Temperature tunable random laser using superconducting materials

Optics Communications 285 (2012) 1900–1904

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Temperature tunable random laser using superconducting materials Abbas Ghasempour Ardakani a, Mehdi Hosseini a, Ali Reza Bahrampour a,⁎, Seyed Mohammad Mahdavi a, b,⁎⁎ a b

Department of Physics, Sharif University of Technology, Tehran, Iran Institute for Nanoscience and Nanotechnology, Sharif University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 10 October 2011 Received in revised form 9 December 2011 Accepted 10 December 2011 Available online 22 December 2011 Keywords: Random laser FDTD method Superconductor layer

a b s t r a c t We propose that spectral intensity of superconductor based random lasers can be made tunable by changing temperature. The two fluid model and wavelength dependent dispersion formula have been employed to describe the optical response of the superconducting materials. Random laser characteristics have been calculated using the one dimensional FDTD method. Our simulation results reveal that the emission spectrum can be manipulated through the ambient temperature of the system. It is observed that transition from metal phase to pure superconducting phase leads to the enhancement of the laser emission. Furthermore, spatial distribution of the fields in one dimensional disordered media is very sensitive to the system temperature. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Lasing in disordered media is a subject of growing interest in recent years and has been widely studied both theoretically and experimentally [1–15]. So far many models have been constructed for investigation of random lasers [6–10]. Among them, a time dependent model was presented by Jiang et al. in which Maxwell–Bloch equations are solved numerically by using FDTD method for one dimensional (1D) case [11], and it was extended by Sebah and co authors to two dimensional (2D) case [12]. Many properties of random lasers can be explained by this model [11–15]. To use random lasers, the method used to externally control random laser line-width, intensity and wavelength are important. Tuning of the output of random laser with temperature was demonstrated by Lawandy et al. using a dye dissolved in polymethylmetalcrylate matrix [16]. Another scheme was also reported by K. Lee et al.; they used a novel system based on the lower critical solution temperature (LCST) mixture containing a high gain dye [17]. The LCST materials can be reversibly transformed from a transparent state to a highly scattering colloid, as the temperature is increased above room temperature (41 °C) [17]. So scattering length and diffusion coefficient in disorder active system based on this material as a scatterer are temperature dependent and tuning the line-width of random laser emission is possible. Wiersma et al. made a liquid crystal based random laser, in which minor change in the temperature leaded to significant change in the laser emission behavior [18]. In this paper, a novel random laser using superconducting materials has been proposed that its emission

⁎ Corresponding author. ⁎⁎ Correspondence author to: S.M. Mahdavi, Institute for Nanoscience and Nanotechnology, Sharif University of Technology, Tehran, Iran. E-mail addresses: [email protected], [email protected] (A.R. Bahrampour), [email protected] (S.M. Mahdavi). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.052

behavior is temperature dependent. Superconducting materials were used in photonic crystal (PC) previously [19–23]. Due to the damping of electromagnetic waves in metals, metallic elements in photonic crystal were replaced with superconducting (SC) elements. The dielectric function of SC depends on superconducting gap and SC state which can vary with external parameters such as temperature and magnetic fields. As a result the optical properties of PC can be controlled externally. Using superconducting materials in random lasers provides a method for control of laser emission with temperature. Here, random lasers based on SC element are investigated and lasing emissions are calculated at different temperatures. Our results reveal that lasing spectrum and electric field wave functions depend on temperature and emission intensity is enhanced as superconductive properties increases. Remaining parts of this paper are organized as follows. In Section 2, we introduce a general form of time dependent theory in onedimensional case in the presence of dispersive superconducting element. In Section 3, we calculate spectral intensity and spatial distribution of electric fields for different temperature. Finally, the paper is concluded in Section 4 with some conclusions. 2. Theoretical model For simplicity, we calculated one-dimensional (1D) random lasers which could reveal qualitative properties of 2D and 3D random lasers. As shown in Fig. 1, our 1D system is made of binary layers consisted of two dielectric materials. The black layer with dielectric constant of εSC(ω) and random variable thickness of dsn, simulate the superconducting materials (YBCO). The white layer in Fig. 1, with dielectric constant of ε = 4 and random thickness of dgn simulates active or gain media (Nd:YAG). The value of dsn and dgn is randomly selected in the range [300 nm–480 nm]. The system consists of 76 pairs of binary layers, and the length of system is 60 micrometer.

