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Optical and Quantum Electronics 16 (1984)225-233

Determination of energy and duration of picosecond light pulses by bleaching of dyes

G . G R O N N I N G E R , A . P E N Z K O F E R

Naturwissenschaftliche Fakultat II - Physik, Universitat Regensburg, 8400 Regensburg, Federal Republic of Germany

Received 15 August 1983

T h e transmission of a light pulse through a dye solution is determined by its energy density if the pulse duration is short compared to the absorption recovery time. T h e energy transmission measurement allows the determination of the pulse energy. Simultaneous measurement of energy transmission and input pulse peak intensity makes it possible to calculate the pulse duration.

1. Introduction

The interaction o f intense picosecond light pulses w i t h absorbing dye solutions promotes an appreciable number o f molecules to an excited state and reduces the ground state absorption. F o r pulse durations AtL short compared to the absorption recovery time rF the number o f excited molecules becomes equal to the number o f absorbed photons i f the excited state absorption is weak. A measurement o f the pulse transmission through a dye sample allows one to determine the pulse energy, E. Since the pulse energy is proportional to the peak pulse intensity and to the pulse duration, a measurement o f energy transmission TE and input peak intensity I0L makes it possible to calculate the pulse duration AtL for AtL < rF.

In this article we calculate plots o f TE(E) and TE(I0L, AtL) for energy and pulse duration detection, respectively. Curves are presented for the interaction o f dyes w i t h picosecond light pulses o f a ruby laser (fundamental and second harmonic) and an Nd-glass laser (fundamental, second, third, and fourth harmonic). The calculations are verified experimentally for a mode-locked ruby laser. The energy detec- t i o n is applied to calibrate photodetectors to absolute energy values. The measurement o f TE and I0L

gives an inexpensive possibility o f picosecond pulse duration measurement.

The dye solutions are described b y a realistic four level system as depicted i n F i g . 1 [1]. The laser light excites molecules from the singlet ground state So to a F r a n c k - C o n d o n level 2 i n the Si band (absorp- t i o n cross-section oL). F r o m there the molecules relax quickly to a temporal equilibrium position 3 i n the Si state (relaxation time rFC = 0.7 ps assumed i n our calculations [2, 3 ] ) . E x c i t e d state absorption (cross-section ae x) may promote molecules from Si to higher lying states Sn. A fast relaxation from Sn

to Si is assumed i n the calculations ( re x = 1 0 ~1 3 s [ 4 - 6 ] ) . The molecules i n the Si band relax to the ground state w i t h a time constant rF. Depopulation o f the Si state b y amplified spontaneous emission is negligible for the applied dye concentrations (small signal transmission T0 = 0.01) [ 1 , 5 ] and is not included i n the level scheme and the calculations. Triplet states are neglected since dyes w i t h small intersystem-crossing rates are selected.

The dynamics o f interaction o f dye molecules w i t h laser light is governed b y the following differen- tial equation system

2. Theory

dNjXd)

dt' 3oLcos2e[Ni(d)-N2(d)] + N2(0) + N3(0) Nx(d)-Ni

o r

(0

0306-8919/84 S03.00 + 12 © 1984 Chapman and Hall Ltd. 225

(2)

3

S

'

/

/

^0 Figure 1 Level systems of dyes used in calculations.

3 W ) _ h „

2

{ 3 aL c o s20 [ W i ( 0 ) - W2( 0 ) ] - ffexIW)-^)]}

V F T F C /

W )

- ^ 2 1 z ^

( 2 )

—Tl- - — ~ Oex[N3(P) — N4[d)\+ + (3)

Ot hVL TFC Tp T g x Tor

= ±-o„[N*6

) +

Nm-NM (4)

ot hvL rex

_ rit/2

Nt = Nt(d) sin 0 d<9 / = 1 , 2 , 3 (5)

Jo

T T = ~ / L { 3 aLc o s M ^ i( 0) - ^ 2 ( f l ) ]

+

a « [ ^ 2( 9 )

+ W)]}sinedfl

(6) oz Jo

The transformation t' = t — cz/rj and z — z is used (c is vacuum light velocity; r? is refractive index).

The initial conditions are Nx(d, t' = - °°, z', r ) = TVQ, N2( 0 , f' = - °0, z\ r) = N3( 0 , f' = - °° z\ f) = iV4(<9, r ' = - oos z' , f) = 0 and 7L( f ' , z = 0, r ) = I0Lf(t'/t0)g(r/r0). N0 is the total density o f dissolved dye molecules. f(t'lt0) a n d g ( r / r0) are the temporal and spatial input pulse shapes, respectively.

