac{\alpha}{4} \int\_{-\infty}^{\alpha} \int \Delta \varepsilon^{(q)} \left| \vec{E}\_t^{(p)} \cdot \vec{E}\_t^{(q)} - \frac{\varepsilon^{(p)}}{\varepsilon} \vec{E}\_z^{(p)} \cdot \vec{E}\_z^{(q)} \right| dx \, dy \tag{12}$$

where: *q q* , 2 *q q n* .

The matrix *B* is an *N*x*N* diagonal matrix with phase constants *<sup>i</sup>* (*i* = 1…*N*) of the modes propagating in all the waveguide channels. It should be noted that matrices *C* and *S* are symmetric, what is very important to prove the orthogonality of the supermodes, whereas the matrix *M*, in general, is not necessarily symmetric.

The solution of the coupled mode equations given by Eq.8 leads to the modal field distribution in the entire array for a given propagation constant *<sup>P</sup>*. In the system consisting of *N* coupled waveguides, *P* supermodes are generated (*N* = *P*), which may be written as follows:

$$\begin{aligned} \tilde{E}\_1 &= A\_1 \cdot \left[ a\_1^{(1)}(z) E\_1^{(1)}(x, y) + \dots + a\_N^{(1)}(z) E\_N^{(1)}(x, y) \right] \cdot e^{i\gamma\_1 z} \\ &\vdots \\ \tilde{E}\_P &= A\_P \cdot \left[ a\_1^{(p)}(z) E\_1^{(p)}(x, y) + \dots + a\_N^{(p)}(z) E\_N^{(p)}(x, y) \right] \cdot e^{i\gamma\_P z} \end{aligned} \tag{13}$$

where *Ak* (*k* = 1…*P*) are scaling coefficients, *P* is the order of a supermode, and 1 *Na* indicates the amplitude of a field distribution of the 1st supermode in the *N*th waveguide.

#### **4.3 The model of light generation in planar multi-channel photonic crystal membrane lasers**

In this Section, an approximate model of laser generation in planar multi-channel PhC membranes is described. Fig.19 shows phased array lasers with mirrors made of 1D and 2D photonic crystals processed in a membrane. The effective values of reflection coefficients are denoted with *r1* and *r2*. It is assumed hereafter that the reflection coefficient of an input mirror is *r1* = 1.0.

In the proposed model of light generation, a field distribution in the single- and multichannel photonic crystal membrane lasers is substituted with a field in the equivalent effective planar waveguides (see Section 4.2). To achieve laser characteristics of those

 <sup>1</sup> <sup>ˆ</sup> <sup>2</sup> *q p Cpq E H z dx dy t t*

Another *N*x*N* matrix *K* , applied in Eq.8, consists of coupling coefficients between all the *N*

*<sup>q</sup> p q p q Kpq E E E E dx dy t t z z*

propagating in all the waveguide channels. It should be noted that matrices *C* and *S* are symmetric, what is very important to prove the orthogonality of the supermodes, whereas

The solution of the coupled mode equations given by Eq.8 leads to the modal field distribution

( ) ( )

<sup>1</sup> 1 1 (1) (1)

, ,

*P P P P i z*

*N N*

waveguides, *P* supermodes are generated (*N* = *P*), which may be written as follows:

*P P N N*

where *Ak* (*k* = 1…*P*) are scaling coefficients, *P* is the order of a supermode, and 1

the amplitude of a field distribution of the 1st supermode in the *N*th waveguide.

**4.3 The model of light generation in planar multi-channel photonic crystal** 

*E A a zE xy a zE xy e*

*E A a zE x y a zE x y e*

In this Section, an approximate model of laser generation in planar multi-channel PhC membranes is described. Fig.19 shows phased array lasers with mirrors made of 1D and 2D photonic crystals processed in a membrane. The effective values of reflection coefficients are denoted with *r1* and *r2*. It is assumed hereafter that the reflection coefficient of an input

In the proposed model of light generation, a field distribution in the single- and multichannel photonic crystal membrane lasers is substituted with a field in the equivalent effective planar waveguides (see Section 4.2). To achieve laser characteristics of those

, , *<sup>P</sup>*

 

*p*

(12)

(10)

1 *C C pp qq* (11)

*<sup>P</sup>*. In the system consisting of *N* coupled

*i z*

*<sup>i</sup>* (*i* = 1…*N*) of the modes

(13)

*Na* indicates

It should be emphasized that the electromagnetic field was normalised so as to:

waveguides, which are defined in the following way:

4

the matrix *M*, in general, is not necessarily symmetric.

in the entire array for a given propagation constant

1 1

1 1 1 1

where: *q q* ,

**membrane lasers**

mirror is *r1* = 1.0.

