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Theory of ultrabroadband light amplification in a spatially-dispersive scheme
Volume
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number 1,2
Theory of ultrabroadband light amplification in a spatially-dispersive scheme I.P. Christov
’ and M.B. Danailov
Faculty of Physics, Sofia University, 1126-SoJa, Bulgaria
Received 2 1 November 1990; revised manuscript received 4 March 199 1
The amplification of broad spectral bandwidth light in a spatially-dispersive scheme is analyzed both analytically and numerically. It is shown that an initial spectrum with a width close to the fluorescent bandwidth can be amplified without narrowing and the initially narrow spectrum can be broadened considerably during the amplification. An approximate analytical expression for the output spectral width is derived which could be useful in building ultrabroadband amplifiers.
1. Introduction
In a series of recent papers [ I-41 we have demonstrated experimentally the performance of an optical generator/amplifier in a “spatially-dispersive” scheme. The basic idea is to reduce the competition of the frequency components by spreading them in different spatial areas of the homogeneously broadened active medium. The last was accomplished using the spectral walk-off effect arising when the frequency components pass a pair of dispersive optical components (grating pair [ 1 ] or prism pair [ 2,3] ). In this way, ultrabroadband (up to 35 nm) generation and amplification was observed. Moreover, the spatial spreading of the spectrum enables very flexible control of the amplification of different groups of frequency components by passing them through a pump region with appropriate spatial distribution of the pump. In particular, in refs. [ 3,4] we have demonstrated experimentally that the last can be used for “spectrum-preserving” amplification. In a more general sense, it could be noted that the combination of a dispersive system and a gain medium appears as a powerful shaping device both for the spectrum and the temporal profile of the radiation. In fact the spatially-dispersive amplifier could be considered as an ’ Present address: Max Planck Institute fdr Quantenoptik, W8046 Garching, Germany. 003s4018/91/$03.50
active variant of the passive shaping technique based on attenuation and dephasing of the spectral components [ 5,6 1. In this paper we consider the theoretical aspects of the spatially-dispersive amplifier by solving numerically the governing equations. Approximate analytical results for the output spectrum width are also presented.
2. Design and analytical description Two schematic designs of amplifiers in a spatiallydispersive configuration are drawn in fig. la, b. Fig. 1a shows an input beam with broad spectrum which is dispersed in space by passing a grating pair. In this case after the second grating the spectral intensity is distributed along the x-axis according to [ 71: Zt(X, 0,O)
=Zin(O) exp[ - (X+j?&2)‘/UZ]
,
(I)
where a is the radius of the input beam, a= w- w, is the shift from the central frequency wo, /3=2rrc/ (wgdcos y, ) is the angular dispersion of the grating, d the grating groove spacing, y, the angle of diffraction on the first grating, b the slant distance between the gratings along the path of the central frequency. The intensity I, (x, w, 0) is the input signal (z=O) for the amplifying medium (dye in our case). Here we suppose that the input spectrum “covers” a part
0 1991 - Elsevier Science Publishers B.V. (North-Holland )
61
Volume 84, number 1,2
OPTICS COMMUNICATIONS
WNo-N,/T,
aN,/at= +No
1 July 1991
s
-N,
j- q(w)
G(W) I(z, 0) do >
(3b)
No+N,=N,
/PUMP
Fig. I. Schemes of amplifiers with spatial dispersion: (a) G,, G2, G3, G,, diffraction gratings; a., u,, absorption and emission cross sections; (b) L,, L2, lenses with common focal plane F, placed between diffraction gratings G, and GZ.
