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Quantum efficiency and signal bandwidth of thallium atomic line filters
Optics Communications 90 ( 1992 ) 245-250 North-Holland
OPTICS COMMUNICATIONS
Quantum efficiency and signal bandwidth of thallium atomic line filters Andreas F. Molisch, Bernhard P. Oehry, Walter Schupita and Gottfried Magerl Institut ]~r Nachrichtentechnik und Hochfrequenztechnik, Technische Universittit Wien, Guflhausstrafle 25/389, A- I 040 Wien, Austria
Received 3 December 1991; revised manuscript received 3 March 1992
We derive a new, simplified method for the computation of trapping effects in coupled three-level atomic systems. The linearity of the Holstein radiation trapping equation allows the reduction of the general rate equation to the solution of the basic two-level Holstein equation and to an algebraic eigenvalue problem. The method is applied to the numerical simulation of the quantum efficiency and the signal bandwidth of thallium atomic line filters. With a pumping scheme that achieves population inversion between the ground state and the metastable state, 90% quantum efficiency and l 0 MHz signal bandwidth can be achieved, while in noninverting pumping schemes, even quite high pump intensities result in less than 60% quantum efficiency and 8 MHz signal bandwidth. A tradeoffbetween quantum efficiency, signal bandwidth and pump power is possible.
A t o m i c line filters ( A L F s ) are ultra-narrow-band, wide-field-of-view optical filters for the detection o f weak optical signals e m b e d d e d in strong background noise; for an excellent review o f the underlying physics, see ref. [ l ]; a theoretical model for r u b i d i u m A L F s was d e v e l o p e d in ref. [ 2 ]. M a i n applications are underwater c o m m u n i c a t i o n s [3], deep space c o m m u n i c a t i o n s [ 4 ], satellite tracking [ 5 ] and atmospheric lidar systems [6 ]. One o f the most p r o m ising candidates for these applications is the thallium ALF, because the input transition overlaps the emission wavelength o f a well-developed solid-state laser, the frequency d o u b l e d N d : B E L (BEL stands for l a n t h a n u m beryllate, LaEBe205 [ 7 ] ). The basic operating principle o f T1-ALFs can be u n d e r s t o o d from the term scheme o f thallium in fig. I. Since T1-ALFs are active systems, the metastable 6P3/2 level has to be p o p u l a t e d by some p u m p i n g mechanism [ 8,6,9 ]. An incoming signal photon with the wavelength 535 nm can then be absorbed, and it excites a metastable a t o m to the 7s state. The desired upconversion o f green photons to uv photons is achieved when the 7s level decays to the ground state with the emission o f a 378 n m photon. The cell containing the atomic vapor is located between two dyedglass filters: a longwave-pass filter in front o f the in-
"; -7.5ns
\
\
F-1
~
input
output \ 378nm \
..
6 2P3/2"'~
F=2 F=I
,.'" ." 1.281.tm
, ,,"" t
82pll2~
F=I F=O
Fig. 1. Thallium energy level diagram (hyperfine splitting not to scale ). put window a n d a uv b a n d p a s s in front o f the detector ( p h o t o m u l t i p l i e r tube, P M T ) . In c o m b i n a tion, these filters are completely opaque. Only photons that are wavelength-shifted by the v a p o r can reach the detector. As the absorption b a n d w i d t h o f the T1 metastables is extremely small (only a few G H z ) , the system acts as a very-narrow-band optical filter. However, the above description is overly simplified. Actually, a 7s a t o m will decay with about 50%
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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Volume 90, number 4,5,6
OPTICS COMMUNICATIONS
probability either to the ground state or to the metastable state, emitting a 378 nm uv or a 535 nm green photon, respectively. These fluorescence photons may escape from the cell or be reabsorbed by the vapor. A uv photon is reabsorbed by ground-state atoms with the probability pUV, while green photons will be reabsorbed with a probability pg. In either case, the reabsorption result in the creation of another 7s atom which will then decay again to either the ground state or to the metastable state. This process will be repeated until a uv or a green photon finally leaves the cell. As 535 nm radiation is blocked from the PMT, all green photons leaving the cell are lost for the detection. Thus it is obvious that pU~ and pg determine the quantum efficiency q of the filter; q is defined as the number of uv photons leaving the cell over the number of absorbed signal photons. The probabilities pUV and p g a r e proportional to the ground state density r/gnd and the metastable state density nm, respectively. Furthermore, the radiation trapping effect described above will strongly influence the response time of the filter. A signal photon that goes through i absorption-reemission processes will take