The matrix *B* is an *N*x*N* diagonal matrix with phase constants

*q q n* .

2

where

structures, energy theorem is used (Szczepanski et al., 1989; Szczepanski, 1988). It allows us to represent a normalised small-signal gain saturation of the laser as a function of a saturation power, a distributed losses coefficient and a laser's geometry. The field distribution of the modes generated in the membrane was obtained in two ways: applying the FDTD method (Taflove & Hagness, 2005; QWED) and using analytical formulas derived for the effective planar waveguides. As it is shown in the next Section, discrepancy between lasers characteristics achieved with both methods remains below ca. 10 %.

Fig. 19. A schematic view of phased array photonic crystal lasers processed in a membrane with a square photonic crystal lattice with mirrors made of a) a 1D photonic crystal and b) a 2D photonic crystal.

The field distribution in the laser is written in the following way:

$$\begin{aligned} \operatorname{E}\left(\mathbf{x}, y, z\right) &= \operatorname{R}\left(\mathbf{x}, y, z\right) + \operatorname{S}\left(\mathbf{x}, y, z\right) = f\_{\operatorname{R}}\left(z\right) E\_{\operatorname{R}}\left(\mathbf{x}, y, z\right) e^{i\boldsymbol{\beta}\cdot\boldsymbol{z}} + \\ &+ f\_{\operatorname{S}}\left(z\right) E\_{\operatorname{S}}\left(\mathbf{x}, y, z\right) e^{-i\boldsymbol{\beta}\cdot\boldsymbol{z}} \end{aligned} \tag{14}$$

where is a phase constant of a laser mode, *fR(z)* and *fS(z)* denote complex amplitudes of forward and backward propagating waves, whereas *ER(x,y,z)* and *ES(x,y,z)* represent a transverse distribution of the laser mode.

The coupled-mode equations for the considered laser structures may be written as (Szczepanski, 1994):

$$\begin{cases} \frac{df\_R(z)}{dz} + \left(\alpha\_L - i\delta\right) f\_R\left(z\right) = \frac{1}{N\_R} \int \left[\int g\_o \circ D \, d\_\alpha \, f\_R\left(z\right)\right] \mathrm{E}\_R\left(x, y, z\right) \Big|^2 \, d\mathrm{xdy} + \\\\ + \frac{1}{2N\_R} \int \left[\int g\_o \circ D \, d\_\kappa \, f\_S\left(z\right) \, \mathrm{E}\_S\left(x, y, z\right) \, \mathrm{E}\_R^\*\left(x, y, z\right) \, d\mathrm{xdy}, \\\\ - \frac{df\_S\left(z\right)}{dz} + \left(\alpha\_L - i\delta\right) f\_S\left(z\right) = \frac{1}{N\_S} \int \left[\int g\_o \circ D \, d\_\alpha \, f\_S\left(z\right)\right] \mathrm{E}\_S\left(x, y, z\right) \Big|^2 \, d\mathrm{xdy} + \\\\ + \frac{1}{2N\_S} \int \left[\int g\_o \circ D \, d\_\kappa^\* \, f\_R\left(z\right) \, \mathrm{E}\_R\left(x, y, z\right) \, \mathrm{E}\_S^\*\left(x, y, z\right) \, \mathrm{dxdy}, \end{cases} \tag{15}$$

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 115

<sup>2</sup> <sup>2</sup> <sup>2</sup>

<sup>2</sup> <sup>2</sup> <sup>2</sup>

<sup>1</sup>

An approximate expression relating the normalised small-signal gain coefficient *goL* to the

0 0 2 2 2 2

*t L t RS*

The transverse electric field distribution of the laser mode in the photonic crystal membrane *ER* = *ES* = *Et* was calculated numerically (see Section 3.3) and analytically using the effective