of the emission cross section a, ( w ), plotted in fig. 1a together with the absorption cross section a,(w). After amplification in a transversely pumped cell, the spread frequency components are recollimated by a second grating pair. Then the output radiation will be spectrally pure, i.e. the spectral dependence will be the same over the beam cross section. The configuration shown in fig. lb acts similarly to that in fig. la but it assures better resolution and higher intensity of the frequency components near the focal plane F. Moreover, this system is nondispersive as a whole [ 81 and therefore it is especially suitable for amplification of ultrashort pulses. Since in this case the frequencies are well selected only near the F-plane, a thin cell or jet should be used. The spatial distribution of the spectrum is given by Z*(X, W, O)=Zin(0)
exp[-
(x+fPQ)2a2/a?1
9
62
(3c)
where No and N, are the population densities of the lower and upper lasing level, respectively, W is the pump rate, T, is the life time of the upper level, n is the index of refraction of the host medium, N is the total concentration of the active particles and z is directed along the amplified beam. Here we have neglected the singlet-triplet conversion, the amplified spontaneous emission, and the non-saturable photoabsorption losses. For simplicity, in eqs. (3) the argument x is omitted. The set of equations (3a-c) could be solved numerically in the present form. However, here we make some simplifying assumptions suitable for the numerical as well as for analytical treatment of the problem. First, for relatively high pump rates (about 1O*s‘ ’ and higher) the steady state regime is reached very quickly, in less than 2 ns [ 91. As far as in our experiments [ l-41 we have used pump pulses with duration 15 ns, the steady-state description is certainly a good approximation. Then, neglecting the time derivatives in eq. (3a, b) we find one equation for the propagating photon flux: az(z, W)/az=z(z, x
o,(w)
+a,(~) (2)
where af= 2nf/ak, (k= o,,/c) is the monochromatic spot size (see [6]). We should note that in both above shown schemes the grating pairs could be replaced by prism pairs since they are cheaper and introduce considerably lower losses [ 2-41. The description of the amplification process is based on the known set of rate equations for the photon flux and the population densities:
Z(z, co) dw
-G(W)
x
0) N
W-o,(w)lT,
s s
G,(W)z(z, w) dw u,(w) z(z, w) dw
>
[G(W) +G(w> 1 z(z, 0) dw -I
+W+l/T,
>
.
(4)
Volume 84, number
I,2
OPTICS
In the following calculations the spectral dependence of the emission and absorption cross sections r&(w) and ~~(0) are assumed to be gaussian functions of central frequencies with the frequency w nm) and wao= 0,,=3.31 x 10’5 s-’ (&=570 3.55x 1OL5s-’ (hao=530 nm), and widths at e-’ level Aw,=Aw,=3.4~10’~ s-’ (M=50 nm), respectively. The peak values of the cross sections are a,,=1.8x10-‘6cm2andcr,o=2.7X10-‘6cm2 [lo]. The pump radiation is supposed monochromatic, with carrier frequency wP= wao and photon flux Zp=Pp/ (fiw,A), where P, is the pump power and A is the pumped area. Then, the pumping rate is given by W=Zpaa(wp). AvalueofT,=5.5~10-9sisused
[lOI. In order to obtain analytical estimations of the output spectral width we perform some additional simplifying assumptions concerning the basic equation (4). Namely, we neglect the absorption losses (i.e. a,(w)=O) which reduces eq. (4) to
am 0) -=
m(w)
az
I(& w)
(5)
(l/W)Jo,(w)Z(z,w)dw+a!’
wherea=l+(l/WT,). The formal solution
of eq. (5) is given by
Z(z, 0) =Z(O, 0) exp Na,(w) ( z dz’ ’ so (l/W)Jo,(w) Z(z’, w) dw+a Then, the two frequencies limiting half maximum are determined by Z(z, w)=O.5
(6)
>’ the spectrum
at
I (AwJAwo)~}-“~,
where Z,(z) =Jo=(w) ” Furthermore
- Z,(z) -Z,(O) o!w
Z(z, w) dw/a,(w,)
d (wo) e >.
We should note that eqs. (8) and (9) are similar to eqs. (24) and (20) in ref. [ 111. The only difference is in the averaging of the stimulated emission cross section over the spectrum made in ref. [ 111. Now, we will consider the more interesting problem of amplification in the scheme where the input radiation is preliminarily dispersed spatially (fig. 1). In this case it is easy to realize that with increasing the dispersion, the different spectral components are amplified in weakly overlapping “channels” of the dye cell which reduces their interaction in a saturation regime. We will suppose that the dispersion is sufficiently high to eliminate the intermode competition at all. This means mathematically that the exponents in eqs. ( 1) and (2) act as a &function under the integral in eq. (5), yielding
Z(z, w)=Z(O,
(10)
possesses a solution: Na,(w) (y
w) exp
z
- I(& 0) -I(& WI G(W) . aW
(11)
>
From eqs. (7) and width in this case: Aw=Aw,
+Z(O, wo) [l+ (8) #I.
we assume that the integral over frequencies here can be replaced by a sum in order to save the dimensions of both Z,(z) and Z(z, w) as photon fluxes (cm-* s-l).