iv seconds to reach the detector (r is the lifetime of the 7s state) and the signal bandwidth, SBW, is lowered accordingly. In the following, we develop a method that allows for the computation of the quantum efficiency q and the SBW in a typical TI-ALF cell in dependence of ng, d and nm. The basic assumption for the following derivation is that the problem is linear, i.e. the incoming signal photons must not change ng.d or n m appreciably. Since ALFs are designed for the detection of very low light levels, this assumption is valid in all practical cases. The geometry of the cell considered is shown in fig. 2. The cylindrical vapor cell has mirrored side walls. In order to enhance quantum efficiency, the front and back walls are reflecting for 378 nm and 535 nm, respectively. Hence, uv photons can only escape toward the PMT, and green photons are only lost through the front window; all mirrors are assumed to be ideal. This geometry allows for a large receiving area of the filter. A cylindrical cell with arbitrary cross section and perfectly reflecting side walls is equivalent to an infinite slab, since we are only interested in the z-component of the atoms' position and of the photons' direction. 246
15 June 1992
The transient rate equation (excitation cut off at t - - 0 ) for the 7s atoms is a generalized Holstein equation:
On(z, t) - -Ot
l -n(z,t) z
L/2
+fl- f V
n ( z ' , t ) G U V ( z , z ' ) dz'
--L/2
L/2
+
(l-r)
|
d
V
n(z',t) Gg(z,z')dz ',
(1)
--L/2
where n(z, t) is the density of 7s atoms and GUV(z, z' ) is the probability that a 378 nm photon that is emitted at z' is reabsorbed at z; an explicit expression for G(z, z' ) is given in ref. [ 10]. Superscript uv denotes the 378 nm transition, superscript g denotes the 535 nm transition, and fl is the branching ratio of the uv transition (fl=47%). The solution for the two-level problem
On(z, t) 1 - _--n(z,t) Ot r L/2
if
+ -
V
n(z', t) G(z, z') dz' ,
(2)
--L/2
has been studied extensively. Eq. (2) is a Fredholm integral equation of the second kind. Solutions are of the form 1
n(z, t ) = ~ c t j ~ ( z ) e x p ( - t / g j v ) , gj= 1 - 2 j ' J
(3) where the 2j and the %(z) are the eigenvalues and the eigenfunctions, respectively. They can either be found by dividing the slab into stripes with constant ~,(z), i.e. approximating ~,(z) by a sum of pulse functions with unknown amplitudes and solving the resulting algebraic eigenvalue problem [10,11 ], or by using tabulations and approximations (see e.g. refs. [ 12-14] ). Thus, in the following we will assume that 2 "v, 2g, ~ f f ( z ) , and ~,~(z) are known. The c~j are the expansion coefficients of the initial distribution of 7s atoms in terms of the ~,j. Note that the ~,j constitute an orthonormal system of functions. If we denote the integral operator fdz' G(z, z' ) as A, and assuming that eq. ( 1 ) has solutions of the form
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS z
15 June 1992
z
t/'n
r,,,n0
spatial lowest-order mode
0;i'"'/'1.5*o ,,.-
/ out
reflecting for green
side walls I
a)
z--L/2
',~
...........................................
~ out
b)
c)
Fig. 2. (a) Geometry of the TI-ALFvapor cell. (b) Equivalent infinite slab geometry. (c) Spatialdistribution of the lowest-ordermode: only green transition trapped with 2k~L = 4 (dash-dotted); only uv transition trapped with 2k~ L = l0 (dotted); coupled mode (solid).
eq. (3), we are looking for solutions of the equation
2c~C(z)=flAUV~C(z)+(1-fl)Ag~C(z) ,
(4)
where superscript c denotes the coupled system. Introducing the expansions
~,*(z) = Y c,*k~y,V(z), ~u~V(z)= Y, cUZcO(z), k
k
~C(z)= Y~ c ~ V ( z ) ,
(5)
after N terms. We thus have to find the N eigenvalues 2,c. and eigenvectors ci% ( i = 1 to N) of this system. In practice, N = 3 will give sufficient accuracy, and the eigenvalue problem can be solved analytically. The g ~ = l / ( 1 - 2 ~ ) are the decay constants of the coupled system, and the ci% determine the coupled eigenfunctions ¢t~(z) via
~ ( z ) = ~, c,~w~,V(z).
We wish to find out how many 378 nm photons will leave the slab. Denoting the expansion coefficients of the initial distribution in terms of the coupled eigenmodes as a~, the distribution of 7s atoms at any time is
into eq. (4), we get
+(1--,)AS(~CCk~CY, TCt,(z)).
(6)
We insert the solution of the two-level problem _ uv .4 u v ~ u v (z)-2j ~ u v (z) (and analogously for the green transition) into eq. (6), multiply by 9'~V(z) and integrate over the slab. Thus we get the solution of the coupled problem from
2c.,--fl2,.c.,+(l--fl) Y" ~. CkCkt2~C~,.. c
c
_
uv
(8)
k
k
¢
c
k
uv
(7)
I
Basically, eq. (7) is an infinite system of linear homogeneous algebraic equations, which we truncate
n(z, t) = ~, a~ q/~(z) e x p ( - t / g ~ z ) . Using the expansions of eq. (8), the number of 378 nm photons that are created at a point z within a time interval [t, t + d t ] is
_p T
ot~ ~ C)~k~U~,v(Z) e x p ( - - t / g ~ z )
dt.
