The Spectrum Characters Of Medium Driven By A Standing Wave And Slow Light Induced By Coherent Hole-burnings

In this thesis for doctorate that consists of three parts, we study coher-ently induced stop-bands in resonantly absorbing and inhomogeneously broad-ened doped crystals, the coherent hole-burnings induced by the bichromatic laserfield, and the new theory about slow light by the coherent hole-burnings.The spectrum and stop-bands of optical lattices inresonantly absorbing and inhomogeneously broadenedsolidsIn this part, we investigate that the inhomogeneously broadened doped solidssupporting electromagnetically induced transparency are viable materials whereoptically tunable photonic band gaps may be observed and results are shown forPr:YSO and N-V color center materials. Such crystals, when coherently driven bya strong pump beam in a suitably designed standing wave configuration, exhibit a tunable stop-band with rather di?erent values of the re?ectivity. The di?erentpropagating behavior is investigated and characterized for a weak probe acrossthe gap region of the two materials.(1) Coherently induced stop-bands in resonantly absorbing and inho-mogeneously broadened doped crystalsIn Fig. 1, a weak probe beam with frequencyωp and Rabi frequency ?ppropagates in the x-direction in the presence of a strong pump or coupling beamof frequencyωc and Rabi frequency ?c.Γij (i,j = a,b,c) are the populationrelaxation rates and the coherence decay rates. The pump is here retro-re?ectedupon impinging on a mirror of re?ectivity Rm forming a standing wave (S.W.)pattern within the sample. The resulting pump Rabi frequency varies periodicallyalong x, with a spatial periodicity a which is half the pump beam wavelengthλc.For a perfect (Rm = 1) standing wave ?c2(x) varies periodically and vanishes at thenodal positions where the medium remains absorbing. By slightly reducing themirror re?ectivity the pump intensity can however be made nowhere vanishing,and the nodes will be replaced by quasi-nodes. In terms of the parameterη=(1 ?√Rm)/(1 +√Rm) The strong pump intensity pattern modifies in a periodicfashion the probe absorption and dispersion as one moves along x from quasi-nodes to antinodes. In particular, at the quasi-nodes nearly complete suppressionof the probe absorption with a concomitant steep dispersion occurs within anarrow window centered around the b?a transition frequency. At the antinodes,instead, the Autler-Townes splitting is so large that the dielectric function is nearly exactly unity within the narrow frequency range where the photonic bandgap develops.In the framework of semiclassical theory, by solving the density matrix equa-tions in steady state, and averaging over the entire frequency range of the corre-sponding transitions in order to obtain the relevant macroscopic susceptibilityχ. Owing to the spatially periodic modulation induced by the pump (3.7), aprobe with frequencyωpωab propagates as in a one-dimensional photoniccrystal with periodicity a =λc/2, whereλc is approximately equal to the proberesonant transition wavelengthλab. The opening of a photonic stop band canbe anticipated to occur at the Brillouin zone boundaryπ/a. Using the transfer-matrix method, we can obtain the structure of the stop-band.Fig. 2 show the re?ectivity, transmission, and absorption spectra are shownwhen (A) no pump, (B) a c.w. pump and (C,D) s.w pump, respectively, are used.Considerably larger re?ectivity as in (C), are instead obtained using a configura-tion in which the forward and backward control beams are slightly misaligned byan angleα. The spatial periodicity in this case is given by (λc/2)/cos(α/2). Bycomparing the three results, it appears that one attains an appreciable band-gapre?ectivity when pumping in a standing wave configuration occurs.When we instead increase the Rabi frequency to ?0 = 10 MHz and unbalancethe standing wave potential (η= 0.05), yet keeping the same misalignment (α=0.98 mrad), a well developed band-gap characterized byκπ/a andκ= 0appears as shown in Fig. 3, which is derived from the real and imaginary partsof the Bloch complex wavevector (Eq. 8). Within the gap,κ= 0 correspondsto re?ection rather than absorption of the probe, althoughκis not large enoughto stop the probe within a few centimeters long sample.Figure 4 shows the re?ectivity spectra of N-V diamond respectively when(A) no pump and (B) a modified standing wave pump is on. It is clear thatwithout the pump, the re?ectivity is very low and the solid medium is completelyopaque to the signal near resonance. The peak of the re?ected beam becomeshowever a substantial fraction of the input signal beam, and much larger thanin the case of Pr:YSO above, when a standing wave coherent driving pump ison. In both cases the transmissivity is extremely small and is not reported here.Furthermore, when a continuous wave (C.W.) pump is present we can show thatthe re?ectivity becomes even smaller than in (A) while the transmission exhibits a very small peak due to the usual electromagnetically induced transparency.Both values of the re?ectivity and transmissivity are however pretty much van-ishing and will not be reported here as well. The incipient photonic band-gapstructure around the lowest photonic gap corresponding to the case of Fig. 6(B)is reported instead in Fig. 5.