- Email: [email protected]
Analogy between hole shape in two-step pulsed spectral hole burning and time-dependent zero-phonon line shape
1 March 1998
Optics Communications 148 Ž1998. 36–40
Analogy between hole shape in two-step pulsed spectral hole burning and time-dependent zero-phonon line shape Inna Rebane
1
Institute of Physics, Riia 142, EE-2400 Tartu, Estonia Received 28 May 1997; revised 16 October 1997; accepted 21 October 1997
Abstract The two-step persistent spectral hole burning ŽPSHB. of a hole in the inhomogeneous distribution function ŽIDF. of the zero-phonon line ŽZPL. optical transition frequencies in a three-level system by two successive light pulses, in the case of an exponentially increasing and decreasing first coherent pulse and an infinitely short second pulse, is considered. The time-dependent ZPL in the luminescence spectrum in the recording scheme with the time gate in front of the spectrometer with high resolution in the case of exponential opening and closing of the time gate is calculated. It is shown that the time-dependent ZPL in the luminescence spectrum is described by a formula which is analogous to the formula describing the spectral hole in IDF. The effect of compensation of the natural linewidth of the excited level and the parameter describing the exponential increase of the transparency of the time gate is presented. q 1998 Elsevier Science B.V. PACS: 78.47 Keywords: Two-step persistent spectral hole burning; Time-dependent zero-phonon line; Line narrowing
1. Introduction Persistent spectral hole burning w1,2x ŽPSHB. is a method of very high resolution spectroscopy of impurities in solid matrices providing a number of applications. The capability of PSHB materials for applications is determined by the ratio G inh : G hom , where G inh is the inhomogeneous and G hom the homogeneous linewidth of the ZPL. The minimum width of a PSHB hole in IDF in the one-step PSHB with monochromatic light is G hom and the minimum width of a spectral hole measured in conventional spectra is 2 G hom w2,3x. A theory of PSHB with light pulses of arbitrary shape and duration was proposed in Refs. w3–5x. In a three-level system w3,4x, it is possible to eliminate the energy-relaxation-induced hole width g 1. As
1
E-mail: [email protected]
a result the holes in IDF are narrower than G hom s g 1 q Gphase , where Gphase is the pure phase relaxation broadening Žmodulation broadening.. In the theory of time-dependent Žtransient. spectra the scheme used for measurement of the spectrum is important. Two schemes of one-gate measurement have been considered: Ži. time gate placed behind the spectral device Žthe first scheme was considered in Refs. w6–9x., Žii. time gate in front of the spectral device Žthe second scheme was considered in Ref. w10x.. The first scheme includes a spectral device followed by a point-like photon detector, which is supplied by a fastoperating time gate and the time-dependent spectrum determined as the photon counting rate at the time t with the spectral device turned on the frequency V with spectral resolution h. In this scheme the linewidth of the ZPL in the luminescence spectrum depends Žapart from t and other parameters. on < g 1 y h <, and as a result, lines with ; ty1 linewidth can be observed at h s g 1. ŽLet us note
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 6 3 0 - 5
I. Rebaner Optics Communications 148 (1998) 36–40
that it is so for the model where G hom s g 1. It is more precise to suppose that G hom s g 1 q Gphase . Then the limit linewidth is < g 1 y h < q Gphase . Below we take Gphase equal to zero.. The effect of compensation of spectrometer resolution h has been theoretically considered in Refs. w6–9x, and in Refs. w11–13x for absorption. For experiments on time-dependent ZPL spectra see Ref. w14x Žand references therein., where measurements on the Mossbauer ZPL are ¨ reported. The second recording scheme comprises the time gate in front of the spectral device with high resolution Žh s 0., the photon counter counts all arriving photons w10x. The linewidth depends on the regime of opening and closing of the time gate. In this paper the case of hole-burning by two subsequent pulses of excitation light for the three level model of impurity centre is studied theoretically while a specific time dependence of the first pulse is supposed: its intensity increases exponentially from t s y` to t s t 1, is constant from t 1 until t 2 and decreases exponentially after t 2 ; the second pulse is infinitely short and arrives at the time t 3. The time-dependent ZPL luminescence spectrum is considered for the case of the second recording scheme, an infinitely short excitation at t s 0 and the time dependence of the gating is the following: exponential increase from t s y` to t s T1, constant from T1 to T2 , exponential decrease after T2 . The formulae describing the PSHB and ZPL are in these approximations analogous. If the time dependence of the first pulse in PSHB is the following: zero from t s y` to t s t 1, after t 1 exponential decreasing with parameter Do , the hole width in IDF decreases monotonically with the growth of the time interval t 3 y t 1 to the limit width < Do y g 1 < Žcompensation effect. w3,4x. Analogously if the time dependence of the gating is the following: exponential increasing with parameter g o from t s y` to T1, after T1 zero, the ZPL width in the luminescence spectrum decreases monotonically with the growth of the time T1 to the limit width < g o y g 1 < Žcompensation effect.. So, if pure phase relaxation is not taken into account Ž Gphase s 0. the ultimate limit for both the hole width in IDF and the ZPL width in the luminescence spectrum are zero if Do s g 1 and g o s g 1. This means that compensation of the natural linewidth of the ZPL g 1 and the exponential increase rate of the gate transparency g o take place up to the full compensation. In the case of the spectral hole compensation between g 1 and Do describing the exponential decrease of the first excitation pulse takes place. The compensations in line Žhole. width are achieved at expense of the signal strength: long time has to pass after the excitation before the compensation reaches some considerable value. In Žconventional. experiments this means very weak luminescence and very shallow holes. Thus the experiments are difficult.
