Amplified spontaneous emission lifetime based on the different phase matching modes in BaAlBO3F2 crystal
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JOURNAL OF OPTICS
J. Opt. 13 (2011) 035205 (6pp)
Amplified spontaneous emission lifetime based on the different phase matching modes in BaAlBO3F2 crystal Hong-ying Wang1,2 , Kang Li2,3 and N J Copner2 1 Physics Department, Xi’an University of Arts and Science, Xi’an 710065, People’s Republic of China 2 Faculty of Advanced Technology, University of Glamorgan, Pontypridd, Mid-Glamorgan CF37 1DL, UK
E-mail: firstname.lastname@example.org and email@example.com
Received 18 November 2010, accepted for publication 25 January 2011 Published 24 February 2011 Online at stacks.iop.org/JOpt/13/035205 Abstract Amplified spontaneous emission lifetime (ASEL) distribution characteristics of the new BaAlBO3 F2 (BABF) crystal have been calculated via the equations of emitted light intensity and amplified signal intensity gain. The ASEL distribution was calculated and discussed for three types of injection into the BABF crystal based on an optical parametric amplifier. We found ASEL was first distributed mainly in an ellipsoid and then in a torus with the increase of the pump phase matching angle when a monochromatic signal is injected. For the polychromatic signal pulse injection, ASEL is changed from a scattered distribution to a concentrated distribution. For the broadband pump, the ASEL distribution range is greatly expanded because of the increasing phase matching range. Keywords: parametric fluorescence, BaAlBO3 F2 crystal, fluorescence lifetime
(Some figures in this article are in colour only in the electronic version)
a type I LBO crystal around degeneracy . Later, OPF spectra have been calculated and demonstrated experimentally in a periodically poled medium presented by Vladislav Beskrovnyy et al [10, 11]. In 2004, Brustlein and co-workers obtained images of different fluorescence lifetimes in the picosecond range using parametric image amplification . Recently, an experimental study has been done investigating the characteristics of ultra-weak fluorescence using the picosecond optical parametric amplification [13, 14] technique and work using the fluorescence lifetime imaging amplification by Shifeng Du et al . Aluminum borate difluoride, BaAlBO3 F2 (BABF) [16–20], is an interesting and new nonlinear optical crystal belonging to the fluoroborate family. It is a negative uniaxial crystal, has a wide range of transparency at 165–3600 nm, has a relatively large nonlinear optical coefficient and has a relatively high optical damage threshold. Recently, the nonlinear optical properties of this new BABF crystal have been investigated , which is chemically stable, is not
1. Introduction During the non-collinear parametric generation, the amplified quantum noise is usually called amplified spontaneous emission or optical parametric fluorescence (OPF). Parametric fluorescence down-conversion is the nonlinear process whereby two photons (called the idler and signal) are created from a parent photon (called the pump photon). OPF can usually be observed in the absence of signal input and with a condition of high gain. Owing to OPF important applications in the medical and microbiology domains, its characteristics have attracted a great deal of attention in the past 40 years. Parametric fluorescence was first predicted in 1961 by Louisell et al . Since that time, comprehensive theoretical treatments have appeared [2–4] along with numerous experimental observations [2, 5–8]. Devaux and co-workers reported a theoretical and experimental study of the spatio-temporal properties of the spontaneous parametric emission generated in 3 Author to whom any correspondence should be addressed.
