**2. Fundamentals of quantum optics phenomena in photonic crystals**

Optical processes in nano-structured materials with a period close to the light wavelength are essentially different from those in bulk uniform media. It is due to the regularities of propagation of Bloch optical waves in such periodic structures ("photon confinement").

#### **2.1 Luminescence**

Consider the spontaneous emission transition in two-level system presented in Fig. 1.

Fig. 1. Spontaneous emission transition in two-level system

The downward transition probability *Wnm* is determined in accordance with Fermi's golden rule as follows

$$\mathcal{W}\_{nm} = \frac{2\pi}{\hbar} \cdot \left| \left< m \middle| \hat{H}\_{\text{int}} \middle| n \right> \right|^2 \cdot g(E\_n - \hbar o) \,, \tag{1}$$

where int *mH n* ˆ is a matrix element of the perturbation operator, ( ) *<sup>n</sup> g E* is a density of final states of micro-object. In case of placing the micro-object into photonic crystal, the ( ) *<sup>n</sup> g E* spectrum is characterised by a density of optical states *g*(*ω*) in photonic crystal. The spontaneous emission spectrum *S*(*ω*) is completely determined by the spectral distribution of transitions frequencies *ωnm* and the density of optical states *g*(*ω*) within a region of these frequencies (Vats et al., 2002).

When frequencies *ωnm* are in photonic band gap, where *g*(*ω*) = 0, spontaneous emission must be completely inhibited. In general case, a dip in spontaneous emission spectrum should appear. The spectral position of this dip is correspondent to positions of reflection spectrum maximum and transmission spectrum minimum (Gaponenko et al., 1999). As a result of spontaneous emission inhibition the localisation of photon near by irradiative atom inside photonic crystal becomes possible if the transition frequency is within a band gap region or in the vicinity of band gap edges (John, 1987). In this case, bonded atom-photon state is coming into being. The photon emitted returns to the atom due to Bragg reflection and is reabsorbed by this atom. The existence of a group of such atoms may result in forming narrow photonic impurity band like an impurity band in semiconductor at sufficient concentrations of impurity atoms. Kinetics of luminescence in the vicinity of band gap edges demonstrates non-exponential behaviour (John, 1987).

#### **2.2 Enhanced Raman scattering**

88 Quantum Optics and Laser Experiments

allows modifying optical properties of such systems by filling the pores with various

Synthetic opal photonic crystals containing nonlinear optical substances give a good chance to observe quantum optics phenomena in spatially nonuniform media where the photon mean free path is close to the light wavelength. Moreover, in this case the input optical power that is necessary to observe phenomena may be lower than the power required usually for observing the same phenomena in uniform nonlinear substances. The reason for it is the existence of diffuse transfer of photons that can result in photon accumulating inside photonic crystals and, consequently, in local optical power increasing. In particular, the possibility of experimental manifestation of Raman scattering and spontaneous parametric down-conversion in synthetic opals is discussed (Gorelik, 2007). The latter phenomenon is of special interest as it is convenient method to obtain bi-photon fields consisting of correlated photon pairs (Kitaeva & Penin, 2005). In the recent years, crystals with chirped structure of quadratic susceptibility (Kitaeva & Penin, 2004), and materials with spatially regular and stochastic distribution of quadratic optical susceptibility (Kalashnikov et al., 2009), are considered as sources of bi-photons. It is quite possible synthetic opal photonic

**2. Fundamentals of quantum optics phenomena in photonic crystals** 

Consider the spontaneous emission transition in two-level system presented in Fig. 1.

Fig. 1. Spontaneous emission transition in two-level system

Optical processes in nano-structured materials with a period close to the light wavelength are essentially different from those in bulk uniform media. It is due to the regularities of propagation of Bloch optical waves in such periodic structures ("photon confinement").

The downward transition probability *Wnm* is determined in accordance with Fermi's golden

int <sup>2</sup> <sup>ˆ</sup> ( ) *W mH n gE nm <sup>n</sup>*

of final states of micro-object. In case of placing the micro-object into photonic crystal, the

where int *mH n* ˆ is a matrix element of the perturbation operator, ( ) *<sup>n</sup> g E*

2

spectrum is characterised by a density of optical states *g*(*ω*) in photonic crystal.

, (1)

is a density

substances.

crystals will be ranked with these sources.

