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Theory of the superposition of a train of short laser pulses in a dye medium
Volume
12, number
1
THEORY
OPTICS COMMUNICATIONS
OF THE SUPERPOSITION
OF A TRAIN
September
OF SHORT
LASER
1974
PULSES
IN A DYE MEDIUM* S.L. CHIN Laboratoire de Recherches en Optique et Laser, B’partement de Physique, UniversitC Laval, Qukbec IO, Canada Cl K 7P4
Received
15 July 1974
A new way is proposed in which a train of picosecond pulses is superposed tions and stimulated emissions in a suitably prepared dye medium.
1. Introduction Recently, Kastler [3] gave an in-depth analysis of the effect of a Fabry-P&rot interferometer on a short light pulse during the International Symposium on Optics and Lasers. After that talk, Stoicheff gave a short comment by introducing the following idea. He said that a train of short pulses could be superimposed if there existed a “magic” Fabry-P&ot interferometer which could trap the pulses in the following sense. One sent in the pulse train perpendicular to the mirror surface. The entrance mirror would “magically” let pass the first pulse which is then totally reflected by the second mirror back to the first mirror at the same point of entrance. The spacing of the mirrors could be adjusted such that the reflected first pulse arrived at the first mirror as the second pulse did. Now, if the first “magic” mirror would transmit the second pulse but reflect the first, the two pulses would be superimposed, propagating together towards the second mirror which reflected them back to the first mirror, at which time the third pulse arrived. The same process repeated for the rest of the pulses until at last all the pulses were superimposed. Then the second mirror suddenly became transparent and one would obtain a
* The research
for this paper was supported in part by the National Research Council of Canada, Grant number A-95 11 and in part by the Defence Research Board of Canada, Grant number 95 1 O-l 16.
on one another
via successive
absorp.
single “fat” pulse at the output. This “wild” idea of obtaining a single short pulse from a pulse train has an obvious economical advantage over the presently known technique of switching out a single pulse from a pulse train. The above idea was indeed not “wild”, for the author has found a way to modify it theoretically making it physically possible. The theory is presented in this paper.
2. Theory The basic idea is illustrated by considering a train of two pulses. By sending the pulses through a suitably prepared fluorescing medium (to be discussed later), the energy of the first pulse is absorbed and in the form of a population inversion of the medium. This energy is released via stimulated emission when the second pulse arrives. Hence, two pulses become one after passing through the medium. If there are 4 pulses, every adjacent pair of pulses will collapse into one pulse. By sending the pulses through a series of identical media of suitable lengths, the output of the lst, 2nd, 3rd ,... medium will contain 4/2, q/4, q/8 ,... pulses. In general, 4 = 2” + h, /I being an integer, 0 5 h < 2m. If 11= 0, all the pulses collapse into one pulse at the output of the (m - 1)th medium. If Il # 0, only the first 2m pulses become one pulse at the output of the (m - 1)th medium, the rest being absorbed. In
Volume
12. number
I
OPTICS
-
Dye
COMMUNICATIONS
medium
-x x =o (b)
(a)
I$?. 1. (a) Fnergy levels (schematic) inside the ground and first excned singlets of a dye medium. (b) Two short pulses being sent into the dye medium.