37
2. Two-step pulsed spectral hole burning in three-level systems We consider PSHB in an optically thin layer of a dilute solid solution of photochromic molecules at perpendicular incidence of the burning light. The function D Ž V ,t . describes the time-dependent IDF during the hole burning w15x. In the case of the three-level systems under consideration it is necessary to introduce a two-dimensional IDF which reflects the inhomogeneous distribution of the frequencies V 01 and V 12 of the ZPL transitions 0 ™ 1 and 1 ™ 2 ŽFig. 1.. The function D 0 Ž V 01, V 12 . describes a two-dimensional inhomogeneous initial distribution existing before the light pulses arrive, we suppose that the frequency domains V 01 and V 12 do not overlap. Let two pulses fall successively onto the system, the first transferring the system from level 0 to level 1, and the second from level 1 to level 2. As a result of the two excitation light pulses the centre is phototransformed with probability a into a new state, which is no more in resonance with both pulses. The two-dimensional IDF changes exponentially in time if certain conditions are fulfilled w15x and we have
D Ž V 01 , V 12 ,t . s D 0 Ž V 01 , V 12 . exp w yP3 Ž V 01 , V 12 ,t . x ,
Ž1. where P3Ž V 01, V 12 ,t . shows how many impurity centres have been burned out during time t w3x. t
P3 Ž V 01 , V 12 ,t . s a
X
t
X
t
X
Hy` d t Hy`Hy`d t d t
=
1
X 1
S2 Ž t1 ,tX1 .
X
t1
t1
Hy` d t Hy`d t 2
X 2
S1Ž t 2 ,tX2 . F3 Ž tX ,t 1 ,tX1 ,t 2 ,tX2 . ,
Ž2.
where S1Ž t 2 ,tX2 . and S2 Ž t 1,tX1 . are the correlation functions ŽCF. of the first and the second light pulses, CF F3Ž tX ,t 1,tX1,
Fig. 1. Two-step hole burning in the three-level system 0–1–2. The first pulse with main frequency v 0 excites the system 0 ™1 and makes a spectrally narrow transient hole in IDF; the second Ž d-pulse. performs the transition 1™2, fixes the narrow hole as permanent and provides the possibility to accumulate hole depth.
I. Rebaner Optics Communications 148 (1998) 36–40
38
t 2 ,tX2 . describes the three-level impurity centre. Here we use the model where the relaxation processes of the first and the second excited energy levels are described by the rates of energy relaxation, g 1 and g 2 . The corresponding CF of the three-level system is as follows: F3 Ž tX ,t 1 ,tX1 ,t 2 ,tX2 . s C3 exp w yg 2 tX y i V 12 Ž t 1 y tX1 . qg 2 Ž t 1 q tX1 . r2 y i V 01Ž t 2 y tX2 . yg 1Ž t 1 q tX1 y t 2 y tX2 . r2 x ,
P31Ž x . s 2 a C3 S10 S20 eyg 1Žt 3yt 2 .
Ž3.
where C3 is constant. ŽPhonon wings have not been taken into account.. Let us suppose that CF of the first pulse is as follows: S1Ž t 2 ,tX2 . s S10 ´ ) Ž t 2 . ´ Ž tX2 . ,
°expwyi v t y D Žt y t .r2x , ¢ 0
i
1
´ Ž t . s exp Ž yi v 0 t . , exp w yi v 0 t y Do Ž t y t 2 . r2 x ,
if t - t 1 , if t 1 F t F t 2 , if t 2 - t ,
where the parameters Di and Do define, respectively, the rates of switch-on and switch-off of the holeburning light and v 0 is the main frequency of the first pulse. In the second step of the excitation 1 ™ 2 a short light pulse Ž d-pulse. at the time t 3 , described by the following CF, is used: S2 Ž t 1 ,tX1 . s S20 d Ž t 1 y t 3 . d Ž tX1 y t 3 . ,
t2-t3.
P3 Ž x . s a C3 S10 S20 eyg 1Žt 3y t 2 . q ey D o Žt 3y t 2 . y2eyŽ g 1q D o .Žt 3y t 2 .r2 cos Ž x Ž t 3 y t 2 . . rc 1
Let us determine the time-dependent fluorescence spectrum I2 Ž V . as follows. The spectrometer is tuned to frequency V Žinfinitely high spectral resolution h s 0 is supposed.. The time gate is placed in front of the spectrometer. The photons, after passing through the time gate, enter the spectrometer, and those of frequency V are selected and counted. Then the spectrometer is tuned to another frequency V X and the photons of frequency V X are counted. Repeating this procedure for a number of frequencies we obtain the spectrum w10x `
I2 Ž V . s
HHy` d t d t 1
=
X 1
exp w i V Ž t 1 y tX1 . x G Ž t 1 ,tX1 .