(9)
j
The escape probability of a 378 nm photon belonging to the kth mode of the uv transition is 1/g~V. Thus the number of photons that are created between t and t+dt and leave the slab without reabsorption is, 247
Volume 90, number 4,5,6
-r j
J . cik
OPTICS COMMUNICATIONS
~'~V(z)dzg~ v
× e x p ( - t / g ~ z ) dt .
(10)
Eq. (10) gives the time behavior of the emergent radiation. If we integrate this over time and compare it to the number of 7s atoms at t=0, we finally get the formula for the quantum efficiency q ofa TI filter
gk
/
--I
The g~V and the ~'~V(z) are the solutions of the basic Holstein equation, eq. (2). The g~ and cj5 are the solutions of eq. (7) and the a~ are the expansion coefficients of the initial distribution in terms of the ~,~(z), which are known from eq. (8). We have thus decoupled the trapping effects of the green and the uv transition. In contrast to the direct solution of the generalized Holstein equation, where the solution depends on two parameters (kSVL and k~L, where the ko are the absorption coefficients at line center), our approach allows to solve the two one-parameter basic Holstein equations for the green and the uv transitions separately, and then to combine the results by eq. (7). This allows for a much more efficient simulation of the trapping effects. If we want to know r/and SBW of qUVq~combinations of uv and green opacities, our approach requires only q,V+ q~ solutions of the basic Holstein equation instead of qUVqgsolutions of the generalized Holstein equation. We made the following assumption for the numerical simulation of T1-ALFs: - In order to have minimum optical bandwidth, the vapor contains only one isotope, Z°STI; the hyperfine structure of this isotope is taken into account
[15]. -A Voigt profile with a=0.946 for the green transition and a = 0.9 for the uv transition is assumed (a is the Voigt parameter (In 2) ~/2 × Lorentz-fwhm/ Doppler-fwhm), corresponding to 100 mbar argon gas pressure [ 16,17 ] and a temperature of 700 K in the T1 cell. This buffer gas pressure gives a high lifetime of the metastable state (see ref. [18] ) and re248
15 June 1992
duces pump power requirements. It is assumed that there is no collisional intermixing between the two hfs levels of the 7s state. In this case the ratio of green to uv opacity is about twice the population ratio of metastable over ground state density. - The cell is completely reflecting for 378 nm at the top and for 535 nm at the bottom. Thus the effective cell length is 2L, and the eigenfunctions for the uncoupled trapping problems have their maximum at L/2 and - L / 2 , respectively (see fig. 2 for the shape of the lowest-order modes). - The distribution of metastables is assumed to be spatially uniform. The validity of this assumption depends on the pumping geometry and will have to be examined in each case; however, it will always give a good estimate. - We define the SBW as 1/(2nrg~), i.e., we neglect higher-order modes for the time decay to make the definition independent of the initial distribution of 7s atoms. In a low-opacity vapor, g~ = 1 and SBW is 21 MHz. SBW will be reduced by a factor of 1/g~ in a radiation trapped ALF. With the first four assumptions, we solve eq. (2) for the uv and the green transitions by dividing the slab into 81 stripes and using the procedure of ref. [ 11 ] described above. As an example, simulations for the coupled threelevel system were done for three values of k~L. In the case 2k~L=4, the input was tuned to line center. For the other two line center opacities (2k~L= 10 and 2k~L = 50) the input was detuned in such a way that the opacity at input frequency remained 4. This was done to avoid a high concentration of 7s atoms near the input window, as green photons originating in this region can escape easily. In both cases, the signal absorption amounted to 98.2% and the initial distribution of 7s atoms was n (z, 0) = exp [ - 2 (z + 0.5L) ] +exp[ - 2 ( 1 . 5 L - z) ]. In a direct pumping scheme [9], nm/ng,d is limited to values less than 2, reaching 2 in the limit of infinite pumping intensity. In an indirect pumping scheme, e.g. when pumping intermittently via the 7s level [6], complete inversion can be achieved and the ratio of green to uv opacity can become very large. Thus, depending on the pumping scheme and the available pump power, the uv opacity may vary in a large range. In our simulation, we varied 2k~VL between 0.1 and 10.
Volume 90, number 4,5,6
OPTICSCOMMUNICATIONS
Fig. 3 shows the dependence of r/and SBW (as computed from eqs. ( l 1 ) and (7)) on the opacity of the uv transition at constant green opacities. The number of modes that were taken into account is 3; the results were checked with a Monte Carlo simulation and agreement within 0.5% was found. From fig. 3 and from further simulations, we can infer the following design rules for T1-ALFs for optimized q: - In direct pumping schemes, only a low opacity ratio between green and uv transition can be reached. When the 1.28 Hm transition is pumped with its saturation intensity (on the order of 1 W/cmZ), the opacity ratio is about 2. According to the simulations, quantum efficiency is much lower for lower opacity ratios. Hence indirect pumping schemes, were population inversion can be reached, have the potential for higher quantum efficiency. In a direct pumping scheme, quantum efficiency is limited to about 60% for reasonable SBW and pump power. As an example, q=56% and g~=2.57 when the green opacity is 4 and we pump with the saturation intensity so that the uv opacity is 2). In an indirect pump-
7
....