In this Section, exemplary gain characteristics of phased array lasers processed in defect photonic crystal membranes are given. The transverse field distribution of the

1 2 1 1 ln

*C E x y z dxdy E x y z f z f z dxdydz*

*E xyz f z f z*

2 2 2 2

*dxdydz <sup>P</sup> E xyz f z f z P K*

2 2 2

1 2 1 01 *K fL r f r R S* (33)

2 2 2

*f rf R S* 0 0 <sup>2</sup> (25)

*f L rf L S R* <sup>1</sup> (27)

*fR f zA z* exp (29)

<sup>2</sup> *fS f z Ar z* exp (30)

<sup>2</sup> *<sup>f</sup> L rr* (31)

, (32)

, 2 0 1 , ,0 *PS out S <sup>S</sup> <sup>f</sup> r Ex <sup>y</sup> dxdy* (26)

, 1 <sup>1</sup> , , *<sup>R</sup> P f L r E x y L dxdy R out R* (28)

The solution of Eq.24 requires boundary conditions to be specified:

where *PP P out R out S out* , , is a total power generated by the laser.

output power and the parameters of the planar laser is given as follows:

,, 2 , ,

, ,

*out <sup>d</sup> t RS*

0 0 0

*eff eff*

*b b L*

<sup>2</sup> <sup>1</sup> <sup>1</sup> , ,

*S*

*d d <sup>o</sup> <sup>b</sup> <sup>L</sup> t RS*

0

planar waveguide model (see Section 4.2).

**4.4 Laser gain characteristics**

0 0

*eff*

In a threshold approximation, *fR(z)* and *fS(z)* are equal to:

where

*g L*

where:

where *go* is a small-signal gain coefficient, *L* stands for laser's distributed losses, denotes a frequency-shift parameter understood as the discrepancy between an oscillating frequency in passive and active resonators. The normalisation factors *NR* and *NS* may be calculated as:

$$N\_R = \iint \left| E\_R \left( \mathbf{x}, \mathbf{y}, \mathbf{z} \right) \right|^2 d\mathbf{x} dy \,, \ N\_S = \iint \left| E\_S \left( \mathbf{x}, \mathbf{y}, \mathbf{z} \right) \right|^2 d\mathbf{x} dy \,\tag{16}$$

The shape of a gain spectral line applied in Eq.15 is given by:

$$D = \left(\chi + i\left(\alpha - \upsilon\right)\right)^{-1} \tag{17}$$

whereas the other parameters are given as follows:

$$C\_i = 1 + \left[ \left| f\_R(z) E\_R(\mathbf{x}, y, z) \right|^2 + \left| f\_S(z) E\_S(\mathbf{x}, y, z) \right|^2 \right] \cdot \frac{L}{I\_S} \tag{18}$$

$$\mathcal{C}\_c = 2\left[ \left| f\_R \left( z \right) E\_R \left( x, y, z \right) \right|^2 \cdot \left| f\_S \left( z \right) E\_S \left( x, y, z \right) \right|^2 \right] \cdot \frac{L}{I\_S} \tag{19}$$

$$d\_{\alpha} = \left(\mathbf{C}\_{i}^{2} - \mathbf{C}\_{c}^{2}\right)^{-0.5} \tag{20}$$

$$d\_{\kappa} = -\mathbb{C}\_{c} \, e^{i\theta} \left[ \sqrt{\mathbf{C}\_{i}^{2} - \mathbf{C}\_{c}^{2}} \left( \mathbf{C}\_{i} + \sqrt{\mathbf{C}\_{i}^{2} - \mathbf{C}\_{c}^{2}} \right) \right]^{-1} \tag{21}$$

where *L* denotes the length of the laser (see Fig.19).