X
t1
t1
Hy` d t Hy`d t 2
X 2
F2 Ž t 1 ,tX1 ,t 2 ,tX2 . S Ž t 2 ,tX2 . ,
Ž9.
where GŽ t 1,tX1 . is the CF of the time gate, CF F2 Ž t 1,tX1,t 2 ,tX2 . describes the two-level impurity centre, SŽ t 2 ,tX2 . is the CF of the exciting pulse. Let the excitation from level 0 to level 1 take place by a short light pulse Ž d-pulse. at time t s 0: S Ž t 2 ,tX2 . s S0 d Ž t 2 . d Ž tX2 . ,
Ž 10.
y g 1Žt 3 y t 2 .
qe
yg 1Ž2 t 3 y t 1 y t 2 .r2
y2e
In this case with increase of the time interval t 3 y t 1 the holewidth decreases monotonically to zero.
Ž6.
In fact the pulse has to be essentially shorter than the duration of the first pulse and the relaxation times of levels 1 and 2, and spectrally broader than the frequency distribution of V 12 of the transition 1 ™ 2. By substituting Ž3. – Ž6. into Ž2. and after integration at t ™ ` we obtain the frequency dependent PSHB efficiency, which determines the hole shape in IDF,
q e
Ž8.
3. The time-dependent ZPL in the luminescence spectrum
Ž5.
yg 1Žt 3 y t 1 .
= 1 y cos Ž x Ž t 3 y t 2 . . r Ž g 2 x 2 . .
Ž4.
where S10 is constant, and
~
Di ™ ` Ži.e. momentary switch-on of the first step excitation.. Then the hole in the IDF D Ž V 01 . is described by the first term of Ž7. and the holewidth decreases monotonically with the growth of the time interval t 3 y t 1 to the limit holewidth < Do y g 1 < Žcompensation effect.. The holedepth decreases also. If we choose Do s g 1 the first term in Ž7. is
and the time gate is opened and closed exponentially. Then the CF of the time gate is
cos Ž x Ž t 2 y t 1 . . rc 2
qeyg 1Žt 3yt 1 .rc 3 q Iinterf Ž x . 4 rg 2 .
Ž7.
Here x s V 01 y v 0 , c 1 s x 2 q Žg 1 y Do . 2r4, c 2 s x 2 q g 12r4, c 3 s x 2 q Žg 1 q Dc . 2r4, and Iinterf Ž x . describes the interference terms Žsee the Appendix.. In Refs. w3,4x the special case t 1 s t 2 , Di ™ ` with Gphase different from zero and in Ref. w16x the case t 1 s 0, t 2 ™ ` Žor, the same, t 2 s t 3 ., Di ™ `, and Gphase s 0 was calculated. The contributions of the second term, the third term and Iinterf Ž x . in Ž7. are zero, if we choose t 1 s t 2 and choose
G Ž t 1 ,tX1 . s G 0 g Ž t 1 . g Ž tX1 . ,
Ž 11 .
where G 0 is constant, and
° ¢expwyg Ž t y T .r2x ,
exp w yg o Ž T1 y t . r2 x , g Ž t . s 1,
~
c
2
if t - T1 , if T1 F t F T2 ,
Ž 12.
if T2 - t.
g o and g c define, respectively, the rates of opening and closing of the time gate.
I. Rebaner Optics Communications 148 (1998) 36–40
39
Fig. 2. The shape of the time-dependent ZPL in the luminescence spectrum Žexcitation 0 ™ 1 at t s 0 by d-pulse, transition frequency V 01 . in the second recording scheme, where the time gate is in front of the spectrometer, which is scanned over the frequency V . Opening Žat Žsee T2 s 100gy1 . we T1 s 0. and closing Žat T2 . of the time gate is momentary Ž g o ™ `, g c ™ `.. For long gate open times T2 4 gy1 1 1 obtain a conventional stationary ZPL spectrum with width g 1. y s V y V 01 .
To describe the two-level system Ž0 ™ 1. we use a model, where the relaxation process of the excited level 1 is described by the rate of energy relaxation, g 1. Then the corresponding CF is F2 t 1 ,tX1 ,t 2 ,tX2
Ž
. s C2 exp w
yi V 01 t 1 y tX1
Ž
yg 1 t 1 q tX1 y t 2 y tX2
Ž
.
y i V 01 t 2 y tX2
. r2 x ,
Ž
.
Ž 13.
where C2 is constant. Substituting Eqs. Ž10. – Ž13. into Ž9., we obtain after integration the time-dependent ZPL luminescence spec-
trum in the second scheme, I2 Ž y . s C2 G 0 S0 eyg 1T1 q eyg o T1 y2eyŽ g 1qg o .T1 r2 cos Ž yT1 . rj 1 q eyg 1T1 q ey g 1T 2 y 2ey g 1ŽT1qT 2 .r2 =cos Ž y Ž T2 y T1 . . rj 2 qeyg 1T2rj 3 q Iinterf Ž y . 4 ,
Ž 14.
where y s V y V 01, j 1 s y q Žg 1 y g o r4, j 2 s y 2 q 2
.2
Fig. 3. The shape of the time-dependent ZPL in the luminescence spectrum Žexcitation 0 ™ 1 at t s 0 by d-pulse, transition frequency V 01 . in the second recording scheme, where the time gate is in front of the spectrometer, which is scanned over the frequency V . The time gate is opened exponentially with rate g o s g 1 and at T1 closed momentarily ŽT1 s T2 , g c ™ `.. Ultimate limit Žat T1 ™ `. is zero. ZPL intensity decays exponentially. y s V y V 01 .