2
1974
enters the medium at x = 0. Only those levels higher than level 1 (i.e. level 3) will absorb causing a transition from 3 to 4 which decays in < - 1 psec to level 2 (condition 2). Hence, the energy of the first pulse is accumulated in level 2 resulting in an inversion between N, and hl,. Since all the levels are in a steady state due to the cw pumping, the rate of change of the population of any level is zero before the arrival of the first pulse. When the first pulse arrives, level 3 absorbs the photons of the first pulse according to the rate eq. dN$X> f) dt
practice, the in media can be combined into one. It follows from the above considerations that the conditions for the molecules in such a medium to work are: I ) initially, the populations N, , N2 of the two levels (2 being the upper level) responsible for the stimulated emission are equal; 2) the absorption of the first pulse is between two other levels 3 and 4 of the same molecule. Fig. I a. (4 being the upper level) in which level 4 decays radiationlessly to level 2 in a time short compared to the time T between two adjacent pulses; 3) the fluorescence lifetime 7 of level 2 is longer than i(q .- h)T = Irnml T. The above conditions can be satisfied by a system of fluorescing organic dye molecules. We can choose levels 1 and 3 to be inside the ground electronic singlet band, 3 being higher. They lie close to the lowest vibrational- rotational level (within about a few 0.01 eV) such that at room temperature, levels 1 and 3 are still significantly populated, assuming thermal equilibrium. Levels 2 and 4 (4 being higher) are inside the first excited electronic singlet band, with level 2 at the lowest vibrational-rotational level. These levels satisfy the I:‘, and I:‘,~ E, = hu. I: conditions: lf4 ~ I:‘, = I:‘, being energy and v the &an (carrier) frrequency of the pulses. Consider the medium to be of unit cross section lying along the x-axis (fig. 1t)). Initially, the dye molecules are uniformly pumped by a cw light source or by a pulse long compared to the pulse train such that NI = N, is always maintained (condition 1). Amplified spontaneous emission between 2 and 3 is neglected by controlling the parameters of the dye molecules [ 11. Meanwhile, we send in a picosecond pulse train with hv = E2 ~ I:‘, , and T - nanoseconds. Consider the first two pulses. At time t = 0, the first pulse
September
= ~ oaN3(x, t)n, (x, t)>
(1)
where N always denotes the number density of a level and the subscript, the level concerned. Similarly, n denotes the photon flux, and the subscript, the order of a pulse. u, is the absorption cross section. Due to the uniform cw pumping, the initial density NjO of level 3 is uniform along x. Hence the first pulse effectively sees a constant density N30 of level 3 as it passes through the medium and the photon flux is reduced by the absorption by level 3. Hence, at the position x, the photon flux change is dn, (x, t) = -u,&~J~~
(x, t)dx.
(2)
For simplicity, we assume the pulses to be triangular in shape with halfwidth 71. Integrating eqs. (1) and (2) and combining, we get N3(x)
=N30exp[-~a~,0~1exp(-oaN,,x)1,
(3)
where 11~”is the peak flux of the first pulse. Hence, there are IV,,, ~ N3(x) molyles raised to level 4 which decays to level 2 in a time - 1 ps < r, where r is the fluorescence lifetime of the molecules. Since 7, < T and the transition time between 3 a 4 is also short compared to T, we can assume t = 0 to be the moment when the molecules have arrived at 2 from 4. Then, the initial density of level 2 at t = 0 is
N2,(x) r N, + NjO ~- N3(x),
(4)
where N, is a constant equal to the density of level 1 before the arrival of the pulses. The rate equation of level 2 is (before the 2nd pulse arrives)
Volume
12, number
1
OPTICS COMMUNICATIONS
fl,(X> t) =_-N2(x>t>
(5)
2
at
7
where amplified spontaneous emission is neglected as mentioned earlier. Also neglected are excited singlet absorption and singlet-triplet cross over. Integrating eq. (5) between t = 0 and t = T, we have -
N2(x) = ]Nr + N,,
(6)
Nj(x)l exp(-T/+
Now if the pulse train were not there, the loss of N,, given by N, - N, exp(-T/r), would be compensated for by the extra pumping maintaining the system at N2 = N, . Thus, taking the extra pumping into account, this loss should be added to eq. (6) resulting in N2(x) = N, + ]N3u -
N3(x)1ew-WI.
(7)
Using eq. (3), we get N2(x) -N,
=N30 (1 -
exp[-o,nlo71exp(-oa~~x)l}
X exp (-T/r)
(8)
which is positive. An inversion thus exists. Now (at t = T) the second pulse arrives and is amplified according to the following equation, provided r > T (condition 3) dn, = aen2(N2
- N,)dx
- uaN3n2dx,
(9)
where u, is the stimulated emission cross section [l] . Here, for simplicity, only the loss due to the absorption by level 3 is included, others such as triplet loss and internal conversion being neglected. The second pulse will be amplified up to a maximum distance L at which the gain equals the loss, i.e. from eq. (9), ue(N2 - N1) - aaN
= 0.