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H-Y Wang et al
Quantum mechanical treatment of the optical parametric interaction gives, in the undepleted pump approximation with no input signal, the parametric amplification gain G(φm , θm , λm ) which for the mode m is defined by the expression [9, 22] G(φm , θm , λm ) = ⎧
2 2 2 ⎪ k k ⎪ ⎪ g sinh L g2 − g2 − ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ k ⎪ ⎪ g> ⎨ 2
2 2 2 ⎪ ⎪ ⎪ k k ⎪ ⎪ g sin L − g2 − g2 ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k ⎩ g 2 (3) where (φm , θm ) are the propagation angles of the wavevector km associated with a spatial mode m with respect to the crystal axes. L represents the length of the crystal and the parametric gain coefficient g is written as
hygroscopic and possesses a moderate birefringence suitable for high-power solid-state ultraviolet (UV) light generation. In this paper, we focus on theoretical calculations and analysis of the emitted ASEL of the BABF crystal generated by a monochromatic signal, polychromatic signal and broadband pump in the optical parametric generation. When a monochromatic signal is injected into the optical parametric amplifier, the fluorescence lifetime is first distributed mainly in an ellipsoid and then in a torus. In addition, for the polychromatic signal pulse, ASEL is over a wide wavelength range with increasing phase matching angle of the pump. It is dispersed in the vicinity of 0.8 μm and 1.6 μm at the phase matching angle of 18.9◦ , respectively, and then it reduces to 650 ps over a 200 nm wavelength range concentrated on 1– 1.2 μm at a particular phase matching angle (θp = 20.9◦ ). When quantum noise is pumped by a wide spectrum laser, the parametric fluorescence phase matching range is greatly expanded. Its lifetime is distributed within a wide pump wavelength range.
2. Theoretical model of amplified spontaneous emission lifetime
g = 4πdeff [Ip /(2ε0 n mp n ms n mi cλms λmi )]1/2 .
Three-wave nonlinear optical frequency conversion processes are the analysis evidence of ASEL. According to the conservation of energy and momentum during the three-wave optical parametric interaction in media, if the phase match conditions are satisfied between the signal pulses and pump pulses, only the part of parametric fluorescence which goes through the pump pulse time window is propagated and amplified . Also, there is a fluorescence intensity change within the overlapping time of the signal and pump pulses. The intensity of parametric fluorescence is given by the expression d I (t) = −
1 I (t) dt τ (t)
For the relation (4), indices p, s and i refer to the pump, signal and idler, respectively. Ip is the pump intensity which is assumed to be constant during the interaction time duration between the pump and fluorescence signals. n and λ are the refractive index and wavelength of a mode m. The refractive indices of BABF are cited as follows:
n 2o (λ) = 2.6213 + 0.013 53/(λ2 − 0.012 04) − 0.010 55λ2 n 2e (λ) = 2.4833 + 0.011 78/(λ2 − 0.009 96) − 0.004 47λ2 . (5) For the BABF crystal with P 6¯ point-group symmetry the effective nonlinear coefficient deff for type I phase matching (o + o → e) is given by
in which τ (t) is the time of 1/e intensity of the maximum and is called the parametric fluorescence lifetime. When τ (t) = τ is a constant, the intensity of parametric fluorescence versus time is written as
I (t) = I0 exp(−t/τ )
deff = (d11 cos 3φ − d22 sin 3φ) cos θ
where φ and θ are the phase matching (PM) angles. The values of the d11 and d22 coefficients were calculated to be d11 = 0.165 pm V−1 and d22 = 1.32 pm V−1 , respectively. The phase velocity mismatch factor k in equation (3) is given by the following formula:
where I0 is the maximal value of parametric fluorescence intensity at time t = 0. In the optical parametric generation, the propagation time t of the amplified parametric fluorescence is limited by the pulse width of the pump. During the optical parametric amplification, parametric fluorescence is achieved when a strong pump pulse is phasematched with a weak input signal in a quadratic crystal. The amplified parametric fluorescence intensity is closely connected with the phase matching value. Therefore, the amplified optical parametric fluorescence lifetime depends on the phase matching value for the determined pump intensity and pump pulse width. The different phase matching modes are used to calculate and analyze ASEL. OPF intensity can be obtained through the achieved OPF gain, which can be calculated from equation (3). Therefore, the ASEL distribution image can be acquired by use of equation (2).