(2)Dynamic control of the photonic stop-bands formed by standingwave in inhomogeneous broadening solidsFigure 6 shows the two e?ects of the intensity enhancement of SW Rabifrequency: the width of the band gap will increase, whileκwithin the gap edgesis reduced, with the increment of the intensity of SW Rabi frequency. So, we needto combine these"good"and"bad"e?ects in order to obtain the better gap.Next we consider the function of tunable parameterηand the misalignmentα, while keeping pump Rabi frequency. It can be found from Fig. 7(a) that theband gap in the first Brillouin zone boundary is barely defined forη= 0, butwhen we slightly changeη, a well-developed band gap characterized byκ=π/aandκ= 0 appears as shown in Fig. 7(b). The explanation to this phenomenonis likely to the EIT disappears at the nodes of standing wave forη= 0, which isdue to the Rabi frequency equals zero at those points. It is known that withoutthe pump field, the re?ectivity is very low and the solid medium is completelyopaque to the signal light near the resonance. Figure 8 shows that the gap widthandκwithin the gap become smaller than ones in Fig. 7(b) by reducing thetwo counter-propagating beams misalignmentαalong x, by which one can easilychange in fact the spatial periodicity given by a = (λc/2)/cos(α/2).Finally, we consider the e?ect of incident angleθof the probe (nonnormalincidence). In Fig. 9 fixing all other parameters and only changing the angle ofincidence we plot the imaginary part of the Bloch wave vector near the Brillouinzone boundary. From this figure, we can see that the position and width ofPBG in a solid SW-EIT photonic crystal can be made relatively sensitive to theincident angle. (3)E?ects of resonant absorption and inhomogeneous broadening onre?ection and absorption spectra of optical lattices in diamond NVcentersUsing the transfer-matrix method, the e?ects of absorption and inhomoge-neous broadening, in one-dimensional optical lattice constructed from inhomo-geneously broadened spin transitions of nitrogen-vacancy color centers in singlecrystal diamond (NV diamond), on the re?ection and absorption spectrum arepresented.Further analysis show that, in realistic periodic stacks of the NV diamondcolor centers, modulating the geometrical configuration of the external opticalpotential, the absorption lineshape scale, and the inhomogeneous broadening, onecould easily access the diverse gap structures and a high band-gap re?ectivity.These pretty useful calculations hold more potential for e?ective control of thelight-matter interaction and realization in practice.Figure 10: (a) the gap could survive in the presence of strong dissipationexcept for the width of gap decreasing; (b) for narrow absorption profile (blue)the re?ectivity within the resonance region is higher than the ones in broadenedcases (red and black), which is due to the corresponding higher refractive index(η)shown under Fig. 10(a). This e?ect will be remarkable for the whole gapdeveloping within the NV resonance region, which may be achieved through ap-preciable modifications of the NV lattice periodicity a as shown in Fig. 10(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 absorp-tion profile. So the band-gap only survives for very small absorption bandwidth(blue).we emphatically study the e?ect of inhomogeneous width on the re?ectionspectrum. This can be carried out by reasonably reducing the number of activecolor centers in NV diamond, which will lead to the decreases of inhomogeneousbroadening. According to the actual value Wab 375 GHz corresponding to the From Fig. 11(a, b) it is clear that decreasing the broadening width one canincrease the gap re?ectivity, however, the price to be paid for this is a corre-sponding sharp reduction in the width of gap and the stop bands are smearedout. Even for the actual large value Wab 375 GHz, about 80% re?ectivity cansurvive at resonance. Especially, notice there are many oscillations between twopeaks adjacent to resonant frequency shown in the insets of Fig. 11(b, d), whichis directly related to departures of the photon dispersion from linear (Di?erent spacing corresponds indeed to di?erent local slopes of the dispersion around theband edges, as clearly shown in Fig. 12(b, d)), and they are considered to degradewith the larger inhomogeneous broadening. So, the broadening of NV diamonddoes not as significantly or fatally a?ect the high band-gap re?ectivity as we evertook for.The re?ectivity for a longer NV diamond stacks, L 6.55×104a (2 nm), isalso plotted in Fig. 11(c, d). Now, case 2 shows a distinct predominance in thespectral response, i.e., the perfect band-gap (re?ectivity in the gap goes up tounity) appears to be symmetric with respect to resonance within the inhomoge-neous width of NV. Whereas case 1 exhibits the same behavior as ones in Fig.11(a) except for the region outside its wide gap where the absorption is increasingdue to the longer sample. One can understand this phenomenon by calculatingthe typical values of absorption (labs) and the extinction (lext) length decided bytheir Bloch modes shown in Fig. 12. Either of both above cases for NV diamondhas its merits.Coherent hole-burnings induced by bichromatic laserIn a Doppler broadened three-levelΛ-type system driven simultaneously bya coupling laser and two equal-amplitude saturating laser fields with a frequencyseparation 2δ, the absorption spectrum of a weak probe laser exhibits multipledeep coherent hole-burnings (CHBs) with controllable numbers, widths, depths,and positions. More significant, changingδor lasers directions, CHBs can de-generate into narrower and deeper hole-burnings where the slope of the refractiveindex is very steep. The multiple narrow spectral CHBs in a single absorptionprofile are expected to have potential applications in high density storage, opticalinformation processes, and slow-light.The three-levelΛ-type atomic system under consideration is shown in Fig.13. Intermediate bichromatic laser field acts on the same transition as the probefield, and the two field