I. Rebaner Optics Communications 148 (1998) 36–40
40
g 12r4, j 3 s y 2 q Žg 1 q g c . 2r4 and Iinterf Ž y . describes the interference terms given in the Appendix. Let us note that in Ref. w10x the special case T1 s T2 and g o s g c , i.e. closing of the time gate started immediately after opening was finished and the rates of opening and closing were equal, was calculated. Formula Ž14. coincides with formula Ž7., if we substitute y by x, g o by Do , g c by Di , T1 by Žt 3 y t 2 ., and T2 by Žt 3 y t 1 .. In the special case g o ™ `, g c ™ `, i.e. momentary opening and closing of the time gate the ZPL is described by the second term of Ž14.. For small DT s T2 y T1 the ZPL is broad, with increasing DT the line narrows monotonically to the limit width g 1 ŽFig. 2.. The contribution of the second term, the third term and Iinterf Ž y . in Ž14. is zero if we choose T1 s T2 and g c ™ `, i.e. momentary closing of the time gate. Then ZPL is described by the first term of Ž14. and the width of the ZPL decreases monotonically with increasing T1 to the limit width G lim s < g o y g 1 < Žcompensation effect.. Its intensity decays exponentially. If we choose g o s g 1 the first term in Ž14. is I21Ž y . s 2C2 G 0 S0 eyg 1T1 w 1 y cos Ž yT1 . x ry 2 .
Ž 15 .
In this case with the growth of the time T1 the ZPL width decreases monotonically to zero ŽFig. 3..
Acknowledgements The author is indebted to Professor K. Rebane for valuable discussions. This work was supported, in part, by the Estonian Science Foundation grant 2268, and in part, by the International Center for Scientific Culture – World Laboratory.
Appendix A The interference term in Ž7. is given by Iinterf Ž x . s 2 R Ž eŽyi xyŽ g 1q D o .r2.Žt 3y t 2 . y ey g 1Žt 3y t 2 .1 . qeŽyi xy g 1 r2.Žt 2y t 1 .
ž
1
1 y
a 1 a 2)
1 q
a 1 a 3)
a 2 a 3)
qeŽyi xy g 1 r2.Žt 3yt 1 .y D o Žt 3yt 2 .r2 1 1 1 = y q y eyg 1Žt 3y t 1 . , ) ) a1a2 a1 a3 a 2 a 3)
ž
/
1
a 1 a 2)
/
where a 1 s ix y Žg 1 y Do .r2, a 2 s ix y g 1r2 and a 3 s ix y Žg 1 q Di .r2. The term Iinterf Ž y . in Ž14. can be obtained from Iinterf Ž x . if we replace x by y, Do by g o , Di by g c , t 3 y t 2 by T1 and t 3 y t 1 by T2 .
References w1x A.A. Gorokhovskii, R.K. Kaarli, L.A. Rebane, JETP Lett. 20 Ž1974. 216. w2x K.K. Rebane, L.A. Rebane, in: W.E. Moerner ŽEd.., Persistent spectral hole-burning: science and applications, Topics in current physics, vol. 44, Springer, Berlin, 1988, pp. 17–77. w3x I.K. Rebane, phys. stat. sol. Žb. 145 Ž1988. 749. w4x I.K. Rebane, J. Phys. B 22 Ž1989. 2411. w5x I. Rebane, Optics Comm. 76 Ž1990. 225. w6x V.V. Khizhnyakov, I.K. Rebane, Sov. Phys. JETP 47 Ž1978. 463. w7x I.K. Rebane, A.L. Tuul, V.V. Khizhnyakov, Sov. Phys. JETP 50 Ž1979. 655. w8x V. Hizhnyakov, I. Rebane, Proc. Estonian SSR Acad. Sci. 35 Ž1986. 406. w9x V.V. Hizhnyakov, in: O. Sild, K. Haller ŽEds.., Zero-phonon lines and spectral hole burning in spectroscopy and photochemistry, Springer, Berlin, 1988, pp. 37–54. w10x I. Rebane, V. Hizhnyakov, Proc. Estonian SSR Acad. Sci. 30 Ž1981. 1 win Russianx. w11x F. Shimizu, K. Shimizu, H. Takuma, Phys. Rev. A 28 Ž1983. 2248. w12x W. Zinth, M.C. Nuss, W. Kaiser, Optics Comm. 44 Ž1983. 262. w13x P.E. Coleman, D. Kagan, P.L. Knight, Optics Comm. 36 Ž1981. 127. w14x E. Realo, in: O. Sild, K. Haller ŽEds.., Zero-phonon lines and spectral hole burning in spectroscopy and photochemistry, Springer, Berlin, 1988, pp. 164–182. w15x L.A. Rebane, A.A. Gorokhovskii, J.V. Kikas, Appl. Phys. B29 Ž1982. 235. w16x I.K. Rebane, phys. stat. sol. Žb. 167 Ž1991. 301.