The saturation power in the active region can be written as:

$$P\_S = I\_S \cdot A\_I \tag{22}$$

where *IS* is a saturation intensity

$$I\_S = \frac{h\nu}{\sigma\tau} \tag{23}$$

and *Al* denotes a cross-section of the laser, *h* is the Planck constant, is the frequency of a laser mode, is an emission cross-section, represents recombination lifetime in the active region. Operations on Eq.15 lead to (Szczepanski et al., 1989; Szczepanski, 1988):

 2 2 22 22 2 2 2 2 2 2 22 22 22 22 , , 2 2 , , , , , , 2 2 , , , , <sup>2</sup> . *R o R R S LR S R i c S S R o S o S R S R i c i ci i c R S o R S S i ci i c <sup>L</sup> g f z E xyz <sup>d</sup> fz fz fz fz dxdy dz <sup>N</sup> C C L L g f z E xyz g f zE xyz f zE xyz dxdy dxdy N N C C C CC C C <sup>L</sup> g f zE xyz f zE xyz dxdy <sup>N</sup> C CC C C* (24)

The solution of Eq.24 requires boundary conditions to be specified:

$$\left|f\_R\left(0\right)\right| = r\_2 \left|f\_S\left(0\right)\right|\tag{25}$$

$$P\_{S,out} = \left| f\_S \begin{pmatrix} 0 \end{pmatrix} \right|^2 \left( 1 - r\_2^2 \right) \iint \left| E\_S \begin{pmatrix} \mathbf{x}\_\prime \mathbf{y}\_\prime \mathbf{0} \end{pmatrix} \right|^2 d\mathbf{x} dy \tag{26}$$

$$\left| f\_S \left( L \right) \right| = r\_1 \left| f\_R \left( L \right) \right| \tag{27}$$

$$P\_{R,out} = \left| f\_R \left( L \right) \right|^2 \left( 1 - r\_1^2 \right) \left[ \iint \left| E^R \left( \mathbf{x}\_\prime \mathbf{y}\_\prime L \right) \right|^2 d\mathbf{x} dy \right] \tag{28}$$

where *PP P out R out S out* , , is a total power generated by the laser.

In a threshold approximation, *fR(z)* and *fS(z)* are equal to:

$$f\_{\mathcal{R}}(z) = |A| \exp\left(\gamma\_f z\right) \tag{29}$$

$$f\_S(z) = |A|r\_2^{-1} \exp\left(-\gamma\_f z\right) \tag{30}$$

where

114 Photonic Crystals – Innovative Systems, Lasers and Waveguides

frequency-shift parameter understood as the discrepancy between an oscillating frequency in passive and active resonators. The normalisation factors *NR* and *NS* may be calculated as:

*L* stands for laser's distributed losses,

<sup>1</sup> *D i* (17)

*S*

*S*

(20)

*P IA S Sl* (22)

2 2

*i c*

*R*

2 2

represents recombination lifetime in the active region.

, , 2

*o R*

<sup>2</sup> , , *N Ex R R <sup>y</sup> z dxdy* , <sup>2</sup> , , *N Ex S S <sup>y</sup> z dxdy* (16)

(18)

(19)

<sup>1</sup>

(21)

denotes a

(23)

(24)

is the frequency of a laser

2 2 <sup>1</sup> , , , , *i RR SS*

2 2 <sup>2</sup> , , , , *c RR SS*

*c i ci i c d Ce C C C C C*

*S <sup>h</sup> <sup>I</sup>*

*<sup>L</sup> C f z E xyz f z E xyz <sup>I</sup>*

 0.5 2 2 *i c d CC*

*i* 22 22

*<sup>L</sup> C f z E xyz f z E xyz <sup>I</sup>*

where *go* is a small-signal gain coefficient,

The shape of a gain spectral line applied in Eq.15 is given by:

whereas the other parameters are given as follows:

where *L* denotes the length of the laser (see Fig.19).

where *IS* is a saturation intensity

is an emission cross-section,

*o R S*

, , , , <sup>2</sup> .

*<sup>L</sup> g f zE xyz f zE xyz dxdy <sup>N</sup> C CC C C*

22 22

*R S LR S R*

2

2 2

*S R*

mode, 

*S*

The saturation power in the active region can be written as:

and *Al* denotes a cross-section of the laser, *h* is the Planck constant,

22 22

*i ci i c*

*R S*

Operations on Eq.15 lead to (Szczepanski et al., 1989; Szczepanski, 1988):

*L L g f z E xyz g f zE xyz f zE xyz dxdy dxdy N N C C C CC C C*

 