A. Ghasempour Ardakani et al. / Optics Communications 285 (2012) 1900–1904

1901

where α = 1 − t β is the superconducting fluid fraction; ωp ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e nt =mε0 is the plasma frequency corresponding to the total electron Fig. 1. Schematic for one dimensional random system in which the blue and white layers simulate the SC and active layers respectively.

density. It should be noted that some authors have applied Eq. (4) for tunable SC photonic crystals in the visible frequencies [19,35]. The dielectric constants of two layers in one-dimensional random system are given as: 

For isotropic, non-magnetic and active media, the Maxwell's equations are ∂Eðx; t Þ ∂Hðx; t Þ ¼ μ0 ∂x ∂t

ð1Þ

∂H ðx; t Þ ∂Dðx; t Þ ∂P transition ðx; t Þ ¼ þ ∂x ∂t ∂t

ð2Þ

where Ρtransition is the polarization density due to the specific atomic transition from which gain or amplification is obtained; by setting Ptransition = 0 the above equations can be used for passive medium; D is the electric displacement. In the SC materials for simplicity, we consider the electromagnetic property by using the two-liquid Gorter–Kazimir model which describes the system as an admixture of two independent carrier liquids, the super and the normal electrons [22–25]. The dielectric constant of the superconductors is a complex function of the exciting frequency [22–26]: " # 2 2 2 e nn τ ns e nτ  s  εðωÞ ¼ 1− þ −i mε0 1 þ ω2 τ2 ω2 mε0 ω 1 þ ω2 τ2

ð3Þ

where e, m, nn, ns and τ are the electron charge, electron mass, normal electron concentration, super electron concentration and relaxation time of normal electron, respectively. The normal electron concentration (nn ðT Þ ¼ nt t β ) is an increasing function of the normalized temperature  (t = T/Tc ), while the super electron concentration (ns ðT Þ ¼ nt 1−t β ) is a decreasing function of t [27–29]. Here nt is the total electron concentration of the superconductor; Tc is the critical temperature of the superconductor and β is about 2 and 4 for the high temperature (HTS) and low temperature superconductors (LTS), respectively. The superconductor gap is a decreasing function of temperature and approaches to zero as the temperature goes to the critical value Tc. Below the critical temperature (T≪Tc), at the frequency corresponding to the superconductor gap, the superconductor resistance has a sharp threshold. The superconductor gap of YBCO is about 30 meV [30], and the absorption coefficient which is proportional to imaginary part of the right hand side (RHS) of the Eq. (3) is approximately zero for frequencies below the superconductor gap and increases for photon with energy larger than the superconductor gap. However variation of imaginary part of RHS of Eq. (3) is negligible compared to its real part. This effect makes superconductor as a suitable material for tunable photonic crystals in the near infrared region [31,32]. The superconductor threshold wavelength is the vanishing refractive index wavelength. For the YBCO high temperature SC, the threshold wavelength is in the range of (1260 nm to 1880 nm) [31,33,34], which are in the infrared region. In the vicinity of the optical band gap of superconductor the absorption energy is not negligible and has the significant effects on the optical behavior of superconductor. The optical gap of YBCO is in the range of (1.4 eV–1.9 eV) [31,33,34] which is larger than the photon energy of Nd:YAG lasers. So, in the near infrared region, the dielectric constant of the superconductor layer is approximately expressed as:

2

ε SC ðωÞ ¼ 1−ωp



α 1−α þ ω2 ω½ω−iγ

 ð4Þ

εðx; ωÞ ¼

4 for active medium εSC ðωÞ for supercinducting medium

ð5Þ

Therefore the dielectric constant of the superconducting materials can be tuned by changing the temperature. In order to simulate the superconductor layers as a dispersive media in FDTD, Eq. (5) must be put into the sampled time domain. The detail of transformation from frequency to time domain is given in the appendix. The active medium (Nd:YAG) is considered as a four-level system [36]; the third level (N3) and the second level (N2) are called the upper and lower lasing level respectively. The life time of levels N4, N3 and N2 are τ43, τ32 and τ21 respectively. The rate equations for this system are as follows [11,37]: dN4 ðx; t Þ N ðx; t Þ ¼ P r N1 ðx; t Þ− 4 dt τ43

ð6Þ

dN3 ðx; t Þ N4 ðx; t Þ N3 ðx; t Þ E dP transition ¼ − þ dt τ43 τ32 ℏωl dt

ð7Þ

dN2 ðx; t Þ N ðx; t Þ N 3 ðx; t Þ E dP transition ¼− 2 þ − dt τ21 τ32 ℏωl dt

ð8Þ

dN1 ðx; t Þ N ðx; t Þ ¼ −P r N1 ðx; t Þ þ 2 dt τ 21

ð9Þ

where Ni (i = 1–4) are the population density in level i; Pr is pumping rate which is proportional to the pump power and is assumed to be constant here. The ωl = (E3 – E2)/ℏ is lasing frequency between level E dP transition 2 and 3; the stimulated emission is given by the term ℏω . For dt l single electron case, the polarization density is obtained from the following equation of motion [11,37]: 2

d P transition ðx; t Þ dP ðx; t Þ 2 þ ωl P transition ðx; t Þ ¼ kΔNðx; t ÞEðx; t Þ ð10Þ þ Δωl transition dt dt 2

where Δωl = 1/τ32 + 2/T2 is full width at half maximum linewidth of atomic transition; T2 is the mean time between dephasing events; ΔN = N2 – N3 is the population inversion and k = 6πε0c 3/ωl2τ32 is a constant. The amplification line-shape derived from Eq. (10) is Lorentzian homogeneously broadened when ΔN is independent of time. The values of those parameters that will be used in our simulation are taken as: T2 = 2 × 10 − 14 s, τ43 = 1 × 10 − 13 s, τ32 = 1 × 10 − 10 s, 4 P Ni ¼ 3  1024 m−3 and νl = ωl/2π = 2.82 × τ21 = 1 × 10 − 11 s, N ¼ i¼1

10 14 Hz (λl = 1064 nm) [36]. The life time of level 3 is chosen shorter than real value in order to reduce the computation times as performed in previous works [12]. The high temperature superconductor YBCO is considered in the following calculations. Its parameters in normal state are ωp = 1.67 × 10 15 rad/s, γp = 1.34 × 1013 rad/s and Tc = 91 K [38]. When the active system is pumped, the electromagnetic fields can be calculated using different methods such as transfer matrix method, effective refractive index method and FDTD method. In this paper, we apply FDTD method to solve Eqs. (1–2) and (6–10). In order to simulate an open system, the Liao method is used to impose absorbing boundary condition (ABC). Due to numerically solving of Maxwell's equations, boundary conditions for the fields at the interface between two media are automatically satisfied. The space and time increment are taken to be Δx = 10 nm and Δt = 1.67 × 10 − 17 s respectively. A

A. Ghasempour Ardakani et al. / Optics Communications 285 (2012) 1900–1904

Gaussian electromagnetic pulse with an arbitrary amplitude and duration time of 10 − 15 s is used to simulate spontaneous emissions in the system. The electric field is recorded after 4 × 10 6 Δt, during a time window of length of 10 5 Δt at all nodes in the system. By taking the Fourier transform of recorded fields, emission spectrum is obtained. 3. Results and discussion