(11)
one obtains
[0.7aW/o,(wo)+0.5Z(z,
x{ WNZ-0.5
[0.7aW/a,(wo)]-“’
+ [~!W/G(WO)
ff
(
(
(7)
+zt(Q)
of
Z,(z)=Z,(O) exp Na,(wo)z
This equation
Z(z, 0) .
x{ WNZ-l,(z)
From eq. (5) it follows that Z,(z) is a solution the transcendental equation:
am,4 m?(w) Z(z,0) ~ az = f&(w) Z(z, w)/ w+a .
Assuming that Z(0, w) =Z(O, wo) exp( -sZ2/Awg), from eqs. (6) and (7) it follows for the output spectral width: Aw=Aw,
1 July 1991
COMMUNICATIONS
the spectral
w~)]“~
Z(z, 00)
(Aw,/A~o)~l
+ [~W/G,(WO)I (Awe/A~o)~) -“2,
(12)
where Z(z, wo) satisfies the same eq. (9) as Z,(z). It isseenfromeqs. (5) and (10) thatZ,=aW/u,(w,) is the saturation photon flux, and if I( z, w) K Z, we have a small signal gain regime for which (from (11)): 63
Volume 84, number I,2
Qz, 0) =I(% w) expk(~)zl
OPTICS COMMUNICATIONS
,
1 July 1991
(13)
where g(w) =Na,(w)/a is the small signal gain coefficient. In this case eqs. (8) and ( 12) yield equal results for the small-signal spectral width: i/2 0.7 Aa= Aw, (14) ’ g(wo)z+ (Aw/A~o)* > This equation shows that, as could be expected, the small-signal amplification develops identically both for the schemes with and without spatial dispersion. In a saturation regime, however, we will see that eqs. ( 8 ) and ( 12 ) give considerably different results.
3. Numerical results In order to test our analytical expressions describing the output spectral width in the case of spatiallydispersive scheme (eq. ( 12) ) we have solved numerically the basic equation (4). Besides the spectral characteristics at the output we .have also calculated the spatial intensity distribution along the xaxis (see fig. 1a). It is supposed that a region of the dye cell from x= -0.6 cm to x=0.6 cm is homogeneously pumped with rate W= 5x lOa s-i. The zero of the x-axis corresponds to the path of the central input frequency wo. Most of the values used are chosen near the experimental values presented in refs. [ l-41. The total density of the lasing particles is N=3x 10” cmP3 (5x 10e4 mol/l), the cell length along the z-axis is L = 2 cm and the radius of the input gaussian beam before the gratings is a = 400 urn. Two principal cases were investigated. In the first one the input radiation has a photon flux I( 0, mO) = 1O24 cme2 s-’ (3.3 x lo5 W/cm2) and spectral width Aw, equal to the fluorescence width Aw,. Some of the results in this case are presented in fig. 2. Fig. 2a corresponds to the case without spatial dispersion at the input (zero distance between the gratings). It is seen that the beam cross size considerably increases due to the amplification which is a guarantee for reaching a saturation regime. On the other hand the spectrum narrows from 50 nm down to 18 nm (fwhm). Fig. 2b shows the results when the gratings spacing is 6= 1.5 cm. Now, the x-size of the dispersed beam exceeds the input size (dashed line in fig. 2b - left). The output spectrum broadens due to the weaker 64
WAVELENGTH (nm)
(cl
WAVZLENGTH (nm) Fig. 2. Calculated spatial (left) and spectral (right) distributions of the input radiation (dashed) and output radiation (solid) for different distances between the gratings: (a) b=O; (b) b= 1.5 cm; (c) b=3 cm; (d) b= 1.5 cm for shifted input spectrum. Here &5.5~10-~cm(18001/mm),andy,=62deg.
competition between the frequencies but exhibits a specific dip at the center, which means that the competition still exists. Such dip was clearly observed in our experimental works (see e.g. fig. 3 in ref. [ 3 ] ). In fig. 2c (b= 3 cm) the output spectral width reaches that of the input and the dip disappears which in.