Optics Communications 148 Ž1998. 36–40
Analogy between hole shape in two-step pulsed spectral hole burning and time-dependent zero-phonon line shape Inna Rebane
1
Institute of Physics, Riia 142, EE-2400 Tartu, Estonia Received 28 May 1997; revised 16 October 1997; accepted 21 October 1997
Abstract The two-step persistent spectral hole burning ŽPSHB. of a hole in the inhomogeneous distribution function ŽIDF. of the zero-phonon line ŽZPL. optical transition frequencies in a three-level system by two successive light pulses, in the case of an exponentially increasing and decreasing first coherent pulse and an infinitely short second pulse, is considered. The time-dependent ZPL in the luminescence spectrum in the recording scheme with the time gate in front of the spectrometer with high resolution in the case of exponential opening and closing of the time gate is calculated. It is shown that the time-dependent ZPL in the luminescence spectrum is described by a formula which is analogous to the formula describing the spectral hole in IDF. The effect of compensation of the natural linewidth of the excited level and the parameter describing the exponential increase of the transparency of the time gate is presented. q 1998 Elsevier Science B.V. PACS: 78.47 Keywords: Two-step persistent spectral hole burning; Time-dependent zero-phonon line; Line narrowing
1. Introduction Persistent spectral hole burning w1,2x ŽPSHB. is a method of very high resolution spectroscopy of impurities in solid matrices providing a number of applications. The capability of PSHB materials for applications is determined by the ratio G inh : G hom , where G inh is the inhomogeneous and G hom the homogeneous linewidth of the ZPL. The minimum width of a PSHB hole in IDF in the one-step PSHB with monochromatic light is G hom and the minimum width of a spectral hole measured in conventional spectra is 2 G hom w2,3x. A theory of PSHB with light pulses of arbitrary shape and duration was proposed in Refs. w3–5x. In a three-level system w3,4x, it is possible to eliminate the energy-relaxation-induced hole width g 1. As
1
E-mail: [email protected]
a result the holes in IDF are narrower than G hom s g 1 q Gphase , where Gphase is the pure phase relaxation broadening Žmodulation broadening.. In the theory of time-dependent Žtransient. spectra the scheme used for measurement of the spectrum is important. Two schemes of one-gate measurement have been considered: Ži. time gate placed behind the spectral device Žthe first scheme was considered in Refs. w6–9x., Žii. time gate in front of the spectral device Žthe second scheme was considered in Ref. w10x.. The first scheme includes a spectral device followed by a point-like photon detector, which is supplied by a fastoperating time gate and the time-dependent spectrum determined as the photon counting rate at the time t with the spectral device turned on the frequency V with spectral resolution h. In this scheme the linewidth of the ZPL in the luminescence spectrum depends Žapart from t and other parameters. on < g 1 y h <, and as a result, lines with ; ty1 linewidth can be observed at h s g 1. ŽLet us note
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 6 3 0 - 5
I. Rebaner Optics Communications 148 (1998) 36–40
that it is so for the model where G hom s g 1. It is more precise to suppose that G hom s g 1 q Gphase . Then the limit linewidth is < g 1 y h < q Gphase . Below we take Gphase equal to zero.. The effect of compensation of spectrometer resolution h has been theoretically considered in Refs. w6–9x, and in Refs. w11–13x for absorption. For experiments on time-dependent ZPL spectra see Ref. w14x Žand references therein., where measurements on the Mossbauer ZPL are ¨ reported. The second recording scheme comprises the time gate in front of the spectral device with high resolution Žh s 0., the photon counter counts all arriving photons w10x. The linewidth depends on the regime of opening and closing of the time gate. In this paper the case of hole-burning by two subsequent pulses of excitation light for the three level model of impurity centre is studied theoretically while a specific time dependence of the first pulse is supposed: its intensity increases exponentially from t s y` to t s t 1, is constant from t 1 until t 2 and decreases exponentially after t 2 ; the second pulse is infinitely short and arrives at the time t 3. The time-dependent ZPL luminescence spectrum is considered for the case of the second recording scheme, an infinitely short excitation at t s 0 and the time dependence of the gating is the following: exponential increase from t s y` to t s T1, constant from T1 to T2 , exponential decrease after T2 . The formulae describing the PSHB and ZPL are in these approximations analogous. If the time dependence of the first pulse in PSHB is the following: zero from t s y` to t s t 1, after t 1 exponential decreasing with parameter Do , the hole width in IDF decreases monotonically with the growth of the time interval t 3 y t 1 to the limit width < Do y g 1 < Žcompensation effect. w3,4x. Analogously if the time dependence of the gating is the following: exponential increasing with parameter g o from t s y` to T1, after T1 zero, the ZPL width in the luminescence spectrum decreases monotonically with the growth of the time T1 to the limit width < g o y g 1 < Žcompensation effect.. So, if pure phase relaxation is not taken into account Ž Gphase s 0. the ultimate limit for both the hole width in IDF and the ZPL width in the luminescence spectrum are zero if Do s g 1 and g o s g 1. This means that compensation of the natural linewidth of the ZPL g 1 and the exponential increase rate of the gate transparency g o take place up to the full compensation. In the case of the spectral hole compensation between g 1 and Do describing the exponential decrease of the first excitation pulse takes place. The compensations in line Žhole. width are achieved at expense of the signal strength: long time has to pass after the excitation before the compensation reaches some considerable value. In Žconventional. experiments this means very weak luminescence and very shallow holes. Thus the experiments are difficult.