The absorption anisotropy o f electric dipole interaction is taken into account for the ground state absorption [oL(6) = 3oL c o s2 6 where 6 is the angle between the transition dipole moment o f molecules and the direction o f the electric field o f linearly polarized laser light]. The excited state absorption is included b y an effective isotropic absorption cross-section ae x (orientation o f excited state transition dipole moments [7] is u n k n o w n ) . ro r describes the molecular reorientation o f the transition dipole moments.

Equations 1-6 are solved numerically. The intensity transmission is given b y Tj(t\ r) = IL{t\ /, r)j Ih(f\ 0> r). The transmission o f the energy density e is T(r) = e(/, r)/e(0, r) = fZ^ h r)

f-~ 40', 0, r) dt'. Finally the energy transmission is TE = E(l)/E(0) = J " e(/, r)r d r / / ~ e(0, r)r dr.

F o r pulse durations AtL ( F W H M ) i n the region 2 0 rFc < &tL < ro r/ 2 Equations 1-6 may be approxi-

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l 1 — i — i i i i 11

INPUT PEAK INTENSITY I0 L[ W c mz]

200

3

1001

I V '

1 1 1 1 i i i •

y ^ y y ^y^—

- ®

y ^y^ y ^y y ^y

-

y^y y^y

-

yy^ yy

yy

-

-

yy ^y

-

1 ' 1 1 1 1 i i i ~~

0.02 0.04 0.06 0.08 0.1

INVERSE PULSE DURATION A t f1 [ p s "1]

Figure 2 (a) Energy transmission against in- put peak pulse intensity. Curves apply to transmission of ruby laser pulses of temporal and spatial Gaussian shape through DDI dis- solved in glycerol. Small signal transmission T0 = 0.013 (see Table I for dye parameters).

The pulse durations At^ are (1) 520ps, (2) 260 ps, (3) 130ps, (4) 50ps, (5) 25ps, (6) 10 ps, (7) 2 ps and (8) 0.7 ps. Circles are experimental points for At^ = 30 ± 3 ps.

(b) Input peak intensity against inverse pulse duration for TE — 0.07. Solid curve is for data as in 2a (Equations 1-6). Dashed curve is linearized solution (Equations 7-10) responsible for TFC = 0 and ro r = °°.

mated b y setting rFC = 0 and ror =00 (deviation about 5% at upper and lower limits). The resulting equations are

3 aLc o s20 W1( 0 ) + ^ ^ (7)

{3oL c o s ' W V ^ f l ) - ae x[N3(p) -N4(0)]}- + - ^ ^ (8)

7 ex

^[Ns(6)-N,(d)]-^1 (9)

[ 3 aL c o s2^ x ( f J ) + oeKN3(6)] s i n 0 dfl ( 1 0 ) 0

In F i g . 2a TE against I0L is plotted for various pulse durations A / ^ ( F W H M ) . A Gaussian temporal and spatial pulse shape is assumed [AtL = 2(ln 2)1/2tQ]. The curves are calculated using Equations 1-6. The used parameters apply to the passage o f ruby laser pulses through D D I dissolved i n glycerol (r0 = 0 . 0 1 3 , for other data see Table I). F o r AtL > rF the TE(I0L) curves change o n l y slightly w i t h pulse duration.

The detection o f AtL from TE(I0L) measurement is limited to AtL ^ rF/ 2 .

The curves o f F i g . 2a are redrawn i n F i g . 2b to depict the dependence o f I0L on Atj} for a fixed energy transmission o f TE = 0.07 (solid curve). The dashed curve is obtained by solving the reduced

97^(0) _ h dt' " hv}

dt' j i hvL

JL_C

dt' ~ hvL

ML _ T dt'

(4)

Equations 7-10. In this case a linear dependence o f I0L o n At]1 is found, i.e. IQL(Atil) — / O L O +

KpTpAtl1). I0L is the peak intensity value at At]} = 0. KF is a constant near to 1 w h i c h depends slightly o n the temporal pulse shape and is independent o f TE and ae x (KF = 1.43 for Gaussian shape, KF = 1.94 for rectangular temporal shape).

The linear dependence o f I0L o n At]1 for rFC = 0 and ro r 00 (dashed curve i n F i g . 2b) allows one to determine an apparent duration A tL a from a measurement o f TE and I0L i f a calibration curve TEQ'OLI &t'L) for rFC = 0 and ro r 00 is k n o w n . A t a fixed TE value the following relation between AtLtQ and At'L is found b y s o l v i n gI0 L = 70 L( 1 4- KFrF/AtLta) and l'0L = 70L( 1 + KFrFjAttL) t o A rL > a:

In the region 2 0 rF C < A fL < ro r/ 2 the IOL^L) dependence (solid curve o f F i g . 2b) is well approxi- mated b y the linearized situation (dashed curve o f Fig. 2b) and the apparent duration AtLt a is equal to the true pulse duration AtL.