15 10 5

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gain and dielectric media, so quality factor and resonance frequency of the formed cavities depend on the refractive index contrast and absorption of materials. As mentioned above, dielectric constant of superconducting materials is sensitive to environment temperature. Varying the real part of the dielectric constant of SC layers with temperature can adjust the scattering strength. On the other hand external control of the imaginary part of the dielectric constant with temperature changes the absorption of different random laser modes. To clarify the mechanism of tuning the random laser based on superconducting layers, real and imaginary parts of the dielectric constant of YBCO layer as a function of wavelength for different temperatures are presented in Fig. 4(a) and (b) respectively. It can be seen that the real part does not change with temperature in the vicinity of 1064 nm, while the imaginary part is considerably temperature dependent. As the temperature is increased, the imaginary part also increases. For the case T = 91 K, since the modes have larger loss, the quality factors of which are smaller. Only

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Fig. 3. Intensity and wavelength of different modes in random laser system at different temperatures.

wavelength(nm) 3

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T=91K T=81.3K T=64.7K T=40.9K T=0K

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intensity(a.u)

In order to study the influence of the temperature on the random laser characteristics, five samples with different values of α are considered in this section. By employing the method described above, the spectral intensity versus wavelength is calculated with varying α from 0 to 1, while keeping pumping rate to be 2 × 10 7 s − 1. Five typical spectral intensities are presented in Fig. 2, as shown in Fig. 2(a); when the value of α is zero (T = 91 K) (corresponding to metal phase), there are two peaks in the emission spectrum at wavelength of 1037 nm and 1053 nm. As can be seen in Fig. 2(b), with increasing the value of α (α = 0.2 (T = 81.3 K)), three peaks emerge in the emission spectrum at wavelength of λ1 = 1037 nm, λ2 = 1053 nm and λ3 = 1070 nm and the peak value of λ2 is more than that in Fig. 2(a). In addition, it is observed that the intensity of mode λ2 is more than that of two other modes. By further increasing the α value (α = 0.5, 0.8 and 1(T = 64.7 K, 40.9 K and 0 K)), number of peaks and their intensities are increased obviously as shown in Fig. 2(c–d). These above results indicate that increasing the value of α from 0 to 1 leads to the enhancement of laser emission and increase in the number of lasing modes. The enhancement of laser intensity is due to the reduced photon absorption in the superconductor layers with increasing the value of α. In other words, the transition from metal phase to the pure superconducting phase causes changing in the random lasing emission characteristics. To show obviously the effect of temperature on the spectral intensities, the wavelength and intensity of different modes at different temperatures are shown in Fig. 3. Anderson localization of electromagnetic waves in active disordered media provides feedback for different modes. As is well known, photon localization requires very high refractive index contrast between the

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mode intensity(a.u)

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wavelength(nm) Fig. 2. The spectral intensity in arbitrary units versus wavelength for various values of α: (a) α = 0; (b) α = 0.2; (c) α = 0.5; (d) α = 0.8 and (e) α = 1.

A. Ghasempour Ardakani et al. / Optics Communications 285 (2012) 1900–1904

a

are far away from the center frequency have low loss and can be excited. Consequently, number of lasing modes increases with decreasing temperature. To study the effect of temperature on the spatial distribution of electric field and the position of the formed random cavities, the distribution of electric field in the length of random system at 4.1 × 10 6 Δt with α varying from 0 to 1 is calculated and results are shown in Fig. 5. For the case of α = 0 as observed in Fig. 5(a), there are four localization centers. As shown in Fig. 5(b), when the value of α increases to 0.2, number of localization centers is increased. It is in agreement with the increase in the number of lasing modes. As the value of α increases from 0.5 to 1, the magnitude of fields in various localization centers is increased (see Fig. 5(d–e)). The results in Fig. 5 demonstrate that spatial intensity distribution in random lasers based on superconducting materials is sensitive to the temperature. As a result, varying the temperature can be used to externally control random laser characteristics and tune random lasers as shown in Figs. 2, 3 and 5. The temperature tunable random laser may be used in various applications such as remote temperature sensing. Although, the discussion given above is for one dimensional random laser, it is expected that laser action in 2 and 3 dimensional disordered media can be made tunable by means of the temperature. It is well known that there are some differences between operation of random lasing in one, two and three dimensional disordered media [39]. Therefore, investigation of two and three dimensional superconductor based random lasers is interesting and is under investigation.