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OPTICS COMMUNICATIONS
Volume 84, number 1,2
dicates that the competition is almost completely eliminated. It must be noted that using a standard scheme it is impossible to obtain an amplified spectrum with a width near the fluorescence bandwidth (see also ref. [ll]). Fig. 2d shows the case when the input spectrum is shifted towards shorter wavelengths with respect to the fluorescence peak, for a gratings spacing b= 1.5 cm. The output spectrum has, however, nearly the same shape as in fig. 2b and it is positioned at the same place. We should note that all asymmetries in the x- and I-distributions arise due to the larger resonant-absorption losses of the shorter wavelengths. The results for the second principal case, when the input spectrum is 4.5 times narrower than the fluorescence bandwidth, are presented in fig. 3. It is seen that without dispersion the output spectrum is only slightly shifted with respect to the input one (fig. 3a),
WAKLENGlH
(b)
whereas for a gratings spacing b= 5 cm the spectrum considerably broadens owing to the amplification (fig. 3b). If the input spectrum is shifted with respect to the fluorescence maximum (fig. 3c), the effect is similar to that in the previous case (fig. 2d). We should note that the power gain calculated by our computer program was on the order of 100 which is close to the experimentally observed values. In order to compare the numerical results with the analytical ones we have plotted in fig. 4 the output spectral width as a function of the pumping rate for amplification of a weak ultrabroadband input flux Z(0, wO)= lOi cm-’ s-’ and A&,=50 nm in three cases. The solid line presents the computer results for grating separation b=3 cm, whereas the two dashed curves correspond to dispersive (eq. ( 12) ) and nondispersive (eq. (8) ) amplification. It is clearly seen that the behavior of the curves is quite different. The spectral width without dispersion lowers with increasing IV, tending to its small-signal value (eq. ( 14) ). The spectrum in a dispersive regime initially broadens and reaches a width comparable with the fluorescence one. In fact, it is easy to check that the function Aw( W) in eq. ( 12) has a maximum with respect to W so that for very high values of the pumping rate (which are not plotted in fig. 4) the output spectral width passes through max-
(nm)
WhELENGlH (nm)
I
L PUMPING
Fig. 3. The same as in fig. 2 but for narrower input spectrum: (a) b=O; (b) b=5 cm; (c) b=5 cm for shifted input spectrum.
RATE
6 WX 10’
8
10
(set-‘1
Fig. 4. Dependence of the output spectral width (fwhm) on the pumping rate W. Solid line, computer results; dashed (short dash), analytical estimation; dashed (long dash), the case without spatial dispersion.
65
Volume 84, number 1,2
OPTICS COMMUNICATIONS
imum and starts to lower down to its small-signal value (eq. ( 14) ). The difference between the analytical and computer results arises mainly due to both the resonant absorption losses and the residual mode competition accounted for in the computer solution.
4. Conclusion The theory of laser amplification under steady-state conditions has been extended to describe an amplifier with spatial dispersion of the radiation spectrum. It is shown that by a sufficiently large spreading of the input spectrum by a grating pair the competition of the various frequency components into a homogeneously broadened medium (dye) may strongly be reduced. This results in output spectral width comparable with the fluorescence bandwidth even in the case of large gain and saturation. The numerical as well as the analytical estimations show an increase of the output spectral width with increasing pump rate, in contrast with the nondispersive amplification. Finally, we note that another amplifica-
66
1July 1991
tion scheme is presented [ 12 ] which also has the capability to realize spatially-dispersive amplification.
References [ I ] M.B. Danailov and I.P. Christov, Optics Comm. 73 ( 1989) 235. [ 21 M.B. Danailov and I.P. Christov, Optics Comm. 77 ( 1990) 397. [ 3 ] M.B. Danailov and I.P. Christov, Appl. Phys. B 5 1 ( 1990) 300. [4] M.B. Danailov and I.P. Christov, J. Mod. Optics 38 ( 1991) 413. [ 51A.M. Weiner and J.P. Heritage, Rev. Phys. Appl. 22 ( 1987) 1619. [6] M.B. Danailov and I.P. Christov, J. Mod. Optics 36 ( 1989) 725. [7] O.E. Martinez, J. Opt. Sot. Am. 3 ( 1986) 929. [8] O.E. Martinez, IEEE J. Quantum Electron. QE-23 ( 1987) 59. [ 91 U. Ganiel, A. Hardy, G. Neumann and D. Traves, IEEE J. Quantum Electron. QE-1 I (1975) 881. [lo] F.P. Schafer, ed., Dye lasers (Springer, Berlin, 1973). [ 111 M.D. Rotter and R.A. Haas, Optics Comm. 76 ( 1990) 56. [ 121 S. Szatmari, P. Simon and H. Gerhardt, Optics Comm. 79 (1990) 64.