37
2. Two-step pulsed spectral hole burning in three-level systems We consider PSHB in an optically thin layer of a dilute solid solution of photochromic molecules at perpendicular incidence of the burning light. The function D Ž V ,t . describes the time-dependent IDF during the hole burning w15x. In the case of the three-level systems under consideration it is necessary to introduce a two-dimensional IDF which reflects the inhomogeneous distribution of the frequencies V 01 and V 12 of the ZPL transitions 0 ™ 1 and 1 ™ 2 ŽFig. 1.. The function D 0 Ž V 01, V 12 . describes a two-dimensional inhomogeneous initial distribution existing before the light pulses arrive, we suppose that the frequency domains V 01 and V 12 do not overlap. Let two pulses fall successively onto the system, the first transferring the system from level 0 to level 1, and the second from level 1 to level 2. As a result of the two excitation light pulses the centre is phototransformed with probability a into a new state, which is no more in resonance with both pulses. The two-dimensional IDF changes exponentially in time if certain conditions are fulfilled w15x and we have
D Ž V 01 , V 12 ,t . s D 0 Ž V 01 , V 12 . exp w yP3 Ž V 01 , V 12 ,t . x ,
Ž1. where P3Ž V 01, V 12 ,t . shows how many impurity centres have been burned out during time t w3x. t
P3 Ž V 01 , V 12 ,t . s a
X
t
X
t
X
Hy` d t Hy`Hy`d t d t
=
1
X 1
S2 Ž t1 ,tX1 .
X
t1
t1
Hy` d t Hy`d t 2
X 2
S1Ž t 2 ,tX2 . F3 Ž tX ,t 1 ,tX1 ,t 2 ,tX2 . ,
Ž2.
where S1Ž t 2 ,tX2 . and S2 Ž t 1,tX1 . are the correlation functions ŽCF. of the first and the second light pulses, CF F3Ž tX ,t 1,tX1,
Fig. 1. Two-step hole burning in the three-level system 0–1–2. The first pulse with main frequency v 0 excites the system 0 ™1 and makes a spectrally narrow transient hole in IDF; the second Ž d-pulse. performs the transition 1™2, fixes the narrow hole as permanent and provides the possibility to accumulate hole depth.
I. Rebaner Optics Communications 148 (1998) 36–40
38
t 2 ,tX2 . describes the three-level impurity centre. Here we use the model where the relaxation processes of the first and the second excited energy levels are described by the rates of energy relaxation, g 1 and g 2 . The corresponding CF of the three-level system is as follows: F3 Ž tX ,t 1 ,tX1 ,t 2 ,tX2 . s C3 exp w yg 2 tX y i V 12 Ž t 1 y tX1 . qg 2 Ž t 1 q tX1 . r2 y i V 01Ž t 2 y tX2 . yg 1Ž t 1 q tX1 y t 2 y tX2 . r2 x ,
P31Ž x . s 2 a C3 S10 S20 eyg 1Žt 3yt 2 .
Ž3.
where C3 is constant. ŽPhonon wings have not been taken into account.. Let us suppose that CF of the first pulse is as follows: S1Ž t 2 ,tX2 . s S10 ´ ) Ž t 2 . ´ Ž tX2 . ,
°expwyi v t y D Žt y t .r2x , ¢ 0
i
1
´ Ž t . s exp Ž yi v 0 t . , exp w yi v 0 t y Do Ž t y t 2 . r2 x ,
if t - t 1 , if t 1 F t F t 2 , if t 2 - t ,
where the parameters Di and Do define, respectively, the rates of switch-on and switch-off of the holeburning light and v 0 is the main frequency of the first pulse. In the second step of the excitation 1 ™ 2 a short light pulse Ž d-pulse. at the time t 3 , described by the following CF, is used: S2 Ž t 1 ,tX1 . s S20 d Ž t 1 y t 3 . d Ž tX1 y t 3 . ,
t2-t3.
P3 Ž x . s a C3 S10 S20 eyg 1Žt 3y t 2 . q ey D o Žt 3y t 2 . y2eyŽ g 1q D o .Žt 3y t 2 .r2 cos Ž x Ž t 3 y t 2 . . rc 1
Let us determine the time-dependent fluorescence spectrum I2 Ž V . as follows. The spectrometer is tuned to frequency V Žinfinitely high spectral resolution h s 0 is supposed.. The time gate is placed in front of the spectrometer. The photons, after passing through the time gate, enter the spectrometer, and those of frequency V are selected and counted. Then the spectrometer is tuned to another frequency V X and the photons of frequency V X are counted. Repeating this procedure for a number of frequencies we obtain the spectrum w10x `
I2 Ž V . s
HHy` d t d t 1
=
X 1
exp w i V Ž t 1 y tX1 . x G Ž t 1 ,tX1 .