**2.1 Luminescence** 

rule as follows

( ) *<sup>n</sup> g E* 

Consider an elementary Stokes Raman scattering process as a disintegration of exciting photon (*ћωex*, *kex*) into scattered photon (*ћω*', *k*') and optical phonon (*ћΩ*, *Κ*) (Fig. 2).

Fig. 2. Elementary Stokes Raman scattering process

The Stokes process probability *W*(*ω*') is determined by the density of optical states *g*(*ω*') in the region of scattered light frequencies (Poulet & Mathieu, 1970)

$$\mathcal{MV}(o') = \frac{2\pi}{\hbar} \cdot \left| \left< m \right| \hat{H}'\_{\text{int}} \left| 0 \right> \right|^2 \cdot \mathcal{g}(o') \,, \tag{2}$$

where int ˆ *m H* 0 is the modulus of the matrix element of the Hamiltonian of the radiationsubstance interaction.

Quantum Optics Phenomena in Synthetic Opal Photonic Crystals 91

where *m* = 1, 2, 3 …, and *q* is a vector of photonic crystal reciprocal lattice (*q* = 2*π*/*d*, *d* is a

Fig. 3. Spontaneous parametric down-conversion process in photonic crystal

0

*J IL* 

the *m*-th order nonlinear diffraction (Kitaeva & Penin, 2004)

as a result of periodical modulation of linear susceptibility.

where *I0* is a pump intensity, (2)

distribution of nonlinear susceptibility *χ*(*2*)

The intensity of each summand is determined by the square of magnitude of corresponding harmonic amplitude. An additional contribution to the parametric down-conversion spectrum should be given by interference of non-coinciding harmonics. At the absence of absorption the parametric down-conversion intensity *JωΦ* per unit spectral *Δω* and angular *ΔΦ* intervals is determined by a magnitude of phase quasi-synchronism turning out *Δm* for

2 (2)

defined via the phase synchronism turning out *Δ* in spatially uniform medium as follows

In contrast to parametric down-conversion spectrum of spatially uniform sample the biphoton field spectrum of photonic crystal should be broadened, and the interference effects may appear in its spectral intensity distribution (Kitaeva & Penin, 2004, 2005; Nasr et al., 2008; Kalashnikov et al., 2009). One of the reasons that cause an additional broadening of the bi-photon spectrum in photonic crystal is the presence of structure disordered domains with period *d* varied along a pump propagation direction. Besides, observable spectrum should be determined by the density of optical states *g*(*ω*) in the region of scattered light frequencies

2

*<sup>m</sup>* is an amplitude of the *m*-th Fourier component of spatial

(8)

. The phase quasi-synchronism turning out *Δm* is

*<sup>m</sup> mqL* . (9)

sin( / 2) ( /2) *<sup>m</sup> <sup>m</sup> m m*

period of photonic crystal structure).

*k k k qk msi p m* 0 , (7)

In general, the density of optical states *g*(*ω*) in photonic crystal is defined by a photon dispersion law and has maxima near by the band gap edges, where ( /) 0 *d dk* . In the frame of one-dimensional model it can be determined as follows

$$\log(o\nu) = (k^2 \;/\; \pi^2)(do \;/\; dk)^{-1} \;. \tag{3}$$

Exciting photons with the frequency *ωex* in the vicinity of band gap edges have the velocity values close to zero. It results in increasing the interval int *t* of radiation-substance interaction in photonic crystal compared with the interval int *t* in uniform material, according to the following expression

$$t\_{\rm int}' = t\_{\rm int} \cdot \frac{c}{n\_{\rm eff}} \cdot \left( d\phi \, / \, dk \right)^{-1} \,, \tag{4}$$

where *c* = 3108 m/s and *neff* is an effective refractive index of photonic crystal. Besides, the interval int *t* may be enlarged because of the presence of "diffuse" photons whose motion is like to Brownian motion. Such diffuse photon transfer is most probably due to multiple photon reflections from disordered elements in photonic crystal structure. Both considered mechanisms give a reason to expect the enhancement of Raman scattering by substances infiltrated into photonic crystal.