COMMUNICATIONS
number 1,2
Theory of ultrabroadband light amplification in a spatially-dispersive scheme I.P. Christov
’ and M.B. Danailov
Faculty of Physics, Sofia University, 1126-SoJa, Bulgaria
Received 2 1 November 1990; revised manuscript received 4 March 199 1
The amplification of broad spectral bandwidth light in a spatially-dispersive scheme is analyzed both analytically and numerically. It is shown that an initial spectrum with a width close to the fluorescent bandwidth can be amplified without narrowing and the initially narrow spectrum can be broadened considerably during the amplification. An approximate analytical expression for the output spectral width is derived which could be useful in building ultrabroadband amplifiers.
1. Introduction
In a series of recent papers [ I-41 we have demonstrated experimentally the performance of an optical generator/amplifier in a “spatially-dispersive” scheme. The basic idea is to reduce the competition of the frequency components by spreading them in different spatial areas of the homogeneously broadened active medium. The last was accomplished using the spectral walk-off effect arising when the frequency components pass a pair of dispersive optical components (grating pair [ 1 ] or prism pair [ 2,3] ). In this way, ultrabroadband (up to 35 nm) generation and amplification was observed. Moreover, the spatial spreading of the spectrum enables very flexible control of the amplification of different groups of frequency components by passing them through a pump region with appropriate spatial distribution of the pump. In particular, in refs. [ 3,4] we have demonstrated experimentally that the last can be used for “spectrum-preserving” amplification. In a more general sense, it could be noted that the combination of a dispersive system and a gain medium appears as a powerful shaping device both for the spectrum and the temporal profile of the radiation. In fact the spatially-dispersive amplifier could be considered as an ’ Present address: Max Planck Institute fdr Quantenoptik, W8046 Garching, Germany. 003s4018/91/$03.50
active variant of the passive shaping technique based on attenuation and dephasing of the spectral components [ 5,6 1. In this paper we consider the theoretical aspects of the spatially-dispersive amplifier by solving numerically the governing equations. Approximate analytical results for the output spectrum width are also presented.
2. Design and analytical description Two schematic designs of amplifiers in a spatiallydispersive configuration are drawn in fig. la, b. Fig. 1a shows an input beam with broad spectrum which is dispersed in space by passing a grating pair. In this case after the second grating the spectral intensity is distributed along the x-axis according to [ 71: Zt(X, 0,O)
=Zin(O) exp[ - (X+j?&2)‘/UZ]
,
(I)
where a is the radius of the input beam, a= w- w, is the shift from the central frequency wo, /3=2rrc/ (wgdcos y, ) is the angular dispersion of the grating, d the grating groove spacing, y, the angle of diffraction on the first grating, b the slant distance between the gratings along the path of the central frequency. The intensity I, (x, w, 0) is the input signal (z=O) for the amplifying medium (dye in our case). Here we suppose that the input spectrum “covers” a part
0 1991 - Elsevier Science Publishers B.V. (North-Holland )
61
Volume 84, number 1,2
OPTICS COMMUNICATIONS
WNo-N,/T,
aN,/at= +No
1 July 1991
s
-N,
j- q(w)
G(W) I(z, 0) do >
(3b)
No+N,=N,
/PUMP
Fig. I. Schemes of amplifiers with spatial dispersion: (a) G,, G2, G3, G,, diffraction gratings; a., u,, absorption and emission cross sections; (b) L,, L2, lenses with common focal plane F, placed between diffraction gratings G, and GZ.