X
t1
t1
Hy` d t Hy`d t 2
X 2
F2 Ž t 1 ,tX1 ,t 2 ,tX2 . S Ž t 2 ,tX2 . ,
Ž9.
where GŽ t 1,tX1 . is the CF of the time gate, CF F2 Ž t 1,tX1,t 2 ,tX2 . describes the two-level impurity centre, SŽ t 2 ,tX2 . is the CF of the exciting pulse. Let the excitation from level 0 to level 1 take place by a short light pulse Ž d-pulse. at time t s 0: S Ž t 2 ,tX2 . s S0 d Ž t 2 . d Ž tX2 . ,
Ž 10.
y g 1Žt 3 y t 2 .
qe
yg 1Ž2 t 3 y t 1 y t 2 .r2
y2e
In this case with increase of the time interval t 3 y t 1 the holewidth decreases monotonically to zero.
Ž6.
In fact the pulse has to be essentially shorter than the duration of the first pulse and the relaxation times of levels 1 and 2, and spectrally broader than the frequency distribution of V 12 of the transition 1 ™ 2. By substituting Ž3. – Ž6. into Ž2. and after integration at t ™ ` we obtain the frequency dependent PSHB efficiency, which determines the hole shape in IDF,
q e
Ž8.
3. The time-dependent ZPL in the luminescence spectrum
Ž5.
yg 1Žt 3 y t 1 .
= 1 y cos Ž x Ž t 3 y t 2 . . r Ž g 2 x 2 . .
Ž4.
where S10 is constant, and
~
Di ™ ` Ži.e. momentary switch-on of the first step excitation.. Then the hole in the IDF D Ž V 01 . is described by the first term of Ž7. and the holewidth decreases monotonically with the growth of the time interval t 3 y t 1 to the limit holewidth < Do y g 1 < Žcompensation effect.. The holedepth decreases also. If we choose Do s g 1 the first term in Ž7. is
and the time gate is opened and closed exponentially. Then the CF of the time gate is
cos Ž x Ž t 2 y t 1 . . rc 2
qeyg 1Žt 3yt 1 .rc 3 q Iinterf Ž x . 4 rg 2 .
Ž7.
Here x s V 01 y v 0 , c 1 s x 2 q Žg 1 y Do . 2r4, c 2 s x 2 q g 12r4, c 3 s x 2 q Žg 1 q Dc . 2r4, and Iinterf Ž x . describes the interference terms Žsee the Appendix.. In Refs. w3,4x the special case t 1 s t 2 , Di ™ ` with Gphase different from zero and in Ref. w16x the case t 1 s 0, t 2 ™ ` Žor, the same, t 2 s t 3 ., Di ™ `, and Gphase s 0 was calculated. The contributions of the second term, the third term and Iinterf Ž x . in Ž7. are zero, if we choose t 1 s t 2 and choose
G Ž t 1 ,tX1 . s G 0 g Ž t 1 . g Ž tX1 . ,
Ž 11 .
where G 0 is constant, and
° ¢expwyg Ž t y T .r2x ,
exp w yg o Ž T1 y t . r2 x , g Ž t . s 1,
~
c
2
if t - T1 , if T1 F t F T2 ,
Ž 12.
if T2 - t.
g o and g c define, respectively, the rates of opening and closing of the time gate.
I. Rebaner Optics Communications 148 (1998) 36–40
39
Fig. 2. The shape of the time-dependent ZPL in the luminescence spectrum Žexcitation 0 ™ 1 at t s 0 by d-pulse, transition frequency V 01 . in the second recording scheme, where the time gate is in front of the spectrometer, which is scanned over the frequency V . Opening Žat Žsee T2 s 100gy1 . we T1 s 0. and closing Žat T2 . of the time gate is momentary Ž g o ™ `, g c ™ `.. For long gate open times T2 4 gy1 1 1 obtain a conventional stationary ZPL spectrum with width g 1. y s V y V 01 .
To describe the two-level system Ž0 ™ 1. we use a model, where the relaxation process of the excited level 1 is described by the rate of energy relaxation, g 1. Then the corresponding CF is F2 t 1 ,tX1 ,t 2 ,tX2
Ž
. s C2 exp w
yi V 01 t 1 y tX1
Ž
yg 1 t 1 q tX1 y t 2 y tX2
Ž
.
y i V 01 t 2 y tX2
. r2 x ,
Ž
.
Ž 13.
where C2 is constant. Substituting Eqs. Ž10. – Ž13. into Ž9., we obtain after integration the time-dependent ZPL luminescence spec-
trum in the second scheme, I2 Ž y . s C2 G 0 S0 eyg 1T1 q eyg o T1 y2eyŽ g 1qg o .T1 r2 cos Ž yT1 . rj 1 q eyg 1T1 q ey g 1T 2 y 2ey g 1ŽT1qT 2 .r2 =cos Ž y Ž T2 y T1 . . rj 2 qeyg 1T2rj 3 q Iinterf Ž y . 4 ,
Ž 14.
where y s V y V 01, j 1 s y q Žg 1 y g o r4, j 2 s y 2 q 2
.2
Fig. 3. The shape of the time-dependent ZPL in the luminescence spectrum Žexcitation 0 ™ 1 at t s 0 by d-pulse, transition frequency V 01 . in the second recording scheme, where the time gate is in front of the spectrometer, which is scanned over the frequency V . The time gate is opened exponentially with rate g o s g 1 and at T1 closed momentarily ŽT1 s T2 , g c ™ `.. Ultimate limit Žat T1 ™ `. is zero. ZPL intensity decays exponentially. y s V y V 01 .