F o r pulse durations AtL < 2 0 rF C the p o p u l a t i o n o f the F r a n c k - C o n d o n level 2 is no longer negligible and the input peak intensity necessary to bleach the dye to transmission TE is reduced. F o r experimental situations obeying the conditions AtL < rF/ 2 and AtL <: ro r/ 2 the reorientation o f molecules w i t h i n the pulse duration causes additional molecules to enter the strongly absorbing direction (small angle 0) and the peak pulse intensity necessary to bleach the dye to TE increases. In the regions AtL ^ 2 0 rF C (but AtL > phase relaxation time T2) and AtL > ro r/ 2 (with AtL < rF/ 2 ) the true pulse duration m a y be obtained from the apparent duration A tL a (Equation 11) b y m u l t i p l y i n g w i t h a correction factor Ft

w h i c h is plotted i n F i g . 3b for temporal Gaussian pulse shapes. Ft(AtLjTFC) (lower curve) is indepen- dent o f TE and ae x. Ft{AtLalrox)(upper curve)is also practically independent o f TE. F o r A tLJ Tox- ^ ° °

the ratio Ft = AtL/AtLa approaches the value o f approximately 1.8.

The TE(I0L, AtL) curves o f F i g . 2a are redrawn i n F i g . 4a to present the dependence o f TE o n normalized pulse energy El(jtrl) [E = 2TTI0L SQ f^fit9/to)g(r/r0)r dt' dr]. A temporal and spatial Gaussian shape is assumed. In F i g . 4b the normalized pulse energy is plotted against pulse duration for a fixed energy transmission o f TE = 0.07. The solid curve (Equations 1-6) is w e l l approached b y the dashed curve (Equations 7 - 1 0 , rFC = 0, ro r = °°) i n the time region 2 0 rF C < AtL < ro r/ 2 . The dashed curve obeys the equation Ea(AtL) = E0[\ + AtL/(KFTF)]. In this linear region the pulse energy Ea(AtL) belonging to TE is related to E'(At'L) at the same transmission TE b y

result o f E q u a t i o n 12 w i t h a correction factor FE w h i c h is diagrammed i n F i g . 3a.

3. Calibration curves

Energy calibration curves TE(E') and pulse duration calibration curves TE{l'QL) are presented i n Figs. 5 and 6, respectively. A Gaussian temporal pulse shape is assumed. The spatial pulse shape is either rectangular w i t h i n a circular aperture o f radius r0 (Figs. 5a, 6a) or Gaussian (Figs. 5b, 6b). Curves are calculated for the fundamental and second harmonic o f the ruby laser (Curves l ' and 2 ' , At'L = 25 ps) as well as for the fundamental, second, t h i r d and fourth harmonic o f the Nd-glass laser (Curves 2, 1 , 3 , 4 , At'L = 6 ps). The curves are calculated for rFC = 0 and ro r =00 (Equations 7-10) w i t h a small signal dye transmission o f T0 = 0.01. The other applied pulse and dye parameters are listed i n Table I.

The pulse durations are obtained b y measuring TE and I0L and reading IoL(TE) from F i g . 5. Then E q u a t i o n 11 is applied to obtain the apparent duration A tL a. Finally the correction factors o f F i g . 3b are used to calculate AtL = Ft(AtLfalTFC)Ft(AtLa/r0r)AtLia. Similarly, the pulse energy densities are determined b y measuring TE and reading the corresponding ^ ' / ( ^ o ^ v a l u e from F i g . 6. E q u a t i o n 12 is used to correct for different pulse durations between the experiment and calibration curve. F o r pulse

(ii)

(12)

(5)

RATIO A tL/ xo r

1 10

1 10 RATIO AtL / xF C

RATIO A tL a/ xo r

1 10 RATIO A tL a / xF C

KK)

100

Figure 3 (a) Correction factors for true energy determination, E, from apparent values Ea obtained for T^Q = 0 and ro r = 0 0 (Equation 12). (b) Correction factors for determination of true pulse durations Af^

from apparent values Afj, fl (Equation 11).