components of which are used as the saturating fields. In the framework of semi-classical theory, by solving equation of densityoperator, utilizing linear response, Laplace transform, and quantum regressiontheories, we can obtain the Doppler-broadened probe absorption spectrum. It isclear that eight CHBs in total could be observed if no degeneracy occurs, which istrue for the case ofδ= 50 MHz(blue, solid) and 40 MHz(red, thin solid) as shownin Fig. 14. Whenδ= 20 MHz(black, dashed), however, there will exist only fourdeeper CHBs in the probe absorption spectrum because the di?erence betweentwo adjacent CHBs as shown by black dots is so small compared to the width ofhole-burning that they are not separately distinguishable. [?] We note that thesedegenerate CHBs are not a simple combination of two adjacent hole-burnings,but deeper due to the interference enhancement of the saturating amplitude thatoccurs when the two field components of bichromatic laser come together. Fromthis figure, we can also see that the CHBs related toωs1(ωs2) will move to positive(negative) detuning direction withδdecreasing.The solid and dotted curves in Fig. 15(a) correspond to the absorption spec-trum with both coupling and monochromatic saturating beams on. We can findthat when we increase the intensity of the monochromatic saturating field, theEIT window remains unchanged but the four CHBs become deeper and wider asexpected (curve b). Instead, if we use the bichromatic saturating fields with fre-quency di?erence 2δ= 40 MHz while keeping the saturating intensity unchanged, we find from curve c that the four CHBs become much deeper and clearer due tothe enhancement of the saturating intensity which results from the interferencebetween the two field components. If simply adding a second monochromaticsaturating field with same intensity, we could only obtain the wide and ?at CHBs (curve d) due to the absence of coherence between two monochromatic saturat-ing fields. Obviously, the deeper and narrower CHBs induced by the bichromaticsaturating field are very useful for the application in high density and precisionoptical information storage.We also find from Fig. 15(b) that the sum of the phase of the two fieldcomponents Es1, Es2 plays a crucial role in the absorption spectrum. It is clearlyshown that whenΦ1+Φ2 = 0 the spectrum is not changed, and whenΦ1+Φ2 =πthe two CHBs near the EIT window become very ?at and almost disappear. Thisphase sensitivity can be beneficial to the control of the hole-burnings in order torealize reading and writing of optical information.Slow-light induced by the coherent hole-burningsWe show that the simultaneous application of a saturating co-propagatingbeam and a coherent counter-propagating beam, allows us to burn a very nar-row spectral hole in the inhomogeneous absorption line of the optical transitionin a Doppler-broadened medium. The resulting rapid spectral variation of therefractive index leads to a large value of the group index. In atomic vapors, wecalculate group indices of the order of 104 with a transmission of 60%. Especially,it is benefit from the bichromatic saturating field to obtain much larger slow groupvelocities with gain simultaneously for light pulses at di?erent frequencies. Thecalculations include all coherence e?ects. The three-levelΛ-type atomic system under consideration is shown in Fig.16. An intermediate laser of wavelengthωs and Rabi frequency ?s as shown inFig. 16(a), which acts on the same transition as the probe field, is used as themonochromatic saturating field. Figure 16(b) show another excitation schemewhere the monochromatic saturating field is replaced by a bichromatic field withtwo frequency componentsωs1 andωs2 and the frequency di?erence is 2δ. The calculated group index ng as a function of the detuning of the probe fromthe atomic transition is shown in Fig. 17(a). The deepest one at the resonantfrequency is so narrow that bring the highest slow-light, which is larger than thecapability of general hole-burning in the same condition. The region of resonanceis bowed up in Fig. 17(a'-c'). The position of slow-light will move to the detuningof saturating field if we change the resonant condition. Simultaneously changingthe detuning of coherent pump will get higher ng shown in Fig. 17(a"-c"), Whichis due to more saturated atoms.In Fig. 18, using the bichromatic laser fields to replace the monochromaticsaturating filed, we can obtain six holes. Thereinto there are two deep and sym-metrical coherent hole-burnings near the resonant frequency, which induce twolarge slow-light frequency. The two slow-light position will move to resonancewith the frequency di?erence between the two components of bichromatic laser,at the same time, the group index ng become the order of 104. Clearly form the Fig. 19 the slow-light induced by the bichromatic coherent hole-burnings hasgreat advantage under this condition comparing with EIT with the same widthand depth of transparency: higher slow-light with gain at two frequencies can beslow down.In conclusion, we discussed in detail how to realize the dynamic control ofthe photonic stop-bands formed by standing wave in inhomogeneous broadeningdoped crystals. We also investigated the new theory about coherent hole-burnings induced by the bichromatic laser fields in a Doppler broadened three-levelΛatomic system, which is expected to have potential applications in high densityof storage, optical information processes, and slow-light. Finally, we studied indetail the slow-light realized by the coherent hole-burnings. Our work can help tofurther understanding of various atomic coherent phenomena in the interactionof atoms and coherent fields.