3.0
5
.........
4.2
N ,,,.,..
o~
.~_
4
~ + ~
5.3
e-
3
~,.I~
7.1
.~
2
,o.6
1
......
0.1 I="
100%
F-
___..~_~_
e-
m
"
ing scheme, r/=90% and S B W = I 0 MHz (corresponding to g~=2.1) can be achieved (e.g. with 2k~VL=0.1, 2kilL= 50). - For pumping schemes where only a low greento-uv opacity ratio can be achieved, a low uv opacity is recommended. In order to fulfill this requirement and to provide for sufficient signal absorption, 2kilL should be about 3 to 4. - If high opacity ratios can be achieved, it is advantageous to have a high green opacity, and to detune the input from the center, as in the example above. The high green opacity keeps the photons in the vapor until they are converted into uv output. If, for example, an opacity ratio 100" 1 can be realized, 2 k [ L = 5 0 and 2k~vL=0.5 yield r/=87%, while 2 k [ L = 4 and 2k~vL=0.04 yield only r/=72%. - It is interesting to note that 2k[L/2k~VL= l results in a quantum efficiency q lower than 50% (e.g. 48% if 2 k [ L = 4 ) . This is due to the fact that the initial distribution of 7s atoms has a maximum at the input window. However, q is not the only important parameter of an ALF. High green opacity increases the optical bandwidth and decreases the SBW. Trapping, and hence the quantum efficiency q, is also influenced by the pressure of a noble buffer gas, as are the lifetime of the metastable state, pump power requirements, and the optical bandwidth. A tradeoff between these performance characteristics depends on the individual application. Considering that one has to simulate many combinations of green opacity k~L and uv opacity k~VL to find the optimum combination for a specific application, we believe that the new method is especially valuable.
lO
~
--'---'-
~"
21.2
1
15 June 1992
~,l
Part of this work was done for the European Space Agency under ESTEC Contract No. 9 5 1 6 / 9 1 / N L / PB(SC).
]i
50% E
--
i
,
2kgL---J
,
I'
References
i 3
'
o"
I 0% 0,1
1
2
10
k~'k
Fig. 3. Quantum efficiencyq and trapping factorg[ for effective green opacity 2k[L=4, 2k~L=l0 and 2k[L=50. The signal bandwidth is defined as SBW=21 MHz/g[ (see text).
[I]J.A. Gelbwachs, IEEE J. Quantum Electron. 24 (1988) 1266. [2] T.M. Shay,Optics Comm. 77 (1990) 368. [3] J.B. Marling, J. Nilsen, L.C. West and L.L. Wood,J. Appl. Phys. 50 (1979) 610. [4] J.R. Lesh, L.J. Deutsch and W.J. Weber, Proc. Soc. PhotoOpt. Instrum. Eng. 1522 (1991) 27. 249
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS
[ 5 ] E. Korevaar, M. Rivers and C.S. Liu, Proc. Soc. Photo-Opt. lnstrum. Eng. 1059 (1989) 111. [6] B.P. Oehry, W. Schupita and G. Margerl, Optics Lett. 16 (1991) 1620. [7] R. Scheps, Appl. Optics 28 (1989) 5288. [ 8 ] C.S. Liu, P.J. Chantry and C.L. Chen, Proc. Soc. Photo-Opt. lnstrum. Eng. 709 (1986) 132. [ 9 ] T.M. Colbert, B.J. Feldman and B.L. Wexler, LEOS'91: IEEE Lasers and Electro-Optics Society 1991 annual meeting, San Jose, CA, paper OSM 7.4. [ 10] L.M. Biberrnan, Zh. Eksp. Teor. Fiz. 17 (1947) 416. [ 11 ] A.V. Phelps, Phys. Rev. 110 (1958) 1362.
250
15 June 1992
[ 12] T. Holstein, Phys. Rev. 72 (1947) 1212. [ 13 ] C.W. Scherr, Phys. Rev. B 4 ( 1971 ) 3885. [ 14] R.P. Blickensderfer, W.H. Breckenridge and J. Simons, J. Phys. Chem. 80 (1976) 653. [ 15 ] A.I. Odintsov, Opt. Spectrosc. 9 (1960) 75. [16] R.S. Dygdala, R. Bobkowski and E. Lisicki, J. Phys. B 22 (1989) 1563. [17 ] R.S. Dygdala, R. Bobkowski, E. Lisicki and J. Szudy, Z. Naturforsch. a 42 (1987) 559. [ 18 ] G. Magerl, B.P. Oehry and W. Ehrlich-Schupita, Final report to the European Space Agency under ESTEC contract No. 8488/89/NL/PM(SC) ( 1991 ).