of the emission cross section a, ( w ), plotted in fig. 1a together with the absorption cross section a,(w). After amplification in a transversely pumped cell, the spread frequency components are recollimated by a second grating pair. Then the output radiation will be spectrally pure, i.e. the spectral dependence will be the same over the beam cross section. The configuration shown in fig. lb acts similarly to that in fig. la but it assures better resolution and higher intensity of the frequency components near the focal plane F. Moreover, this system is nondispersive as a whole [ 81 and therefore it is especially suitable for amplification of ultrashort pulses. Since in this case the frequencies are well selected only near the F-plane, a thin cell or jet should be used. The spatial distribution of the spectrum is given by Z*(X, W, O)=Zin(0)
exp[-
(x+fPQ)2a2/a?1
9
62
(3c)
where No and N, are the population densities of the lower and upper lasing level, respectively, W is the pump rate, T, is the life time of the upper level, n is the index of refraction of the host medium, N is the total concentration of the active particles and z is directed along the amplified beam. Here we have neglected the singlet-triplet conversion, the amplified spontaneous emission, and the non-saturable photoabsorption losses. For simplicity, in eqs. (3) the argument x is omitted. The set of equations (3a-c) could be solved numerically in the present form. However, here we make some simplifying assumptions suitable for the numerical as well as for analytical treatment of the problem. First, for relatively high pump rates (about 1O*s‘ ’ and higher) the steady state regime is reached very quickly, in less than 2 ns [ 91. As far as in our experiments [ l-41 we have used pump pulses with duration 15 ns, the steady-state description is certainly a good approximation. Then, neglecting the time derivatives in eq. (3a, b) we find one equation for the propagating photon flux: az(z, W)/az=z(z, x
o,(w)
+a,(~) (2)
where af= 2nf/ak, (k= o,,/c) is the monochromatic spot size (see [6]). We should note that in both above shown schemes the grating pairs could be replaced by prism pairs since they are cheaper and introduce considerably lower losses [ 2-41. The description of the amplification process is based on the known set of rate equations for the photon flux and the population densities:
Z(z, co) dw
-G(W)
x
0) N
W-o,(w)lT,
s s
G,(W)z(z, w) dw u,(w) z(z, w) dw
>
[G(W) +G(w> 1 z(z, 0) dw -I
+W+l/T,
>
.
(4)
Volume 84, number
I,2
OPTICS
In the following calculations the spectral dependence of the emission and absorption cross sections r&(w) and ~~(0) are assumed to be gaussian functions of central frequencies with the frequency w nm) and wao= 0,,=3.31 x 10’5 s-’ (&=570 3.55x 1OL5s-’ (hao=530 nm), and widths at e-’ level Aw,=Aw,=3.4~10’~ s-’ (M=50 nm), respectively. The peak values of the cross sections are a,,=1.8x10-‘6cm2andcr,o=2.7X10-‘6cm2 [lo]. The pump radiation is supposed monochromatic, with carrier frequency wP= wao and photon flux Zp=Pp/ (fiw,A), where P, is the pump power and A is the pumped area. Then, the pumping rate is given by W=Zpaa(wp). AvalueofT,=5.5~10-9sisused
[lOI. In order to obtain analytical estimations of the output spectral width we perform some additional simplifying assumptions concerning the basic equation (4). Namely, we neglect the absorption losses (i.e. a,(w)=O) which reduces eq. (4) to
am 0) -=
m(w)
az
I(& w)
(5)
(l/W)Jo,(w)Z(z,w)dw+a!’
wherea=l+(l/WT,). The formal solution
of eq. (5) is given by
Z(z, 0) =Z(O, 0) exp Na,(w) ( z dz’ ’ so (l/W)Jo,(w) Z(z’, w) dw+a Then, the two frequencies limiting half maximum are determined by Z(z, w)=O.5
(6)
>’ the spectrum
at
I (AwJAwo)~}-“~,
where Z,(z) =Jo=(w) ” Furthermore
- Z,(z) -Z,(O) o!w
Z(z, w) dw/a,(w,)
d (wo) e >.
We should note that eqs. (8) and (9) are similar to eqs. (24) and (20) in ref. [ 111. The only difference is in the averaging of the stimulated emission cross section over the spectrum made in ref. [ 111. Now, we will consider the more interesting problem of amplification in the scheme where the input radiation is preliminarily dispersed spatially (fig. 1). In this case it is easy to realize that with increasing the dispersion, the different spectral components are amplified in weakly overlapping “channels” of the dye cell which reduces their interaction in a saturation regime. We will suppose that the dispersion is sufficiently high to eliminate the intermode competition at all. This means mathematically that the exponents in eqs. ( 1) and (2) act as a &function under the integral in eq. (5), yielding
Z(z, w)=Z(O,
(10)
possesses a solution: Na,(w) (y
w) exp
z
- I(& 0) -I(& WI G(W) . aW
(11)
>
From eqs. (7) and width in this case: Aw=Aw,
+Z(O, wo) [l+ (8) #I.
we assume that the integral over frequencies here can be replaced by a sum in order to save the dimensions of both Z,(z) and Z(z, w) as photon fluxes (cm-* s-l).
