Using the transfer-matrix method, the effects of absorption and inhomogeneous broadening, in one-dimensional optical lattice constructed from inhomogeneously broadened spin transitions of nitrogen-vacancy color centers in single crystal diamond (NV diamond), on the reflection and absorption spectrum are presented. Further analysis show that, in realistic periodic stacks of the NV diamond, modulating the geometrical configuration of the external optical potential, the absorption lineshape scale, and the inhomogeneous broadening, one could easily access the diverse gap structures and a high band-gap reflectivity. These pretty useful calculations hold more potential for effective control of the light-matter interaction and realization in practice.
©2006 Optical Society of America
In the past few years the theory of the propagation of electromagnetic (EM) waves in periodic (both in two and three dimensions) and/or random dielectric structures (photonic band structures) has been intensively developed. The appearance, in periodic media, of a frequency gap where light does not propagate for all wave vectors - the so-called photonic band gap (PBG) [1, 2], can have a profound impact on several scientific and technical disciplines. Especially, there has been a considerable interest in generating photonic band-gap materials using nonlinear optical processes that arise from coherent quantum control of light-matter interactions. This is an altogether new approach toward tunable photonic crystals and requires the use of electromagnetically induced transparency (EIT)  in the presence of a standing wave optical potential [4–9].
Most of the experiments carried out so far have dealt with atomic media. We would like to note that NV diamond color centers and many rare-earth doped crystals have properties similar to both hot and cold atomic vapors. For many potential applications, however, solid-state media are preferred due to obvious advantages such as high atomic densities, compactness, and absence of atomic diffusion, simplicity, and scalability during assembling. On the other hand, well-known difficulties in solid materials are the resonant absorption and inhomogeneous broadening which play quite a significant role. Yet, we can obtain the narrow band-gap in their EIT window by using the EIT effect to overcome the detrimental effect of dissipation [7, 8]. Unfortunately, the biggest stumbling block to our forthcoming experiment and its widespread application has been the requirement for rather large Rabi frequency and sufficiently long samples due to intrinsically large inhomogeneous broadening.
In view of this, attention is currently focused on the influence of the presence of resonant absorption and inhomogeneous broadening in multilevel solid systems without EIT on the reflection and absorption spectrums. Actually, there were many theoretical calculations taking into account the influence of weak dissipation and inhomogeneous broadening on the photonic band gap structure either in semiconductor optic lattice [10–13] or in periodic quantum well structures [14–17] during the past decade. Especially, in the latest work , a theoretical description of the effects of dissipation on the propagation of light waves through a multilayer periodic mirror built from resonant absorbing atoms was presented.
The purpose of this paper is to extend the above studies into the resonantly absorbing and inhomogeneous broadening solids. In contrast to Ref. , we deal here with a solid dielectric structure with realistic NV diamond parameters, namely an array of thin parallel-sided layers separated by vacuum, and investigate in details the influence of the various factors on the band-gap reflectivity and absorption. The results show that many distinct gap structures can be obtained by modulating the geometrical configuration of the external optical potential and the absorption profiles, more importantly, the inhomogeneous broadening inherent to NV diamond constitutes an additional control parameter we are able to manipulate to tune both the gap structure and the gap reflectivity. This research makes experimental observation and application of high band-gap reflectivity more attractive.
2. The model and equations
A detailed analysis here is to carry out for realistic NV diamond that have a large optical oscillator strength (~ 0.1). Wcb ≃ 30 KHz is the inhomogeneous widths of the optic transition |c→|b〉, where the Raman transition frequency (120 MHz) is determined by the spacing between the S=0 (|b〉) and S=-1 (|c〉) ground-state spin sublevels. This spacing is controlled by the magnitude of the applied magnetic field (~1 KG along the (111) direction is applied) . While the excited states |a〉 and |b〉, where the relevant optical transition at 637 nm has an inhomogeneous width Wab ≃ 375 GHz, are coupled by a weak probe beam with frequency ω, the reflection and absorption of which are the physical quantities we are interested in. All relevant parameters we used come from the published experiment .
To clarify the optical behavior of the periodic dissipative solid structures we have calculated reflectivity and absorption spectra using a transfer matrix approach . The first step is the computation of the first-order susceptibility over the entire range of the frequencies of the corresponding transition, which is determined by the inhomogeneity of the crystal line fields in solids. We hereafter follow the treatment of Ref. [22, 23], assuming that the inhomogeneous broadening could be described by a Lorentzian, and obtain 
where ωij are the frequencies of the corresponding transitions, Δωij represents the detuning of the inhomogeneous broadened line center from an isolated atom line center, and χ̕(ω,ω ab(cb)) corresponds to a single ion with specific detunings ωab and ωcb determined by its position within its host, which is from the off diagonal density matrix elements oscillating ρ ab(ω) . Finally, the optical properties of the single slab are specified by the complex dielectric function
The transfer matrix MN (ω) (N is the number of the periods) for the whole periodic dissipative solid layers, with optical properties specified by the complex dielectric function (3) and otherwise separated by vacuum, is obtained by multiplying together the transfer matrix M(ω) of a single period [6–8]. Combining with the Bloch condition  on the photonic eigenstates
we can get the dispersion of the Bloch modes, i.e., the dependence of the complex Bloch wavevector κ associated with a given incident frequency ω
Here, E + and E - are the electric field amplitudes of the forward and backward (Bragg reflected) propagating probe, a is the periodicity of this structure [27–29].