#### **2.3 Spontaneous parametric down-conversion**

Spontaneous parametric down-conversion is a process of spontaneous disintegration of pump photons (*ћωp*, *kp*) into pairs of signal (*ћωs*, *ks*) and idler (*ћωi*, *ki*) photons. As this process is a second-order nonlinear process it occurs in media with no inversion symmetry. In case of spatially uniform media with a non-zero second-order nonlinear susceptibility *χ*(*2*) the energy and momentum conservation is as follows

$$
\alpha \alpha\_p = \alpha\_s + \alpha\_{i'} \quad \mathbf{k}\_p = \mathbf{k}\_s + \mathbf{k}\_i \tag{5}
$$

In frequency-degenerated ( / 2 *sip* ) and collinear ( / 2 *sip kkk* ) regime at equal polarization of photons in pair (i.e., signal and idler photons are identical) the spectrum of bi-photons is determined by the following expression (Kalashnikov et al., 2009)

$$J(\Omega\_{bp}) \sim \left[\frac{\sin \chi}{\chi}\right]^2,\tag{6}$$

where 2 22 ( /2) ( / ) *bp bp x L dk d* , *Ωbp* is a frequency turning out of / 2 *<sup>p</sup>* , *L* is a sample length in the pump propagation direction.

For spatially nonuniform media with regular structures (photonic crystals) a periodic modulation of linear and nonlinear susceptibilities should be considered in general case. By taking into account the *χ*(*<sup>2</sup>*) periodic modulation the bi-photons spectrum should be determined by an additive sum of single harmonics of *χ*(*<sup>2</sup>*) susceptibility (Kitaeva & Penin, 2005). The spectrum of each of these harmonics is shifted relative to the spectrum in uniform medium, according to the following "quasi-synchronism" condition (see also Fig. 3)

90 Quantum Optics and Laser Experiments

In general, the density of optical states *g*(*ω*) in photonic crystal is defined by a photon

22 1 *g*( ) ( / )( / )

Exciting photons with the frequency *ωex* in the vicinity of band gap edges have the velocity values close to zero. It results in increasing the interval int *t* of radiation-substance interaction in photonic crystal compared with the interval int *t* in uniform material,

> int int ( /) *eff <sup>c</sup> t t d dk n*

where *c* = 3108 m/s and *neff* is an effective refractive index of photonic crystal. Besides, the interval int *t* may be enlarged because of the presence of "diffuse" photons whose motion is like to Brownian motion. Such diffuse photon transfer is most probably due to multiple photon reflections from disordered elements in photonic crystal structure. Both considered mechanisms give a reason to expect the enhancement of Raman scattering by substances

Spontaneous parametric down-conversion is a process of spontaneous disintegration of pump photons (*ћωp*, *kp*) into pairs of signal (*ћωs*, *ks*) and idler (*ћωi*, *ki*) photons. As this process is a second-order nonlinear process it occurs in media with no inversion symmetry. In case of spatially uniform media with a non-zero second-order nonlinear susceptibility *χ*(*2*)

,

polarization of photons in pair (i.e., signal and idler photons are identical) the spectrum of

<sup>2</sup> sin ( ) *bp <sup>x</sup> <sup>J</sup>*

For spatially nonuniform media with regular structures (photonic crystals) a periodic modulation of linear and nonlinear susceptibilities should be considered in general case. By taking into account the *χ*(*<sup>2</sup>*) periodic modulation the bi-photons spectrum should be determined by an additive sum of single harmonics of *χ*(*<sup>2</sup>*) susceptibility (Kitaeva & Penin, 2005). The spectrum of each of these harmonics is shifted relative to the spectrum in uniform

medium, according to the following "quasi-synchronism" condition (see also Fig. 3)

*x*

  *k d dk* . (3)

, (4)

*s i k kk <sup>p</sup> s i* (5)

, (6)

*<sup>p</sup>* , *L* is a sample

) and collinear ( / 2 *sip kkk* ) regime at equal

1

. In the

dispersion law and has maxima near by the band gap edges, where ( /) 0 *d dk*

frame of one-dimensional model it can be determined as follows

according to the following expression

infiltrated into photonic crystal.

**2.3 Spontaneous parametric down-conversion** 

the energy and momentum conservation is as follows

In frequency-degenerated ( / 2

length in the pump propagation direction.

*p* 

*sip* 

bi-photons is determined by the following expression (Kalashnikov et al., 2009)

where 2 22 ( /2) ( / ) *bp bp x L dk d* , *Ωbp* is a frequency turning out of / 2

$$
\Delta \mathbf{k}\_m \equiv \mathbf{k}\_s + \mathbf{k}\_i + mq - \mathbf{k}\_p = \mathbf{0} \,, \tag{7}
$$

where *m* = 1, 2, 3 …, and *q* is a vector of photonic crystal reciprocal lattice (*q* = 2*π*/*d*, *d* is a period of photonic crystal structure).