(10)
Using eqs. (3) and (8), with x = L, we get from eq. (10) uan loT1 In [ 1 +
(~a/oehW~)l
September
1974
the maximum length for combining the third and fourth pulses by simply replacing nlOrl by f23nr3, where n30 is the peak flux and r3, the half-width of the third pulse. In general, the maximum length for combining the (2m - l)th and the 2mth pulses is given by eq. (1 I) with nlOrl replaced by n2m_1,072m_1. Consider the special case in which all the 4 pulses are identical, 4 = 2m + h, 0 5 !I < 2m. As mentioned in the beginning, the last h pulses will be lost after passing through m - 1 media. Hence, we consider only the first 2m pulses and (m ~ 1) identically prepared media of maximum lengths L 1, L 2, - - *, L,_, . The length of the first medium, L 1, is given by eq. (11). For the sake of simplicity, we assume small amplification. This can be achieved by spreading the beam diameter. Thus, the pulse shape is preserved [2] and the output of L 1 is a train of 2m-1 identical triangular pulses of pulse separation 2T. In general, the pulse separation in the pulse train at the output of the kth medium is 2kT. The length Lk is given by eq. (11) with nlOrl and T replaced by ?rkork and 2kT. Here, nkn and rk are the peak flux and the half-width of each pulse in the pulse train at the exit of the kth medium. They can be calculated using a series of equations similar to eq. (9). We shall not do it here. We now use Rhodamine 6G as an example. To show that the above theory is indeed physically possible, it is sufficient to show that L is real and experimentally accessible. Consider first of all a train of eight identical triangular picosecond pulses from a Rhodamine 6G dye laser. Let the wavelength be around 570 nm. We need three successive media of maximum lengths L 1, L,, L3 to combine the 8 pulses into one. Suppose that each pulse is of 1 ps duration with energy 5 mJ and T = 1 ns. Then nlOrl - 1016 cmm2. We choose our pulse combining medium to be Rhodamine 6G in ethanol at a concentration of 10e3 M. At room temperature, NjO - 1Or6 cmm3, corresponding to a level - 0.01 eV above the lowest level of the singlet ground state, assuming a Boltzmann distribution. From ref. [ 11 , ‘I, 10-16cm2,r-10ns.Sub5 X lo-l7 cm2 (5 -1.5X stituting these ;alies into eq. (1 l), we get L 1 z 1 cm. To calculate L,, we should replace nlorl and Tby n20~2 and 2T. Note that (n2072)/2 is the energy each pulse in the pulse train at the exit of the first medium. Its maximum possible value is 2(nr0r1)/2, i.e. twice the energy in a pulse of the pulse train. Hence, by putting 2nr0r1 and 2T in place of nrOrl and Tin eq. (1 l),
b&-,. (11)
This is the maximum length of the medium which will combine the first two pulses into one without substantial loss. A similar expression can be written down for
3
Volume
12, number
1
OPTICS COMMUNICATIONS
an upper limit ofL2; i.e. L, 5 2.3 cm. Similarly, by putting 4n1ur1 yd 4T in place of nlorl and Tineq.(11),wegetL310 cm. Hence, all the lengths are real. In practice, they can be combined into one of length L 1 + L 2 + L j. It can be pumped by a flashlamp or an argon laser, etc. If there are more pulses in the train, the length of the combined medium will be too long to be practical. To overcome this difficulty, the medium can be placed between two totally reflecting surfaces. The pulse train is sent through the side into the medium, reflected back and forth between the mirrors and eventually comes out through the other side, becoming one single pulse, as shown in fig. 2. The path length inside the medium can be adjusted easily by changing the angle of incidence or the spacing between the mirrors. we obtain
September
1974
begins. The disadvantage is the waste of pulse energy in preparing the medium while the advantage is its simplicity. Another way is to use another molecular (or atomic) species to which levels 3 and 4 belong. This species should absorb the first pulse and transfer the energy via collision to level 2 of the original species in a time short compared to the time T between two adjacent pulses.
4. Conclusion We have shown that it is physically possible to combine a train of short pulses into a single short pulse through successive absorption and stimulated emission by a special medium.
Acknowledgement The author thanks Dr. G. BCdard and Dr. A. Zardecki for helpful discussions. Fig. 2. Alternate
design of the dye medium.
References 3. Discussion 111 O.G. Peterson, In principle, there are other alternative ways of combining a train of short pulses. One obvious way is to use the same medium described above except that it is not prepared at N2 = N, . One can simply send in the pulse train. The first few pulses will serve to pump the medium to N2 = N, after which pulse combination
4
J.P. Webb, W.C. McColgin and J.H. Eberly, J. Appl. Phys. 42 (1971) 1917. I21 D. R&s, Lasers, Light Amplifiers and Oscillators (Academic Press Inc., London, 1969), p. 80. 131 A. Kastler, talk presented at the Internat. Symp. on Op tics and Lasers, Universitd de Q&bec, Irois Riviires, Q&bec, Canada (November 9-10, 1973).