OPTICS COMMUNICATIONS
Quantum efficiency and signal bandwidth of thallium atomic line filters Andreas F. Molisch, Bernhard P. Oehry, Walter Schupita and Gottfried Magerl Institut ]~r Nachrichtentechnik und Hochfrequenztechnik, Technische Universittit Wien, Guflhausstrafle 25/389, A- I 040 Wien, Austria
Received 3 December 1991; revised manuscript received 3 March 1992
We derive a new, simplified method for the computation of trapping effects in coupled three-level atomic systems. The linearity of the Holstein radiation trapping equation allows the reduction of the general rate equation to the solution of the basic two-level Holstein equation and to an algebraic eigenvalue problem. The method is applied to the numerical simulation of the quantum efficiency and the signal bandwidth of thallium atomic line filters. With a pumping scheme that achieves population inversion between the ground state and the metastable state, 90% quantum efficiency and l 0 MHz signal bandwidth can be achieved, while in noninverting pumping schemes, even quite high pump intensities result in less than 60% quantum efficiency and 8 MHz signal bandwidth. A tradeoffbetween quantum efficiency, signal bandwidth and pump power is possible.
A t o m i c line filters ( A L F s ) are ultra-narrow-band, wide-field-of-view optical filters for the detection o f weak optical signals e m b e d d e d in strong background noise; for an excellent review o f the underlying physics, see ref. [ l ]; a theoretical model for r u b i d i u m A L F s was d e v e l o p e d in ref. [ 2 ]. M a i n applications are underwater c o m m u n i c a t i o n s [3], deep space c o m m u n i c a t i o n s [ 4 ], satellite tracking [ 5 ] and atmospheric lidar systems [6 ]. One o f the most p r o m ising candidates for these applications is the thallium ALF, because the input transition overlaps the emission wavelength o f a well-developed solid-state laser, the frequency d o u b l e d N d : B E L (BEL stands for l a n t h a n u m beryllate, LaEBe205 [ 7 ] ). The basic operating principle o f T1-ALFs can be u n d e r s t o o d from the term scheme o f thallium in fig. I. Since T1-ALFs are active systems, the metastable 6P3/2 level has to be p o p u l a t e d by some p u m p i n g mechanism [ 8,6,9 ]. An incoming signal photon with the wavelength 535 nm can then be absorbed, and it excites a metastable a t o m to the 7s state. The desired upconversion o f green photons to uv photons is achieved when the 7s level decays to the ground state with the emission o f a 378 n m photon. The cell containing the atomic vapor is located between two dyedglass filters: a longwave-pass filter in front o f the in-
"; -7.5ns
\
\
F-1
~
input
output \ 378nm \
..
6 2P3/2"'~
F=2 F=I
,.'" ." 1.281.tm
, ,,"" t
82pll2~
F=I F=O
Fig. 1. Thallium energy level diagram (hyperfine splitting not to scale ). put window a n d a uv b a n d p a s s in front o f the detector ( p h o t o m u l t i p l i e r tube, P M T ) . In c o m b i n a tion, these filters are completely opaque. Only photons that are wavelength-shifted by the v a p o r can reach the detector. As the absorption b a n d w i d t h o f the T1 metastables is extremely small (only a few G H z ) , the system acts as a very-narrow-band optical filter. However, the above description is overly simplified. Actually, a 7s a t o m will decay with about 50%
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
245
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS
probability either to the ground state or to the metastable state, emitting a 378 nm uv or a 535 nm green photon, respectively. These fluorescence photons may escape from the cell or be reabsorbed by the vapor. A uv photon is reabsorbed by ground-state atoms with the probability pUV, while green photons will be reabsorbed with a probability pg. In either case, the reabsorption result in the creation of another 7s atom which will then decay again to either the ground state or to the metastable state. This process will be repeated until a uv or a green photon finally leaves the cell. As 535 nm radiation is blocked from the PMT, all green photons leaving the cell are lost for the detection. Thus it is obvious that pU~ and pg determine the quantum efficiency q of the filter; q is defined as the number of uv photons leaving the cell over the number of absorbed signal photons. The probabilities pUV and p g a r e proportional to the ground state density r/gnd and the metastable state density nm, respectively. Furthermore, the radiation trapping effect described above will strongly influence the response time of the filter. A signal photon that goes through i absorption-reemission processes will take iv seconds to reach the detector (r is the lifetime of the 7s state) and the signal bandwidth, SBW, is lowered