= |kp − kms − kmi |. k(θp , θs , φp , φs , λp , λs ) = |k|
Type I angle phase matching is used with the signal and idler fields polarized as ordinary waves and the pump field polarized as extraordinary waves in the negative birefringent BABF crystal. Figure 1 shows the orientations of the BABF crystal and non-collinear three-wave interaction fields. The optical axis is parallel to Z (n x = n y > n z ). The angle ρ is defined as the spatial walk-off angle of the pump wave. It is important to consider the case of a different non-collinear phase mismatch geometry which leads to the preferable lifetime for an OPF. The effect of a phase mismatch upon ASEL was systematically studied in the following three cases. In the following description, we investigate ASEL for λs = 800 nm, λp = 532 nm, Ip = 350 MW cm−2 , τp = 6 ns and L = 20 mm. 2
J. Opt. 13 (2011) 035205
H-Y Wang et al
Figure 1. The orientations of BABF crystal and the three-wave interaction fields.
Figure 4. ASEL seeded by a monochromatic signal versus signal phase matching angle in φs = φp = 90◦ plane.
Figure 3 represents the distribution image of ASEL with level lines for two different pump phase matching angles θp = 18.9◦ and 19.2◦ in the (θs , φs ) BABF crystal plane (with the pump plane wave propagation direction φp = 90◦ ). When θp increases, the fluorescence lifetime is first emitted in an ellipsoid (figure 3(a)) and then in a torus (figure 3(b)). These plot the most basic information for ASEL distribution status and its value of around 650 ps within the corresponding phase matching range when the signal phase matching angle and direction angle change. In particular, if the signal plane wave propagation direction φs = 90◦ , ASEL changes versus the signal phase matching angle as depicted in figure 4. It is clear that there is a good agreement between figures 3 and 4. With the increasing of θp , ASEL is firstly concentrated within a small angle range (θs ) of around 0.45◦ (θp = 18.9◦ ) centered on the pump beam, and then ASEL is scattered symmetrically in a gradually increasing phase matching angle range (θs ).
Figure 2. Phase matching configuration of non-collinear optical parametric fluorescence in a (θs , φs ) BABF crystal plane.
3. ASEL in (θs , φs ) plane For the phase mismatch in the (θs , φs ) plane and (θp , φp , λp , λs ) being constants, the effective gain G is determined from the phase mismatch vector k(θs , φs ). We define (θs , φs ) as the quasi-phase matching angular ranges where the amplification efficiency remains greater than 50%. The phase matching configuration of non-collinear optical parametric fluorescence in the (θs , φs ) plane of a BABF crystal is displayed in figure 2.
4. ASEL in (θs , λs ) plane For any given pair of conjugate signal and pump wavelengths, there may exist an optimum pair of emission angles, producing
Figure 3. ASEL distribution seeded by a monochromatic signal with two different propagation directions of the pump beam in the (θs , φs ) BABF crystal plane. (a) θp = 18.9◦ . (b) θp = 19.2◦ .
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Figure 7. The amplification parametric fluorescence lifetime versus different signal wavelengths for the pump phase matching angles of 18.9◦ , 20.9◦ and 21◦ .
Figure 5. Schematic diagram of phase matching geometry of non-collinear OPF in a (θs , λs ) nonlinear BABF crystal plane (φs = φp = 90◦ ).
In this section, ASEL is computed for the entire range of signal wavelength and angle combinations, within the domain of validity of the Sellmeier coefficients for the chosen crystal. This is done by θp incremented from 18.9◦ to 21◦ . A 3D plot can serve as a crude picture of ASEL in the parametric down-conversion as a function of wavelength and angle (see figure 6). For θp = 18.9◦ the ASEL is mainly emitted around 0.8 μm and 1.6 μm, respectively, as shown in figure 6(a). The amplification efficiency will be a maximum for the optimum
perfect phase matching. The phase mismatch velocity factor k(θs , θp , φs , φp , λs , λp ) is a wavelength and angle function of the signal and pump beam. The process of spontaneous parametric down-conversion will be strongest for these optimum combinations of wavelengths and angles. Figure 5 provides a phase matching configuration for noncollinear optical parametric fluorescence in a (θs , λs ) BABF crystal plane (with (φs , φp , θp , λp ) being constant and φs = φp = 90◦ ).
Figure 6. ASEL seeded by a polychromatic signal in a (θs , λs ) phase matching plane. (a) θp = 18.9◦ . (b) θp = 20.9◦ . (c) θp = 21◦ .