(11)
one obtains
[0.7aW/o,(wo)+0.5Z(z,
x{ WNZ-0.5
[0.7aW/a,(wo)]-“’
+ [~!W/G(WO)
ff
(
(
(7)
+zt(Q)
of
Z,(z)=Z,(O) exp Na,(wo)z
This equation
Z(z, 0) .
x{ WNZ-l,(z)
From eq. (5) it follows that Z,(z) is a solution the transcendental equation:
am,4 m?(w) Z(z,0) ~ az = f&(w) Z(z, w)/ w+a .
Assuming that Z(0, w) =Z(O, wo) exp( -sZ2/Awg), from eqs. (6) and (7) it follows for the output spectral width: Aw=Aw,
1 July 1991
COMMUNICATIONS
the spectral
w~)]“~
Z(z, 00)
(Aw,/A~o)~l
+ [~W/G,(WO)I (Awe/A~o)~) -“2,
(12)
where Z(z, wo) satisfies the same eq. (9) as Z,(z). It isseenfromeqs. (5) and (10) thatZ,=aW/u,(w,) is the saturation photon flux, and if I( z, w) K Z, we have a small signal gain regime for which (from (11)): 63
Volume 84, number I,2
Qz, 0) =I(% w) expk(~)zl
OPTICS COMMUNICATIONS
,
1 July 1991
(13)
where g(w) =Na,(w)/a is the small signal gain coefficient. In this case eqs. (8) and ( 12) yield equal results for the small-signal spectral width: i/2 0.7 Aa= Aw, (14) ’ g(wo)z+ (Aw/A~o)* > This equation shows that, as could be expected, the small-signal amplification develops identically both for the schemes with and without spatial dispersion. In a saturation regime, however, we will see that eqs. ( 8 ) and ( 12 ) give considerably different results.
3. Numerical results In order to test our analytical expressions describing the output spectral width in the case of spatiallydispersive scheme (eq. ( 12) ) we have solved numerically the basic equation (4). Besides the spectral characteristics at the output we .have also calculated the spatial intensity distribution along the xaxis (see fig. 1a). It is supposed that a region of the dye cell from x= -0.6 cm to x=0.6 cm is homogeneously pumped with rate W= 5x lOa s-i. The zero of the x-axis corresponds to the path of the central input frequency wo. Most of the values used are chosen near the experimental values presented in refs. [ l-41. The total density of the lasing particles is N=3x 10” cmP3 (5x 10e4 mol/l), the cell length along the z-axis is L = 2 cm and the radius of the input gaussian beam before the gratings is a = 400 urn. Two principal cases were investigated. In the first one the input radiation has a photon flux I( 0, mO) = 1O24 cme2 s-’ (3.3 x lo5 W/cm2) and spectral width Aw, equal to the fluorescence width Aw,. Some of the results in this case are presented in fig. 2. Fig. 2a corresponds to the case without spatial dispersion at the input (zero distance between the gratings). It is seen that the beam cross size considerably increases due to the amplification which is a guarantee for reaching a saturation regime. On the other hand the spectrum narrows from 50 nm down to 18 nm (fwhm). Fig. 2b shows the results when the gratings spacing is 6= 1.5 cm. Now, the x-size of the dispersed beam exceeds the input size (dashed line in fig. 2b - left). The output spectrum broadens due to the weaker 64
WAVELENGTH (nm)
(cl
WAVZLENGTH (nm) Fig. 2. Calculated spatial (left) and spectral (right) distributions of the input radiation (dashed) and output radiation (solid) for different distances between the gratings: (a) b=O; (b) b= 1.5 cm; (c) b=3 cm; (d) b= 1.5 cm for shifted input spectrum. Here &5.5~10-~cm(18001/mm),andy,=62deg.
competition between the frequencies but exhibits a specific dip at the center, which means that the competition still exists. Such dip was clearly observed in our experimental works (see e.g. fig. 3 in ref. [ 3 ] ). In fig. 2c (b= 3 cm) the output spectral width reaches that of the input and the dip disappears which in.