accordingly. In the following, we develop a method that allows for the computation of the quantum efficiency q and the SBW in a typical TI-ALF cell in dependence of ng, d and nm. The basic assumption for the following derivation is that the problem is linear, i.e. the incoming signal photons must not change ng.d or n m appreciably. Since ALFs are designed for the detection of very low light levels, this assumption is valid in all practical cases. The geometry of the cell considered is shown in fig. 2. The cylindrical vapor cell has mirrored side walls. In order to enhance quantum efficiency, the front and back walls are reflecting for 378 nm and 535 nm, respectively. Hence, uv photons can only escape toward the PMT, and green photons are only lost through the front window; all mirrors are assumed to be ideal. This geometry allows for a large receiving area of the filter. A cylindrical cell with arbitrary cross section and perfectly reflecting side walls is equivalent to an infinite slab, since we are only interested in the z-component of the atoms' position and of the photons' direction. 246
15 June 1992
The transient rate equation (excitation cut off at t - - 0 ) for the 7s atoms is a generalized Holstein equation:
On(z, t) - -Ot
l -n(z,t) z
L/2
+fl- f V
n ( z ' , t ) G U V ( z , z ' ) dz'
--L/2
L/2
+
(l-r)
|
d
V
n(z',t) Gg(z,z')dz ',
(1)
--L/2
where n(z, t) is the density of 7s atoms and GUV(z, z' ) is the probability that a 378 nm photon that is emitted at z' is reabsorbed at z; an explicit expression for G(z, z' ) is given in ref. [ 10]. Superscript uv denotes the 378 nm transition, superscript g denotes the 535 nm transition, and fl is the branching ratio of the uv transition (fl=47%). The solution for the two-level problem
On(z, t) 1 - _--n(z,t) Ot r L/2
if
+ -
V
n(z', t) G(z, z') dz' ,
(2)
--L/2
has been studied extensively. Eq. (2) is a Fredholm integral equation of the second kind. Solutions are of the form 1
n(z, t ) = ~ c t j ~ ( z ) e x p ( - t / g j v ) , gj= 1 - 2 j ' J
(3) where the 2j and the %(z) are the eigenvalues and the eigenfunctions, respectively. They can either be found by dividing the slab into stripes with constant ~,(z), i.e. approximating ~,(z) by a sum of pulse functions with unknown amplitudes and solving the resulting algebraic eigenvalue problem [10,11 ], or by using tabulations and approximations (see e.g. refs. [ 12-14] ). Thus, in the following we will assume that 2 "v, 2g, ~ f f ( z ) , and ~,~(z) are known. The c~j are the expansion coefficients of the initial distribution of 7s atoms in terms of the ~,j. Note that the ~,j constitute an orthonormal system of functions. If we denote the integral operator fdz' G(z, z' ) as A, and assuming that eq. ( 1 ) has solutions of the form
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS z
15 June 1992
z
t/'n
r,,,n0
spatial lowest-order mode
0;i'"'/'1.5*o ,,.-
/ out
reflecting for green
side walls I
a)
z--L/2
',~
...........................................
~ out
b)
c)
Fig. 2. (a) Geometry of the TI-ALFvapor cell. (b) Equivalent infinite slab geometry. (c) Spatialdistribution of the lowest-ordermode: only green transition trapped with 2k~L = 4 (dash-dotted); only uv transition trapped with 2k~ L = l0 (dotted); coupled mode (solid).
eq. (3), we are looking for solutions of the equation
2c~C(z)=flAUV~C(z)+(1-fl)Ag~C(z) ,
(4)
where superscript c denotes the coupled system. Introducing the expansions
~,*(z) = Y c,*k~y,V(z), ~u~V(z)= Y, cUZcO(z), k
k
~C(z)= Y~ c ~ V ( z ) ,
(5)
after N terms. We thus have to find the N eigenvalues 2,c. and eigenvectors ci% ( i = 1 to N) of this system. In practice, N = 3 will give sufficient accuracy, and the eigenvalue problem can be solved analytically. The g ~ = l / ( 1 - 2 ~ ) are the decay constants of the coupled system, and the ci% determine the coupled eigenfunctions ¢t~(z) via
~ ( z ) = ~, c,~w~,V(z).
We wish to find out how many 378 nm photons will leave the slab. Denoting the expansion coefficients of the initial distribution in terms of the coupled eigenmodes as a~, the distribution of 7s atoms at any time is
into eq. (4), we get
+(1--,)AS(~CCk~CY, TCt,(z)).
(6)
We insert the solution of the two-level problem _ uv .4 u v ~ u v (z)-2j ~ u v (z) (and analogously for the green transition) into eq. (6), multiply by 9'~V(z) and integrate over the slab. Thus we get the solution of the coupled problem from
2c.,--fl2,.c.,+(l--fl) Y" ~. CkCkt2~C~,.. c
c
_
uv
(8)
k
k
¢
c
k
uv
(7)
I
Basically, eq. (7) is an infinite system of linear homogeneous algebraic equations, which we truncate
n(z, t) = ~, a~ q/~(z) e x p ( - t / g ~ z ) . Using the expansions of eq. (8), the number of 378 nm photons that are created at a point z within a time interval [t, t + d t ] is
_p T
ot~ ~ C)~k~U~,v(Z) e x p ( - - t / g ~ z )
dt.