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matching angle θp . Graphs of the ASEL versus λs can be produced (see figure 7). One may notice the similarity between figures 6 and 7. It is clear that k(θs , λs ) is large except for signal wavelengths of 0.8 and 1.6 μm if θp = 18.9◦ . While if θp = 20.9◦ , then k(θs , λs ) is small between 1 and 1.2 μm. As a result, ASEL will be concentrated on the wavelength of 1–1.2 μm. With the increasing of the pump phase matching angle, the ASEL decreases within the wavelength range of 1– 1.2 μm.
5. ASEL in (θp , λp ) plane The phase matching vector diagram for the non-collinear quasi-phase-matching in the (θp , λp ) plane case for (φp , φs , θs , λs ) being constant and φs = φp = 90◦ is shown in figure 8. The variation in the pump wavevector kp that is due to a change in the pump beam propagation direction and wavelength (θp , λp ) will, in general, lead to a phase mismatch. As a result, the optical parametric fluorescence lifetime τ (θs , θp , φs , φp , λs , λp ) can be defined as τ (θp , λp ) in the (θp , λp ) plane. Figures 9(a) and (b) show in gray levels a numerical calculation of the ASEL calculated for the different pump beam propagation directions and wavelengths in the BABF crystal (θp , λp ) plane (for (φp , φs , θs , λs ) being constant and φs = φp = 90◦ ). For two different values of the central phase
Figure 8. Schematic diagram of phase matching geometry of non-collinear optical parametric fluorescence in a (θp , λp ) nonlinear BABF crystal plane (φs = φp = 90◦ ).
phase matching combinations that result in θp = 20.9◦ and it is over a 200 nm wavelength range (see figure 6(b)). If θp increases, ASEL decreases gradually for the variation of the signal wavelength and angle (see figure 6(c)). Here a special situation will be discussed, namely the signal phase matching angle θs is equal to the pump phase
Figure 9. The ASEL calculation characteristic for the different pump beam propagation directions and wavelengths in the BABF crystal (θp , λp ) plane (φs = φp = 90◦ ). (a) θp0 = 18.9◦ . (b) θp0 = 25◦ .
Figure 10. 2D plot of ASEL versus pump wavelength under the different specific pump phase matching angles (φs = φp = 90◦ ). (a) θp0 = 18.9◦ . (b) θp = 25◦ .
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matching angle θp0 (θp0 = 18.9◦ , θp0 = 25◦ , φp = 90◦ ), it can be seen that the phase matching range has been expanded, corresponding to the increasing of θp0 . When we choose some specific pump phase matching angles, 2D graphs of the ASEL versus λp can be depicted in figure 10 for two central phase matching angles θp0 (θp0 = 18.9◦ , θp0 = 25◦ , φp = 90◦ ). It is clear that there is an optimum pump wavelength as for every group of phase matching parameters, and the ASEL distribution range will increase gradually with the increasing of the pump central phase matching angle.
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6. Conclusions In conclusion, we have calculated and analyzed theoretically the parametric fluorescence lifetime of a BABF nonlinear crystal. The simulation results indicate the sensitive character of ASEL with respect to the phase matching. The ASEL distribution changes from an ellipsoid to a torus corresponding to the pump phase matching angle for the monochromatic signal. For the polychromatic signal pulse, the scattered ASEL distribution becomes gradually a concentrated distribution because of different pump phase matching angles, and then ASEL decreases gradually. Moreover, when we choose the broadband pump pulse, the OPF phase matching range has been expanded corresponding with the increasing of the pump central phase matching angle. Its lifetime is distributed within a wide pump wavelength range. The results provide a theoretical reference for further development and application of the new nonlinear BABF crystal.
Acknowledgments We would like to thank the kind referee for invaluable and positive suggestions, which have improved the manuscript greatly. This project was financially supported by the Young Teacher Natural Science Foundation of Xi’an University of Arts and Science of China (under grant Project no. kyc201018).
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Published on Mar 25, 2014
Amplified spontaneous emission lifetime (ASEL) distribution characteristics of the new BaAlBO3F2 (BABF) crystal have been calculated via the...