12, number
1
THEORY
OPTICS COMMUNICATIONS
OF THE SUPERPOSITION
OF A TRAIN
September
OF SHORT
LASER
1974
PULSES
IN A DYE MEDIUM* S.L. CHIN Laboratoire de Recherches en Optique et Laser, B’partement de Physique, UniversitC Laval, Qukbec IO, Canada Cl K 7P4
Received
15 July 1974
A new way is proposed in which a train of picosecond pulses is superposed tions and stimulated emissions in a suitably prepared dye medium.
1. Introduction Recently, Kastler [3] gave an in-depth analysis of the effect of a Fabry-P&rot interferometer on a short light pulse during the International Symposium on Optics and Lasers. After that talk, Stoicheff gave a short comment by introducing the following idea. He said that a train of short pulses could be superimposed if there existed a “magic” Fabry-P&ot interferometer which could trap the pulses in the following sense. One sent in the pulse train perpendicular to the mirror surface. The entrance mirror would “magically” let pass the first pulse which is then totally reflected by the second mirror back to the first mirror at the same point of entrance. The spacing of the mirrors could be adjusted such that the reflected first pulse arrived at the first mirror as the second pulse did. Now, if the first “magic” mirror would transmit the second pulse but reflect the first, the two pulses would be superimposed, propagating together towards the second mirror which reflected them back to the first mirror, at which time the third pulse arrived. The same process repeated for the rest of the pulses until at last all the pulses were superimposed. Then the second mirror suddenly became transparent and one would obtain a
* The research
for this paper was supported in part by the National Research Council of Canada, Grant number A-95 11 and in part by the Defence Research Board of Canada, Grant number 95 1 O-l 16.
on one another
via successive
absorp.
single “fat” pulse at the output. This “wild” idea of obtaining a single short pulse from a pulse train has an obvious economical advantage over the presently known technique of switching out a single pulse from a pulse train. The above idea was indeed not “wild”, for the author has found a way to modify it theoretically making it physically possible. The theory is presented in this paper.
2. Theory The basic idea is illustrated by considering a train of two pulses. By sending the pulses through a suitably prepared fluorescing medium (to be discussed later), the energy of the first pulse is absorbed and in the form of a population inversion of the medium. This energy is released via stimulated emission when the second pulse arrives. Hence, two pulses become one after passing through the medium. If there are 4 pulses, every adjacent pair of pulses will collapse into one pulse. By sending the pulses through a series of identical media of suitable lengths, the output of the lst, 2nd, 3rd ,... medium will contain 4/2, q/4, q/8 ,... pulses. In general, 4 = 2” + h, /I being an integer, 0 5 h < 2m. If 11= 0, all the pulses collapse into one pulse at the output of the (m - 1)th medium. If Il # 0, only the first 2m pulses become one pulse at the output of the (m - 1)th medium, the rest being absorbed. In
Volume
12. number
I
OPTICS
-
Dye
COMMUNICATIONS
medium
-x x =o (b)
(a)
I$?. 1. (a) Fnergy levels (schematic) inside the ground and first excned singlets of a dye medium. (b) Two short pulses being sent into the dye medium.