(9)
j
The escape probability of a 378 nm photon belonging to the kth mode of the uv transition is 1/g~V. Thus the number of photons that are created between t and t+dt and leave the slab without reabsorption is, 247
Volume 90, number 4,5,6
-r j
J . cik
OPTICS COMMUNICATIONS
~'~V(z)dzg~ v
× e x p ( - t / g ~ z ) dt .
(10)
Eq. (10) gives the time behavior of the emergent radiation. If we integrate this over time and compare it to the number of 7s atoms at t=0, we finally get the formula for the quantum efficiency q ofa TI filter
gk
/
--I
The g~V and the ~'~V(z) are the solutions of the basic Holstein equation, eq. (2). The g~ and cj5 are the solutions of eq. (7) and the a~ are the expansion coefficients of the initial distribution in terms of the ~,~(z), which are known from eq. (8). We have thus decoupled the trapping effects of the green and the uv transition. In contrast to the direct solution of the generalized Holstein equation, where the solution depends on two parameters (kSVL and k~L, where the ko are the absorption coefficients at line center), our approach allows to solve the two one-parameter basic Holstein equations for the green and the uv transitions separately, and then to combine the results by eq. (7). This allows for a much more efficient simulation of the trapping effects. If we want to know r/and SBW of qUVq~combinations of uv and green opacities, our approach requires only q,V+ q~ solutions of the basic Holstein equation instead of qUVqgsolutions of the generalized Holstein equation. We made the following assumption for the numerical simulation of T1-ALFs: - In order to have minimum optical bandwidth, the vapor contains only one isotope, Z°STI; the hyperfine structure of this isotope is taken into account
[15]. -A Voigt profile with a=0.946 for the green transition and a = 0.9 for the uv transition is assumed (a is the Voigt parameter (In 2) ~/2 × Lorentz-fwhm/ Doppler-fwhm), corresponding to 100 mbar argon gas pressure [ 16,17 ] and a temperature of 700 K in the T1 cell. This buffer gas pressure gives a high lifetime of the metastable state (see ref. [18] ) and re248
15 June 1992
duces pump power requirements. It is assumed that there is no collisional intermixing between the two hfs levels of the 7s state. In this case the ratio of green to uv opacity is about twice the population ratio of metastable over ground state density. - The cell is completely reflecting for 378 nm at the top and for 535 nm at the bottom. Thus the effective cell length is 2L, and the eigenfunctions for the uncoupled trapping problems have their maximum at L/2 and - L / 2 , respectively (see fig. 2 for the shape of the lowest-order modes). - The distribution of metastables is assumed to be spatially uniform. The validity of this assumption depends on the pumping geometry and will have to be examined in each case; however, it will always give a good estimate. - We define the SBW as 1/(2nrg~), i.e., we neglect higher-order modes for the time decay to make the definition independent of the initial distribution of 7s atoms. In a low-opacity vapor, g~ = 1 and SBW is 21 MHz. SBW will be reduced by a factor of 1/g~ in a radiation trapped ALF. With the first four assumptions, we solve eq. (2) for the uv and the green transitions by dividing the slab into 81 stripes and using the procedure of ref. [ 11 ] described above. As an example, simulations for the coupled threelevel system were done for three values of k~L. In the case 2k~L=4, the input was tuned to line center. For the other two line center opacities (2k~L= 10 and 2k~L = 50) the input was detuned in such a way that the opacity at input frequency remained 4. This was done to avoid a high concentration of 7s atoms near the input window, as green photons originating in this region can escape easily. In both cases, the signal absorption amounted to 98.2% and the initial distribution of 7s atoms was n (z, 0) = exp [ - 2 (z + 0.5L) ] +exp[ - 2 ( 1 . 5 L - z) ]. In a direct pumping scheme [9], nm/ng,d is limited to values less than 2, reaching 2 in the limit of infinite pumping intensity. In an indirect pumping scheme, e.g. when pumping intermittently via the 7s level [6], complete inversion can be achieved and the ratio of green to uv opacity can become very large. Thus, depending on the pumping scheme and the available pump power, the uv opacity may vary in a large range. In our simulation, we varied 2k~VL between 0.1 and 10.
Volume 90, number 4,5,6
OPTICSCOMMUNICATIONS
Fig. 3 shows the dependence of r/and SBW (as computed from eqs. ( l 1 ) and (7)) on the opacity of the uv transition at constant green opacities. The number of modes that were taken into account is 3; the results were checked with a Monte Carlo simulation and agreement within 0.5% was found. From fig. 3 and from further simulations, we can infer the following design rules for T1-ALFs for optimized q: - In direct pumping schemes, only a low opacity ratio between green and uv transition can be reached. When the 1.28 Hm transition is pumped with its saturation intensity (on the order of 1 W/cmZ), the opacity ratio is about 2. According to the simulations, quantum efficiency is much lower for lower opacity ratios. Hence indirect pumping schemes, were population inversion can be reached, have the potential for higher quantum efficiency. In a direct pumping scheme, quantum efficiency is limited to about 60% for reasonable SBW and pump power. As an example, q=56% and g~=2.57 when the green opacity is 4 and we pump with the saturation intensity so that the uv opacity is 2). In an indirect pump-
7
....