From this we have calculated the reflectivity (|RN |2) and transmissivity (|TN |2) for the L length, namely,
The absorption A is then calculated in the usual way as A=1-|RN |2-|TN |2.
3. Numerical results and discussion
Using the formulas developed in the previous section, we can perform numerical calculations to show the effect of various factors on band-gap reflectivity in realistic periodic stacks of the NV diamond.
In order to calculate the effect of dissipation on the reflectivity, we separate the complex dielectric function into three different instances corresponding to: ε=ε 0(constant),ε=ε′(ω), and ε=ε′(ω) + iε″(ω). The curves in Fig. 1(a) correspond to two antithetic cases: ε 1=εvacuum - Δε 1 and ε 2=εvacuum + Δε 2 (εvacuum =1), where the NV diamond has frequency independent dielectric constant, i.e., non dissipative. It is evident that the band-gaps appear to be symmetric with respect to resonant frequency when Δε 1=Δε 2, in addition, with the value of Δε increasing, the width of gap will be larger. In the presence of dissipation the situation becomes more complicated. The blue curves in Fig. 1(b) correspond to cases where the imaginary part of the frequency-dependent dielectric constants is equal to zero (nonabsorbent materials). We can obtain a perfect traditional photonic band-gap that split right at resonance; in other words, gaps are symmetrically located at both sides of the resonance frequency as shown in . Yet in fact frequency-dependent dielectric constants are always accompanied by rather high absorption (due to a nonzero imaginary part of the dielectric constant), then the corresponding band-gap reflectivity decreases drastically in comparison with the perfect crystal as red curves.
Then, if we modify the Lorentzian absorption profile of NV diamond like done for atomic stack  in an artificial way by a suitable scaling α term of the damping , what will happen? Answer is clearly shown in Fig. 2: (a) the gap could survive in the presence of strong dissipation except for the width of gap decreasing; (b) for narrow absorption profile (blue) the reflectivity within the resonance region is higher than the ones in broadened cases (red and black), which is due to the corresponding higher refractive index (η)shown under Fig. 2(a). This effect will be remarkable for the whole gap developing within the NV resonance region, which may be achieved through appreciable modifications of the NV lattice periodicity a as shown in Fig. 2(c) . We could explain the character properly with the bottom figures under Fig.s (b, c). Very close to resonance the edge of the gap shifts depending on the absorption profile. So the band-gap only survives for very small absorption bandwidth (blue).
All above results not only validate those obtained from atomic stack, moreover, prove the major advantage of NV diamond. One side, the high reflectivity can exist with rather wider gap, ~ 103 GHz, rather larger than ~ 103 MHz in atomic, due to its large inhomogeneous broadening line width. And besides, shorter sample length is needed to stop the probe restricted within the gap. The other side, thus even a weak dissipation in the gap (black curve shown in Fig. 2(c)) leads to a rounding or “smoothing” of the edge of the forbidden zone. Because of this it should be rather difficult to resolve narrow gaps experimentally. However fortunately, the periodicity of the standing wave needed in the experiment should be almost equal to the half-wavelength of the resonant transition from the excited state to the ground-state spin sublevel in NV diamond color centers, i.e., a ≃ 318.5 nm, which is just our case shown in Fig. 2(b). Then even for the black curve, it’s feasible to realize the band-gap and the high reflectivity.
Simultaneously, Fig. 3(a) show the optical properties is also sensitive to the length of period a. Symmetric band-gap splitting is obtained from a solid stack with periodicity a ≃ 318.5 nm that equal to the half-wavelength of the exciton radiation. When a is lager than that one, the gap will move to the frequency region below resonance (Δp > 0) - the so-called red detuning, while moving to above resonance (Δp < 0) - the so-called blue detuning when the period a is reduced. That result does accord with the character of the typical period material. The steep profile for red/blue detuning is due to the resonance absorption, and the other gap is symmetric around mid-gap because of falling far from resonance making the solid periodic structure essentially non-dissipative. The corresponding band-gap reflectivity and absorption spectrum with an array of length L ≃ 6.55×103 a are shown in Fig. 3(b).