2
1974
enters the medium at x = 0. Only those levels higher than level 1 (i.e. level 3) will absorb causing a transition from 3 to 4 which decays in < - 1 psec to level 2 (condition 2). Hence, the energy of the first pulse is accumulated in level 2 resulting in an inversion between N, and hl,. Since all the levels are in a steady state due to the cw pumping, the rate of change of the population of any level is zero before the arrival of the first pulse. When the first pulse arrives, level 3 absorbs the photons of the first pulse according to the rate eq. dN$X> f) dt
practice, the in media can be combined into one. It follows from the above considerations that the conditions for the molecules in such a medium to work are: I ) initially, the populations N, , N2 of the two levels (2 being the upper level) responsible for the stimulated emission are equal; 2) the absorption of the first pulse is between two other levels 3 and 4 of the same molecule. Fig. I a. (4 being the upper level) in which level 4 decays radiationlessly to level 2 in a time short compared to the time T between two adjacent pulses; 3) the fluorescence lifetime 7 of level 2 is longer than i(q .- h)T = Irnml T. The above conditions can be satisfied by a system of fluorescing organic dye molecules. We can choose levels 1 and 3 to be inside the ground electronic singlet band, 3 being higher. They lie close to the lowest vibrational- rotational level (within about a few 0.01 eV) such that at room temperature, levels 1 and 3 are still significantly populated, assuming thermal equilibrium. Levels 2 and 4 (4 being higher) are inside the first excited electronic singlet band, with level 2 at the lowest vibrational-rotational level. These levels satisfy the I:‘, and I:‘,~ E, = hu. I: conditions: lf4 ~ I:‘, = I:‘, being energy and v the &an (carrier) frrequency of the pulses. Consider the medium to be of unit cross section lying along the x-axis (fig. 1t)). Initially, the dye molecules are uniformly pumped by a cw light source or by a pulse long compared to the pulse train such that NI = N, is always maintained (condition 1). Amplified spontaneous emission between 2 and 3 is neglected by controlling the parameters of the dye molecules [ 11. Meanwhile, we send in a picosecond pulse train with hv = E2 ~ I:‘, , and T - nanoseconds. Consider the first two pulses. At time t = 0, the first pulse
September
= ~ oaN3(x, t)n, (x, t)>
(1)
where N always denotes the number density of a level and the subscript, the level concerned. Similarly, n denotes the photon flux, and the subscript, the order of a pulse. u, is the absorption cross section. Due to the uniform cw pumping, the initial density NjO of level 3 is uniform along x. Hence the first pulse effectively sees a constant density N30 of level 3 as it passes through the medium and the photon flux is reduced by the absorption by level 3. Hence, at the position x, the photon flux change is dn, (x, t) = -u,&~J~~
(x, t)dx.
(2)
For simplicity, we assume the pulses to be triangular in shape with halfwidth 71. Integrating eqs. (1) and (2) and combining, we get N3(x)
=N30exp[-~a~,0~1exp(-oaN,,x)1,
(3)
where 11~”is the peak flux of the first pulse. Hence, there are IV,,, ~ N3(x) molyles raised to level 4 which decays to level 2 in a time - 1 ps < r, where r is the fluorescence lifetime of the molecules. Since 7, < T and the transition time between 3 a 4 is also short compared to T, we can assume t = 0 to be the moment when the molecules have arrived at 2 from 4. Then, the initial density of level 2 at t = 0 is
N2,(x) r N, + NjO ~- N3(x),
(4)
where N, is a constant equal to the density of level 1 before the arrival of the pulses. The rate equation of level 2 is (before the 2nd pulse arrives)
Volume
12, number
1
OPTICS COMMUNICATIONS
fl,(X> t) =_-N2(x>t>
(5)
2
at
7
where amplified spontaneous emission is neglected as mentioned earlier. Also neglected are excited singlet absorption and singlet-triplet cross over. Integrating eq. (5) between t = 0 and t = T, we have -
N2(x) = ]Nr + N,,
(6)
Nj(x)l exp(-T/+
Now if the pulse train were not there, the loss of N,, given by N, - N, exp(-T/r), would be compensated for by the extra pumping maintaining the system at N2 = N, . Thus, taking the extra pumping into account, this loss should be added to eq. (6) resulting in N2(x) = N, + ]N3u -
N3(x)1ew-WI.
(7)
Using eq. (3), we get N2(x) -N,
=N30 (1 -
exp[-o,nlo71exp(-oa~~x)l}
X exp (-T/r)
(8)
which is positive. An inversion thus exists. Now (at t = T) the second pulse arrives and is amplified according to the following equation, provided r > T (condition 3) dn, = aen2(N2
- N,)dx
- uaN3n2dx,
(9)
where u, is the stimulated emission cross section [l] . Here, for simplicity, only the loss due to the absorption by level 3 is included, others such as triplet loss and internal conversion being neglected. The second pulse will be amplified up to a maximum distance L at which the gain equals the loss, i.e. from eq. (9), ue(N2 - N1) - aaN
= 0.