3.0
5
.........
4.2
N ,,,.,..
o~
.~_
4
~ + ~
5.3
e-
3
~,.I~
7.1
.~
2
,o.6
1
......
0.1 I="
100%
F-
___..~_~_
e-
m
"
ing scheme, r/=90% and S B W = I 0 MHz (corresponding to g~=2.1) can be achieved (e.g. with 2k~VL=0.1, 2kilL= 50). - For pumping schemes where only a low greento-uv opacity ratio can be achieved, a low uv opacity is recommended. In order to fulfill this requirement and to provide for sufficient signal absorption, 2kilL should be about 3 to 4. - If high opacity ratios can be achieved, it is advantageous to have a high green opacity, and to detune the input from the center, as in the example above. The high green opacity keeps the photons in the vapor until they are converted into uv output. If, for example, an opacity ratio 100" 1 can be realized, 2 k [ L = 5 0 and 2k~vL=0.5 yield r/=87%, while 2 k [ L = 4 and 2k~vL=0.04 yield only r/=72%. - It is interesting to note that 2k[L/2k~VL= l results in a quantum efficiency q lower than 50% (e.g. 48% if 2 k [ L = 4 ) . This is due to the fact that the initial distribution of 7s atoms has a maximum at the input window. However, q is not the only important parameter of an ALF. High green opacity increases the optical bandwidth and decreases the SBW. Trapping, and hence the quantum efficiency q, is also influenced by the pressure of a noble buffer gas, as are the lifetime of the metastable state, pump power requirements, and the optical bandwidth. A tradeoff between these performance characteristics depends on the individual application. Considering that one has to simulate many combinations of green opacity k~L and uv opacity k~VL to find the optimum combination for a specific application, we believe that the new method is especially valuable.
lO
~
--'---'-
~"
21.2
1
15 June 1992
~,l
Part of this work was done for the European Space Agency under ESTEC Contract No. 9 5 1 6 / 9 1 / N L / PB(SC).
]i
50% E
--
i
,
2kgL---J
,
I'
References
i 3
'
o"
I 0% 0,1
1
2
10
k~'k
Fig. 3. Quantum efficiencyq and trapping factorg[ for effective green opacity 2k[L=4, 2k~L=l0 and 2k[L=50. The signal bandwidth is defined as SBW=21 MHz/g[ (see text).
[I]J.A. Gelbwachs, IEEE J. Quantum Electron. 24 (1988) 1266. [2] T.M. Shay,Optics Comm. 77 (1990) 368. [3] J.B. Marling, J. Nilsen, L.C. West and L.L. Wood,J. Appl. Phys. 50 (1979) 610. [4] J.R. Lesh, L.J. Deutsch and W.J. Weber, Proc. Soc. PhotoOpt. Instrum. Eng. 1522 (1991) 27. 249
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS
[ 5 ] E. Korevaar, M. Rivers and C.S. Liu, Proc. Soc. Photo-Opt. lnstrum. Eng. 1059 (1989) 111. [6] B.P. Oehry, W. Schupita and G. Margerl, Optics Lett. 16 (1991) 1620. [7] R. Scheps, Appl. Optics 28 (1989) 5288. [ 8 ] C.S. Liu, P.J. Chantry and C.L. Chen, Proc. Soc. Photo-Opt. lnstrum. Eng. 709 (1986) 132. [ 9 ] T.M. Colbert, B.J. Feldman and B.L. Wexler, LEOS'91: IEEE Lasers and Electro-Optics Society 1991 annual meeting, San Jose, CA, paper OSM 7.4. [ 10] L.M. Biberrnan, Zh. Eksp. Teor. Fiz. 17 (1947) 416. [ 11 ] A.V. Phelps, Phys. Rev. 110 (1958) 1362.
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[ 12] T. Holstein, Phys. Rev. 72 (1947) 1212. [ 13 ] C.W. Scherr, Phys. Rev. B 4 ( 1971 ) 3885. [ 14] R.P. Blickensderfer, W.H. Breckenridge and J. Simons, J. Phys. Chem. 80 (1976) 653. [ 15 ] A.I. Odintsov, Opt. Spectrosc. 9 (1960) 75. [16] R.S. Dygdala, R. Bobkowski and E. Lisicki, J. Phys. B 22 (1989) 1563. [17 ] R.S. Dygdala, R. Bobkowski, E. Lisicki and J. Szudy, Z. Naturforsch. a 42 (1987) 559. [ 18 ] G. Magerl, B.P. Oehry and W. Ehrlich-Schupita, Final report to the European Space Agency under ESTEC contract No. 8488/89/NL/PM(SC) ( 1991 ).