, , , , , , 2 2

*o S o S R*

*<sup>L</sup> g f z E xyz <sup>d</sup> fz fz fz fz dxdy dz <sup>N</sup> C C*

2 2 22 22

*i c i ci i c*

*S S R*

$$\gamma\_f = \frac{1}{2L} \ln \frac{1}{r\_1 r\_2} \tag{31}$$

An approximate expression relating the normalised small-signal gain coefficient *goL* to the output power and the parameters of the planar laser is given as follows:

$$\log L = \frac{\text{C} \int\_{0}^{b} \int\_{\frac{\pi}{2}} \left| E\_{t} \left( x, y, z \right) \right|^{2} dx dy + 2 \alpha\_{L} \int\_{0}^{b} \int\_{\frac{\pi}{2}}^{L} \int\_{0}^{L} \left| E\_{t} \left( x, y, z \right) \right|^{2} \cdot \left( \left| f\_{R} \left( z \right) \right|^{2} + \left| f\_{S} \left( z \right) \right|^{2} \right) dx dy dz}{2 \int\_{0}^{b} \int\_{\frac{\pi}{2}}^{L} \int\_{\frac{\pi}{2}}^{L} \left| E\_{t} \left( x, y, z \right) \right|^{2} \cdot \left( \left| f\_{R} \left( z \right) \right|^{2} + \left| f\_{S} \left( z \right) \right|^{2} \right) }} \,\text{, (32)}$$

where:

$$K = \left| f\_R \left( L \right) \right|^2 \left( 1 - r\_1^2 \right) + \left| f\_S \left( 0 \right) \right|^2 \left( 1 - r\_2^2 \right) \tag{33}$$

The transverse electric field distribution of the laser mode in the photonic crystal membrane *ER* = *ES* = *Et* was calculated numerically (see Section 3.3) and analytically using the effective planar waveguide model (see Section 4.2).

#### **4.4 Laser gain characteristics**

In this Section, exemplary gain characteristics of phased array lasers processed in defect photonic crystal membranes are given. The transverse field distribution of the

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 117

reflection coefficient *r2* is equal to 0.91 and 0.996 for numerical and analytical (effective)

 = 1.55

 = 1.55

*m*). Red curve: computed with the aid of FDTD for (*a/*

blue curve: effective waveguide model with *beff / a* = 0.52.

reflectivity for a single-channel square-PhC membrane laser (*r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.4,

Concluding, exemplary laser small-signal gain characteristics have been shown, which enable the generation of a laser single-mode both in the single- and dual-channel structures

*m*). Red curve: computed with the aid of FDTD for (*a/*

blue curve: effective waveguide model with *beff / a* = 0.49.

reflectivity for a single-channel square-PhC membrane laser (*r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.3,

*m* as a function of cavity mirror

*m* as a function of cavity mirror

*a*) = (0.321, 12

);

 , 

*a*) = (0.340, 13

);

 , 

computations, respectively, leading to ca. 9.4% of the relative discrepancy.

Fig. 21. A normalised small-signal gain at

Fig. 22. A normalised small-signal gain at

*a* = 0.53

*a* = 0.50

laser supermode is calculated numerically with the FDTD method (Taflove & Hagness, 2005; QWED) and analytically, using the non-orthogonal mode theory applied to the calculation of the effective waveguide structure. It is assumed that the distributed losses coefficient is equal to *<sup>L</sup>* = 200 cm-1 (Zielinski et al. 1989; Lu et al., 2008), whereas the output power to saturation power ratio is *Pout / Ps* = 10-6 (Lu et al., 2009; Susaki et al., 2008; van den Hoven, 1996).

Fig. 20. A normalised small-signal gain at = 1.55*m* as a function of cavity mirror reflectivity for a single-channel square-PhC membrane laser (*r/a* = 0.4, *d/a* = 0.4, *b/a* = 0.3, *a* = 0.55*m*). Red curve: computed with the aid of FDTD for (*a/* , *a*) = (0.356, ); blue curve: effective waveguide model with *beff / a* = 0.55.