Fig. 3. Spontaneous parametric down-conversion process in photonic crystal

The intensity of each summand is determined by the square of magnitude of corresponding harmonic amplitude. An additional contribution to the parametric down-conversion spectrum should be given by interference of non-coinciding harmonics. At the absence of absorption the parametric down-conversion intensity *JωΦ* per unit spectral *Δω* and angular *ΔΦ* intervals is determined by a magnitude of phase quasi-synchronism turning out *Δm* for the *m*-th order nonlinear diffraction (Kitaeva & Penin, 2004)

$$J\_{o\Phi} \sim I\_0 \cdot L^2 \sum\_m \left| \mathcal{X}\_m^{(2)} \frac{\sin(\Lambda\_m / \ 2)}{\left(\Lambda\_m / \ 2\right)} \right|^2 \tag{8}$$

where *I0* is a pump intensity, (2) *<sup>m</sup>* is an amplitude of the *m*-th Fourier component of spatial distribution of nonlinear susceptibility *χ*(*2*) . The phase quasi-synchronism turning out *Δm* is defined via the phase synchronism turning out *Δ* in spatially uniform medium as follows

$$
\Delta\_m = \Delta - mqL\tag{9}
$$

In contrast to parametric down-conversion spectrum of spatially uniform sample the biphoton field spectrum of photonic crystal should be broadened, and the interference effects may appear in its spectral intensity distribution (Kitaeva & Penin, 2004, 2005; Nasr et al., 2008; Kalashnikov et al., 2009). One of the reasons that cause an additional broadening of the bi-photon spectrum in photonic crystal is the presence of structure disordered domains with period *d* varied along a pump propagation direction. Besides, observable spectrum should be determined by the density of optical states *g*(*ω*) in the region of scattered light frequencies as a result of periodical modulation of linear susceptibility.

Quantum Optics Phenomena in Synthetic Opal Photonic Crystals 93

Fig. 5. Typical transmission and reflection spectra of initial opals (*D* = 255 nm). Transmission spectrum was measured at normal incidence to (111) plane (*θ* = 00), reflection spectrum was

Photonic crystals based on synthetic opals were obtained by further infiltration of initial opals with organic luminophores (rhodamine 6G, 2,5-bis(2-benzoxazolyl)hydroquinone, pironin G, astrofloksin) or nonlinear optical substances (Ba(NO3)2, LiIO3, KH2PO4, Li2B4O7). In most cases the infiltration was performed by a multiple soaking of samples in corresponding supersaturated solutions at room temperature. For example, synthetic opals were filled with rhodamine 6G by soaking samples in a dilute ethanol solution with laser dye concentrations of 10-4 M or 510-3 M. After soaking the obtained samples were in the air until ethanol was evaporated. In case of infiltration with Ba(NO3)2, LiIO3, KH2PO4 an additional annealing of samples was performed at temperatures lower than melting ones (595 oC for Ba(N03)2 and 120 oC for LiIO3) to remove water. In case of Li2B4O7 the initial opal

Reflection and transmission spectra of opals after infiltration were measured to prove the existence of corresponding substance in pores. Two types of changes in the spectra were registered. First, in opals with organic luminophores, an additional non-transmission band caused by absorption of embedded molecules was observed (Fig. 6). Second, the band caused by Bragg diffraction was shifted if a quantity of embedded substance was enough to change essentially the value of *neff*, according to expression (11) (Podolskyy et al., 2006).

In some experiments, in order to diminish (or exclude) the photonic stop-band effects and to study phenomena in a regular matrix of nano-emitters the opal samples were additionally soaked in water-glycerine solutions or pure glycerine. Opal infiltration with any waterglycerine solution yields in decreasing dielectric contrast in the synthetic opal photonic crystals as the refractive index of a water-glycerin solution *np* (variable from 1.39227 till 1.47399 in our experiments) is close to that of SiO2 globules *nS*. It causes the shift of the stopband center *λc* to the longer wavelengths (10) and the narrowing of stop-band region Δ*λ<sup>g</sup>*

measured at *θ* = 70

was in Li2B4O7 melt at 860 0C.

with increasing glycerin concentration.

22 2 (1 ) *neff <sup>s</sup> <sup>p</sup> f n f n* . (11)