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dicates that the competition is almost completely eliminated. It must be noted that using a standard scheme it is impossible to obtain an amplified spectrum with a width near the fluorescence bandwidth (see also ref. [ll]). Fig. 2d shows the case when the input spectrum is shifted towards shorter wavelengths with respect to the fluorescence peak, for a gratings spacing b= 1.5 cm. The output spectrum has, however, nearly the same shape as in fig. 2b and it is positioned at the same place. We should note that all asymmetries in the x- and I-distributions arise due to the larger resonant-absorption losses of the shorter wavelengths. The results for the second principal case, when the input spectrum is 4.5 times narrower than the fluorescence bandwidth, are presented in fig. 3. It is seen that without dispersion the output spectrum is only slightly shifted with respect to the input one (fig. 3a),
WAKLENGlH
(b)
whereas for a gratings spacing b= 5 cm the spectrum considerably broadens owing to the amplification (fig. 3b). If the input spectrum is shifted with respect to the fluorescence maximum (fig. 3c), the effect is similar to that in the previous case (fig. 2d). We should note that the power gain calculated by our computer program was on the order of 100 which is close to the experimentally observed values. In order to compare the numerical results with the analytical ones we have plotted in fig. 4 the output spectral width as a function of the pumping rate for amplification of a weak ultrabroadband input flux Z(0, wO)= lOi cm-’ s-’ and A&,=50 nm in three cases. The solid line presents the computer results for grating separation b=3 cm, whereas the two dashed curves correspond to dispersive (eq. ( 12) ) and nondispersive (eq. (8) ) amplification. It is clearly seen that the behavior of the curves is quite different. The spectral width without dispersion lowers with increasing IV, tending to its small-signal value (eq. ( 14) ). The spectrum in a dispersive regime initially broadens and reaches a width comparable with the fluorescence one. In fact, it is easy to check that the function Aw( W) in eq. ( 12) has a maximum with respect to W so that for very high values of the pumping rate (which are not plotted in fig. 4) the output spectral width passes through max-
(nm)
WhELENGlH (nm)
I
L PUMPING
Fig. 3. The same as in fig. 2 but for narrower input spectrum: (a) b=O; (b) b=5 cm; (c) b=5 cm for shifted input spectrum.
RATE
6 WX 10’
8
10
(set-‘1
Fig. 4. Dependence of the output spectral width (fwhm) on the pumping rate W. Solid line, computer results; dashed (short dash), analytical estimation; dashed (long dash), the case without spatial dispersion.
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imum and starts to lower down to its small-signal value (eq. ( 14) ). The difference between the analytical and computer results arises mainly due to both the resonant absorption losses and the residual mode competition accounted for in the computer solution.
4. Conclusion The theory of laser amplification under steady-state conditions has been extended to describe an amplifier with spatial dispersion of the radiation spectrum. It is shown that by a sufficiently large spreading of the input spectrum by a grating pair the competition of the various frequency components into a homogeneously broadened medium (dye) may strongly be reduced. This results in output spectral width comparable with the fluorescence bandwidth even in the case of large gain and saturation. The numerical as well as the analytical estimations show an increase of the output spectral width with increasing pump rate, in contrast with the nondispersive amplification. Finally, we note that another amplifica-
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tion scheme is presented [ 12 ] which also has the capability to realize spatially-dispersive amplification.
References [ I ] M.B. Danailov and I.P. Christov, Optics Comm. 73 ( 1989) 235. [ 21 M.B. Danailov and I.P. Christov, Optics Comm. 77 ( 1990) 397. [ 3 ] M.B. Danailov and I.P. Christov, Appl. Phys. B 5 1 ( 1990) 300. [4] M.B. Danailov and I.P. Christov, J. Mod. Optics 38 ( 1991) 413. [ 51A.M. Weiner and J.P. Heritage, Rev. Phys. Appl. 22 ( 1987) 1619. [6] M.B. Danailov and I.P. Christov, J. Mod. Optics 36 ( 1989) 725. [7] O.E. Martinez, J. Opt. Sot. Am. 3 ( 1986) 929. [8] O.E. Martinez, IEEE J. Quantum Electron. QE-23 ( 1987) 59. [ 91 U. Ganiel, A. Hardy, G. Neumann and D. Traves, IEEE J. Quantum Electron. QE-1 I (1975) 881. [lo] F.P. Schafer, ed., Dye lasers (Springer, Berlin, 1973). [ 111 M.D. Rotter and R.A. Haas, Optics Comm. 76 ( 1990) 56. [ 121 S. Szatmari, P. Simon and H. Gerhardt, Optics Comm. 79 (1990) 64.