Fig.20 presents a normalised small-signal gain at = 1.55*m* as a function of output mirror reflectivity *r2* for a single-channel square-PhC membrane laser (*r/a* = 0.4, *d/a* = 0.4, *b/a* = 0.3, *a* = 0.55*m*), the PBG diagram of which is shown in Fig.12. The red curve depicts the gain characteristics calculated for (*a/* , *a*) = (0.356, ) with the aid of FDTD, whereas the blue one indicates the result of analytical computation with Eq.32 for the corresponding effective waveguide model with the channel width *beff / a* = 0.55. In principle, the minimum of the calculated characteristics indicates an optimum value of the mirror reflection coefficient *r2* of an output mirror, for which maximum output power efficiency is achieved. It can be seen from Fig.20 that, although the shape of both curves is substantially different, their minima are in a similar position and the optimum reflectivity *r2* amounts to 0.93 and 0.997 for the red and blue curves, respectively. Consequently, it leads to ca. 7.2% of a relative discrepancy between the optimum values computed with the two approaches.

Similar computations were carried out for dual-channel scenarios with *r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.3 and 0.4. Fig.21,22 show the corresponding laser characteristics for (*a/*, *a*) = (0.340, 13) and (0.321, 12) with *a* = 0.53*m* and 0.50*m*, respectively. In both cases, the

laser supermode is calculated numerically with the FDTD method (Taflove & Hagness, 2005; QWED) and analytically, using the non-orthogonal mode theory applied to the calculation of the effective waveguide structure. It is assumed that the distributed losses

output power to saturation power ratio is *Pout / Ps* = 10-6 (Lu et al., 2009; Susaki et al., 2008;

 = 1.55

*a*) = (0.356,

*m*). Red curve: computed with the aid of FDTD for (*a/*

 , 

between the optimum values computed with the two approaches.

) with *a* = 0.53

*b/a* = 0.3 and 0.4. Fig.21,22 show the corresponding laser characteristics for (*a/*

reflectivity for a single-channel square-PhC membrane laser (*r/a* = 0.4, *d/a* = 0.4, *b/a* = 0.3,

reflectivity *r2* for a single-channel square-PhC membrane laser (*r/a* = 0.4, *d/a* = 0.4, *b/a* = 0.3,

one indicates the result of analytical computation with Eq.32 for the corresponding effective waveguide model with the channel width *beff / a* = 0.55. In principle, the minimum of the calculated characteristics indicates an optimum value of the mirror reflection coefficient *r2* of an output mirror, for which maximum output power efficiency is achieved. It can be seen from Fig.20 that, although the shape of both curves is substantially different, their minima are in a similar position and the optimum reflectivity *r2* amounts to 0.93 and 0.997 for the red and blue curves, respectively. Consequently, it leads to ca. 7.2% of a relative discrepancy

Similar computations were carried out for dual-channel scenarios with *r/a* = 0.4, *d/a* = 0.6,

*m* and 0.50

 = 1.55

*m*), the PBG diagram of which is shown in Fig.12. The red curve depicts the gain

*<sup>L</sup>* = 200 cm-1 (Zielinski et al. 1989; Lu et al., 2008), whereas the

*m* as a function of cavity mirror

*a*) = (0.356,

) with the aid of FDTD, whereas the blue

, 

*m*, respectively. In both cases, the

*a*) = (0.340,

*m* as a function of output mirror

); blue curve:

 , 

coefficient is equal to

van den Hoven, 1996).

Fig. 20. A normalised small-signal gain at

effective waveguide model with *beff / a* = 0.55.

Fig.20 presents a normalised small-signal gain at

*a* = 0.55

*a* = 0.55

13 characteristics calculated for (*a/*

) and (0.321, 12

reflection coefficient *r2* is equal to 0.91 and 0.996 for numerical and analytical (effective) computations, respectively, leading to ca. 9.4% of the relative discrepancy.

Fig. 21. A normalised small-signal gain at = 1.55*m* as a function of cavity mirror reflectivity for a single-channel square-PhC membrane laser (*r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.3, *a* = 0.53*m*). Red curve: computed with the aid of FDTD for (*a/* , *a*) = (0.340, 13); blue curve: effective waveguide model with *beff / a* = 0.49.

Fig. 22. A normalised small-signal gain at = 1.55*m* as a function of cavity mirror reflectivity for a single-channel square-PhC membrane laser (*r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.4, *a* = 0.50*m*). Red curve: computed with the aid of FDTD for (*a/* , *a*) = (0.321, 12); blue curve: effective waveguide model with *beff / a* = 0.52.