(10)
Using eqs. (3) and (8), with x = L, we get from eq. (10) uan loT1 In [ 1 +
(~a/oehW~)l
September
1974
the maximum length for combining the third and fourth pulses by simply replacing nlOrl by f23nr3, where n30 is the peak flux and r3, the half-width of the third pulse. In general, the maximum length for combining the (2m - l)th and the 2mth pulses is given by eq. (1 I) with nlOrl replaced by n2m_1,072m_1. Consider the special case in which all the 4 pulses are identical, 4 = 2m + h, 0 5 !I < 2m. As mentioned in the beginning, the last h pulses will be lost after passing through m - 1 media. Hence, we consider only the first 2m pulses and (m ~ 1) identically prepared media of maximum lengths L 1, L 2, - - *, L,_, . The length of the first medium, L 1, is given by eq. (11). For the sake of simplicity, we assume small amplification. This can be achieved by spreading the beam diameter. Thus, the pulse shape is preserved [2] and the output of L 1 is a train of 2m-1 identical triangular pulses of pulse separation 2T. In general, the pulse separation in the pulse train at the output of the kth medium is 2kT. The length Lk is given by eq. (11) with nlOrl and T replaced by ?rkork and 2kT. Here, nkn and rk are the peak flux and the half-width of each pulse in the pulse train at the exit of the kth medium. They can be calculated using a series of equations similar to eq. (9). We shall not do it here. We now use Rhodamine 6G as an example. To show that the above theory is indeed physically possible, it is sufficient to show that L is real and experimentally accessible. Consider first of all a train of eight identical triangular picosecond pulses from a Rhodamine 6G dye laser. Let the wavelength be around 570 nm. We need three successive media of maximum lengths L 1, L,, L3 to combine the 8 pulses into one. Suppose that each pulse is of 1 ps duration with energy 5 mJ and T = 1 ns. Then nlOrl - 1016 cmm2. We choose our pulse combining medium to be Rhodamine 6G in ethanol at a concentration of 10e3 M. At room temperature, NjO - 1Or6 cmm3, corresponding to a level - 0.01 eV above the lowest level of the singlet ground state, assuming a Boltzmann distribution. From ref. [ 11 , ‘I, 10-16cm2,r-10ns.Sub5 X lo-l7 cm2 (5 -1.5X stituting these ;alies into eq. (1 l), we get L 1 z 1 cm. To calculate L,, we should replace nlorl and Tby n20~2 and 2T. Note that (n2072)/2 is the energy each pulse in the pulse train at the exit of the first medium. Its maximum possible value is 2(nr0r1)/2, i.e. twice the energy in a pulse of the pulse train. Hence, by putting 2nr0r1 and 2T in place of nrOrl and Tin eq. (1 l),
b&-,. (11)
This is the maximum length of the medium which will combine the first two pulses into one without substantial loss. A similar expression can be written down for
3
Volume
12, number
1
OPTICS COMMUNICATIONS
an upper limit ofL2; i.e. L, 5 2.3 cm. Similarly, by putting 4n1ur1 yd 4T in place of nlorl and Tineq.(11),wegetL310 cm. Hence, all the lengths are real. In practice, they can be combined into one of length L 1 + L 2 + L j. It can be pumped by a flashlamp or an argon laser, etc. If there are more pulses in the train, the length of the combined medium will be too long to be practical. To overcome this difficulty, the medium can be placed between two totally reflecting surfaces. The pulse train is sent through the side into the medium, reflected back and forth between the mirrors and eventually comes out through the other side, becoming one single pulse, as shown in fig. 2. The path length inside the medium can be adjusted easily by changing the angle of incidence or the spacing between the mirrors. we obtain
September
1974
begins. The disadvantage is the waste of pulse energy in preparing the medium while the advantage is its simplicity. Another way is to use another molecular (or atomic) species to which levels 3 and 4 belong. This species should absorb the first pulse and transfer the energy via collision to level 2 of the original species in a time short compared to the time T between two adjacent pulses.
4. Conclusion We have shown that it is physically possible to combine a train of short pulses into a single short pulse through successive absorption and stimulated emission by a special medium.
Acknowledgement The author thanks Dr. G. BCdard and Dr. A. Zardecki for helpful discussions. Fig. 2. Alternate
design of the dye medium.
References 3. Discussion 111 O.G. Peterson, In principle, there are other alternative ways of combining a train of short pulses. One obvious way is to use the same medium described above except that it is not prepared at N2 = N, . One can simply send in the pulse train. The first few pulses will serve to pump the medium to N2 = N, after which pulse combination
4
J.P. Webb, W.C. McColgin and J.H. Eberly, J. Appl. Phys. 42 (1971) 1917. I21 D. R&s, Lasers, Light Amplifiers and Oscillators (Academic Press Inc., London, 1969), p. 80. 131 A. Kastler, talk presented at the Internat. Symp. on Op tics and Lasers, Universitd de Q&bec, Irois Riviires, Q&bec, Canada (November 9-10, 1973).