Figure 4 shows theoretical results for the reflectivity coefficient versus frequency for the periodic square lattice correspond to different structure (d/a) of single layer. It is clear that the reflectivity is zero and most probe is absorbed when (d/a)=1 that corresponding to a continuous NV diamond sample without period. While when the ratio of the slab and the period is reduced, i.e., the thickness of solid dielectric slab is decreased, two symmetric band-gaps appear. The more reduced the ratio is, the closer the two band-gap reflectivity spectra are to approach to the resonance frequency, even combined to make one gap.
Finally, we emphatically study the effect of inhomogeneous width on the reflection spectrum. This can be carried out by reasonably reducing the number of active color centers in NV diamond, which will lead to the decreases of inhomogeneous broadening [32, 33]. According to the actual value Wab ≃ 375 GHz corresponding to the density of centers N=3×1018 cm-3 (case 1), we can properly estimate: Wab ≃ Δvjit =100 MHz corresponding to the effective density N=1×1015 cm-3 (case 2). A typical reflectivity and absorption spectrum of such a modification in broadened system is shown in Fig. 5.
From Fig. 5(a, b) it is clear that decreasing the broadening width one can increase the gap reflectivity, however, the price to be paid for this is a corresponding sharp reduction in the width of gap and the stop bands are smeared out. Obviously, that is not a paying business as shown in Fig. 5(b). We can see that even for the actual large value Wab ≃ 375 GHz, about 80% reflectivity can survive at resonance. Especially, notice there are many oscillations between two peaks adjacent to resonant frequency shown in the insets of Fig. 5(b, d), which is directly related to departures of the photon dispersion from linear (Different spacing corresponds indeed to different local slopes of the dispersion around the band edges, as clearly shown in Fig. 6(b, d)), and they are considered to degrade with the larger inhomogeneous broadening. So, the broadening of NV diamond does not as significantly or fatally affect the high band-gap reflectivity as we ever took for.
The reflectivity for a longer NV diamond stacks, L ≃ 6.55×104 a (2 cm), is also plotted in Fig. 5(c, d). Now, case 2 shows a distinct predominance in the spectral response, i.e., the perfect band-gap (reflectivity in the gap goes up to unity) appears to be symmetric with respect to resonance within the inhomogeneous width of NV. Whereas case 1 exhibits the same behavior as ones in Fig. 5(a) except for the region outside its wide gap where the absorption is increasing due to the longer sample. One can understand this phenomenon by calculating the typical values of absorption (labs ) and the extinction (lext ) length decided by their Bloch modes shown in Fig. 6 . Either of both above cases for NV diamond has its merits.
In summary, we have employed a straightforward approach to study the propagation of electromagnetic waves through a one-dimensional model of a periodic solid lattice where the resonantly absorbing and inhomogeneous broadening are taken into account necessarily. Detailed analysis are carried out for realistic NV diamond parameters, and obtain the following significant conclusions. First, with a careful design of the geometrical configuration of the external optical potential, such as the periodicity and the single layer structure, one could easily access these diverse gap structures and spectra are taken from different positions. Second, the results testify the advantage of NV diamond in achieving the high-reflectivity stop-bands, which is much more interesting and has been the focus of the paper. At first it was believed that only a material with a small inhomogeneous broadening could be used to achieve stopgap and high reflectivity, but present calculations are shown to work better than expected. The large inhomogeneous broadening in NV diamond makes for the gap appearing in the very large frequency region, and does not spoil the gap formation so remarkably as it was viewed. In addition, probably due to the use of repump laser in experiment, necessary to prevent reorientation of the NV centers in the diamond lattice, the large inhomogeneous width can instead be used to optically tune both the width of the gap and the degree of high-reflectivity in such lattice.
We note that single crystal diamond is needed for the experimental realization of our theoretical model because the use of high purity crystals is essential to increase decoherence times, and a magnetic field ~ 1 KG should be applied along the (111) direction to lift the degeneracy of the ms =± 1 states so that the spacing of 120 MHz can be got. The fabrication of diamond single crystal layers is difficult but not unfeasible. For example, one can use CVD or MPCVD to grow high quality single crystal diamond film [35, 36]. We also have assumed a diamond superlattice with vacuum as spacer, which seems not realistic to prepare now but is possible with the progressive technique improvement . The accuracy and prompt tenability of these gap structures also should have potential applications in quantum light storage , fast optical switching applications , design of more efficient lasers, and the potential technological device applications.
We would like to thank the support from the NSFC (Grant No. 10334010 and 10404009), and MAE (Grant ST China-Italy). Q. Y. He is grateful for kind help and simulating discussion with Prof. La Rocca from Scuola Normale Superiore in Pisa.
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31. It could be realized by using a rather large misalignment between the two beams. While in Fig. 2(b) the periodicity a ≃ 318.5 nm is the situation in which the two beams are exactly counter propagating, because it is just equal to the half-wavelength of the resonant transition from the excited state |a〉 to the ground-state spin sublevel |c〉 in NV diamond.
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