Concluding, exemplary laser small-signal gain characteristics have been shown, which enable the generation of a laser single-mode both in the single- and dual-channel structures

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 119

Chuang, S.-L., Application of the strongly coupled mode theory to integrated optical

Digonnet, M.J.F. & Shaw, H.J., Analysis of a tunable single mode optical fiber coupler,

Fan, S., Villeneuve, P.R. & Joannopoulos, J.D., Channel drop filters in photonic crystals,

Gwarek, W.K., Analysis of an arbitrarily-shaped planar circuit - a time-domain

Gwarek, W.K., Morawski, T. & Mroczkowski, C., Application of the FDTD Method to

Hardy, A. & Streifer, W., Coupled mode theory of parallel waveguides, *J. Lightwave Technol.*,

Hardy, A., Streifer, W. & Osinski, M., Chirping effects in phase-coupled laser arrays, *Proc.* 

Joannopoulos, J.D., Johnson, S.G., Winn, J.N. & Meade, R.D., *Photonic Crystals. Molding* 

Kapon, E., Lindsey, P., Katz, J., Margalit, S. & Yariv, A., Chirped arrays of diode lasers for supermode control, *Appl. Phys. Lett.*, vol. 45, no. 3, pp. 200-202, 1984 Kapon, E., Lindsey, P., Smith, J.S., Margalit, S. & Yariv, A., Inverted-V chirped phased arrays

Kapon, E., Katz, J., Margalit, S. & Yariv, A., Controlled fundamental supermode operation of

Kapon, E., Rav-Noy, Z., Margalit, S. & Yariv, A., Phase-Locked Arrays of Buried-

Kogelnik, H. & Shank, C.V., Coupled-wave theory of distributed feedback lasers, *J. Appl.* 

Kogelnik, H. & Schmidt, R.V., Switched directional couplers with alternating ", *J. Quant.* 

Kogelnik, H., *Theory of dielectric waveguides*, ch. 2, T.Tamir Edition, New York: Springer-

Lesniewska-Matys, K., *Modelowanie generacji promieniowania w planarnym wielokanałowym* 

Liu, T., Zakharian, A.R., Fallahi, M., Moloney, J.V. & Mansuripur, M., Multimode

*fotonicznego,* Ph.D. Thesis, Warsaw University of Technology, 2011

*laserze sprzężonym fazowo zbudowanym na bazie dwuwymiarowego kryształu* 

Interference-Based Photonic Crystal Waveguide Power Splitter, *J. Lightwave* 

approach, *IEEE Trans. Microwave Theory Tech.*, vol. MTT-33, No.10, pp.1067-1072,

the Analysis of the Circuits Described by the Two-Dimensional Vector Wave Equation, *IEEE Trans. Microwave Theory Tech.*, vol. 41, no. 2, pp. 311-316, Feb.

*the flow of light*, Second Edition, Princeton University Press, ch. 8, pp. 144,

of gain-guided GaAs/GaAlAs diode lasers, *Appl. Phys. Lett.*, vol. 45, no. 12, pp.

phase-locked arrays of gain-guided diode lasers, *Appl. Phys. Lett.*, vol. 45, no. 6, pp.

Ridge InP/InGaAsP Diode Lasers, *J. Lightwave Technol.*, vol. 4, no. 7, pp. 919-925,

devices, *J. Quant. Electron*., vol. 23, no. 5, pp. 499-509, 1987 Collin, R.E., *Field Theory of Guided Waves*, McGraw-Hill Inc., New York, 1960

*J. Quant. Electron.*, vol. 18, no. 4, pp. 746-754, 1982

*Optics Letters*, vol. 3, no. 1, pp. 4-11, 1998

vol. 3, no. 5, pp. 1135-1146, 1985

*IEEE*, vol. 135, no. 6, pp. 443-450, 1988

*Phys.*, vol. 43, no. 5, pp. 2327-2335, 1972

*Electron.*, vol. 12, no. 7, pp. 396-401, 1976

*Technol.*, vol. 22, no. 12, pp. 2842-2846, 2004

1985

1993

2008

1257-1259, 1984

600-602, 1984

Verlag, 1979

1986

in two-dimensional photonic crystal lattices processed in dielectric membranes. It has also been shown that both rigid full-wave and approximate computations of the modal field distributions provide the values of the optimum reflection coefficient of the output mirror, which are in less than 10% agreement.