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those modes whose gain is large enough can lase. The modes whose frequencies are closer to the center frequency of gain line-shape have higher gain. Therefore, meeting the threshold condition for these modes is possible. When temperature is decreased, the loss of modes also decreases. In this case, the modes whose frequencies

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4. Conclusions One dimensional superconductor based random laser has been proposed and numerically investigated using the FDTD method. The results show that with the transition from normal state to superconducting state, when temperature is decreased, the emission spectra exhibit remarkable modifications. Therefore, emission spectrum of random laser can be tuned by adjusting the ambient temperature of the system. It is also observed that spatial distribution of the electric

electric field(a.u)

Fig. 4. The real part (a) and imaginary part (b) of the dielectric constant of YBCO layer versus wavelength.

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length(micron) Fig. 5. Spatial distribution of electric field versus the length of system corresponding to various values of α: (a) α = 0; (b) α = 0.2; (c) α = 0.5; (d) α = 0.8 and (e) α = 1.

1904

A. Ghasempour Ardakani et al. / Optics Communications 285 (2012) 1900–1904

field strongly depends on the temperature of the system. The two and three dimensional superconductor based random lasers are under investigation and will be appeared in near future.

The above integrals can be approximated as a summation in the sampled time domain.

Appendix

References

The electric displacement in the frequency domain is given as follows: ^ ðωÞ ¼ εðωÞE^ ðωÞ ¼ ðε þ χ ^ 1 ðωÞ þ χ ^ 2 ðωÞÞE^ ðωÞ D 0 ε ω2 α − 0ω2p

ðA:1Þ

ε ω2p ð1−α Þ − 0ωðω−iγ Þ .

and χ 2 ðωÞ ¼ The relation between where χ 1 ðωÞ ¼ D and E in the time domain can be written as a convolution integral.     t ′ ′ ′ ε x; t−t E x; t dt Dðx; t Þ ¼ ε0 ∫−∞ ⌢

ðA:2Þ

−iωt ⌢ 1 ∞ where ε ðx; t Þ ¼ 2π ∫−∞ εðx; ωÞe dω. It is easy to show that

χ 2 ðt Þ ¼

 ω2p  −γt U ðt Þ 1−e γ

ðA:3Þ

where U(t) is Heaviside function. To calculate χ1(t), it is sufficient that ^ ðωÞ is replaced by ω(ω − iγ′)where γ′ is a ω2 in the denominator of χ small number which is negligible compared to ω. So χ1(t) can be written as χ 2 ðt Þ ¼

 ω2p  −γ ′ t U ðt Þ: 1−e ′ γ

ðA:4Þ

The Eq. (A.2) can be written as Dðt Þ ¼ F 1 ðt Þ þ G1 ðt Þ þ F 2 ðt Þ þ G2 ðt Þ

ðA:5Þ

where F 1 ðt Þ ¼

G1 ðt Þ ¼

ε0 ω2p α γ′

t   ∫0 E t ′ dt ′

−ε0 ω2p α γ′

t

∫0 e

−γ′ ðt−t ′ Þ

ðA:6Þ   E t ′ dt ′

F 2 ðt Þ ¼

ε0 ω2p ð1−α Þ t  ′  ′ ∫0 E t dt γ

G2 ðt Þ ¼

−ε0 ωp ð1−α Þ t −γðt−t ′ Þ  ′  ′ ∫0 e E t dt : γ

ðA:7Þ

ðA:8Þ

2

ðA:9Þ

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