I. Rebaner Optics Communications 148 (1998) 36–40
40
g 12r4, j 3 s y 2 q Žg 1 q g c . 2r4 and Iinterf Ž y . describes the interference terms given in the Appendix. Let us note that in Ref. w10x the special case T1 s T2 and g o s g c , i.e. closing of the time gate started immediately after opening was finished and the rates of opening and closing were equal, was calculated. Formula Ž14. coincides with formula Ž7., if we substitute y by x, g o by Do , g c by Di , T1 by Žt 3 y t 2 ., and T2 by Žt 3 y t 1 .. In the special case g o ™ `, g c ™ `, i.e. momentary opening and closing of the time gate the ZPL is described by the second term of Ž14.. For small DT s T2 y T1 the ZPL is broad, with increasing DT the line narrows monotonically to the limit width g 1 ŽFig. 2.. The contribution of the second term, the third term and Iinterf Ž y . in Ž14. is zero if we choose T1 s T2 and g c ™ `, i.e. momentary closing of the time gate. Then ZPL is described by the first term of Ž14. and the width of the ZPL decreases monotonically with increasing T1 to the limit width G lim s < g o y g 1 < Žcompensation effect.. Its intensity decays exponentially. If we choose g o s g 1 the first term in Ž14. is I21Ž y . s 2C2 G 0 S0 eyg 1T1 w 1 y cos Ž yT1 . x ry 2 .
Ž 15 .
In this case with the growth of the time T1 the ZPL width decreases monotonically to zero ŽFig. 3..
Acknowledgements The author is indebted to Professor K. Rebane for valuable discussions. This work was supported, in part, by the Estonian Science Foundation grant 2268, and in part, by the International Center for Scientific Culture – World Laboratory.
Appendix A The interference term in Ž7. is given by Iinterf Ž x . s 2 R Ž eŽyi xyŽ g 1q D o .r2.Žt 3y t 2 . y ey g 1Žt 3y t 2 .1 . qeŽyi xy g 1 r2.Žt 2y t 1 .
ž
1
1 y
a 1 a 2)
1 q
a 1 a 3)
a 2 a 3)
qeŽyi xy g 1 r2.Žt 3yt 1 .y D o Žt 3yt 2 .r2 1 1 1 = y q y eyg 1Žt 3y t 1 . , ) ) a1a2 a1 a3 a 2 a 3)
ž
/
1
a 1 a 2)
/
where a 1 s ix y Žg 1 y Do .r2, a 2 s ix y g 1r2 and a 3 s ix y Žg 1 q Di .r2. The term Iinterf Ž y . in Ž14. can be obtained from Iinterf Ž x . if we replace x by y, Do by g o , Di by g c , t 3 y t 2 by T1 and t 3 y t 1 by T2 .
References w1x A.A. Gorokhovskii, R.K. Kaarli, L.A. Rebane, JETP Lett. 20 Ž1974. 216. w2x K.K. Rebane, L.A. Rebane, in: W.E. Moerner ŽEd.., Persistent spectral hole-burning: science and applications, Topics in current physics, vol. 44, Springer, Berlin, 1988, pp. 17–77. w3x I.K. Rebane, phys. stat. sol. Žb. 145 Ž1988. 749. w4x I.K. Rebane, J. Phys. B 22 Ž1989. 2411. w5x I. Rebane, Optics Comm. 76 Ž1990. 225. w6x V.V. Khizhnyakov, I.K. Rebane, Sov. Phys. JETP 47 Ž1978. 463. w7x I.K. Rebane, A.L. Tuul, V.V. Khizhnyakov, Sov. Phys. JETP 50 Ž1979. 655. w8x V. Hizhnyakov, I. Rebane, Proc. Estonian SSR Acad. Sci. 35 Ž1986. 406. w9x V.V. Hizhnyakov, in: O. Sild, K. Haller ŽEds.., Zero-phonon lines and spectral hole burning in spectroscopy and photochemistry, Springer, Berlin, 1988, pp. 37–54. w10x I. Rebane, V. Hizhnyakov, Proc. Estonian SSR Acad. Sci. 30 Ž1981. 1 win Russianx. w11x F. Shimizu, K. Shimizu, H. Takuma, Phys. Rev. A 28 Ž1983. 2248. w12x W. Zinth, M.C. Nuss, W. Kaiser, Optics Comm. 44 Ž1983. 262. w13x P.E. Coleman, D. Kagan, P.L. Knight, Optics Comm. 36 Ž1981. 127. w14x E. Realo, in: O. Sild, K. Haller ŽEds.., Zero-phonon lines and spectral hole burning in spectroscopy and photochemistry, Springer, Berlin, 1988, pp. 164–182. w15x L.A. Rebane, A.A. Gorokhovskii, J.V. Kikas, Appl. Phys. B29 Ž1982. 235. w16x I.K. Rebane, phys. stat. sol. Žb. 167 Ž1991. 301.