T A B L E I Parameters of applied dyes

Laser Wavelength Dye Solvent ^ F - T o r

(nm) [ l O "1 6 cm2] [ l O "1 7 cm2] [ns] [ns]

Ruby 694.3 DDI1 glycerol 4.4 ± 0 . 2 4.8 ± 0.5 0.26s 1.0s

Ruby, SH 347.15 Dimethyl POPOP cyclohexane 1.743 4.9 ± 0.5 1.53 0.27 Nd-glass 1055 B D N I2 1,2-dichloroethane 1.07 ± 0 . 0 5 1.9 l6 0.287

Nd-glass, SH 527.5 Rhodamine 6G ethanol 4.174 54 4.24 0.38

Nd-glass, T H 351.7 Dimethyl POPOP cyclohexane 1.83 4.9 ± 0.5 1.53 0.27 Nd-glass, F H 263.8 9,10-dimethylanthracene cyclohexane 3 . 3 ± 0 . 23 12.5 ± 1 6.13 o . i r

1 l,r-diethyl-2,2'-dicarbocyanine iodide (Kodak dye No. 9618).

2Bis-(4-dimethylaminodithiobenzil)nickel (Eastman dye No. 14015).

3 From Berlman [8].

4 From Falkenstein et ah [4].

5 From Seymour et ah [9].

6 From Drexhage et ah [10].

7Estimated from Debye-Stokes-Einstein hydrodynamic model [11,12].

8 From Lessing et al. [13].

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n i i i i T r

0.011 i i i i 11

103 lO"2

I N P U T E N E R G Y D E N S I T Y E/ (T I T0 2) [ J c m2 ]

10 20 30 40 50 60 70 INPUT PULSE DURATION A tL t p s ]

Figure 4 (a) Energy transmission against normalized input pulse energy. Dye and pulse parameters as in Fig. 2a. Pulse dura- tions A fL are (1) Ops, (2) 2ps, (3) 50ps, (4) 260ps, (5) 520 ps. Circles are exper- imental points for Af^ = 30 ± 10 ps.

(b) Normalized input pulse energy against input pulse duration at fixed energy trans- mission of T% = 0.07. Solid curve, data as in Fig. 4a (Equations 1-6). Dashed curve, linearized solution (Equations 7-10).

durations outside the linear region the apparent pulse energy (Equation 12) is multiplied by the correc- tion factors o f Fig. 3a.

4. Experiments

The dye parameters entering the calculations are taken from the literature where possible (fluorescence lifetimes and reorientation times). The ground state absorption cross-sections are measured w i t h a spec- trophotometer. The excited state absorption cross-sections are obtained from energy transmission measurements at high input laser intensities ( ae x limits bleaching at high pump intensities).

F o r the mode-locked ruby laser the calculated curves are compared w i t h measurements o f energy and pulse duration. The energy was measured w i t h a pyroelectric detector (Gen tec E D 100). The beam radius r0 was determined with a silicon diode array (Tracor D A R S S system). The energy transmission through the dye solutions was measured w i t h two photodetectors and a fast transient digitizer (Tektronix R 7 912). The obtained TE values against the measured normalized energy E/(Trro) are shown in F i g . 4a by the data points (mean pulse duration 30 ps). The experimental points agree w i t h the calculation within ± 10%. Fast photodetectors were calibrated to absolute energy detection by energy transmission measurement w i t h spatial truncated Gaussian pulses. F o r aperture radii rA < 0.5 r0

the calibration curves for rectangular spatial shape ( F i g . 6a) are good approximations.

The pulse durations were measured by the two-photon fluorescence technique [14] (dye: 2.5 x 1 0 "3 molar rhodamine 6 G i n ethanol). The input peak intensity is obtained from nonlinear transmission

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1 F

0.1

to to

to

a 0.01 1

>-

I D

0.1

0.01

l l l U | l I l 111 j ni i i i i u

®

APPARENT INPUT PEAK INTENSITY 1^ [ W c m2l

Figure 5 Calibration curves for pulse duration determination. Temporal pulse shape is Gaussian; spatial pulse shape is rectangular in (a) and Gaussian in (b).

Dashed curves: (1') fundamental ruby laser, (2') second harmonic of ruby laser.

Solid curves: (1) second harmonic, (2) fundamental, (3) third, (4) fourth harmonic of Nd-glass laser. Small signal dye transmission TQ = 0.01. Other data are listed in Table I.

measurements through a two-photon absorbing C d S crystal [15] (the two-photon absorption cross- section o f Blau and Penzkofer [15] corrected to a( 2 ) = 1.3 x 1 0 ~8 cm W "1) . Measured energy transmission points against input peak intensity are included i n F i g . 2a for pulses w i t h measured duration o f 30 ± 3 ps.

The experimental points agree w i t h calculation w i t h i n ± 25%. The accuracy is limited b y the accuracy o f the measurement o f photodetector signals for TE and I0L determination.

T w o photon-absorption techniques are available for the measurement o f the input peak intensity o f the ruby laser and its second harmonic [15] and o f the second [16], t h i r d [16] and fourth [17] harmonic o f the Nd-glass laser. A saturable absorber technique may be used for the intensity detection at the fundamental frequency o f the Nd-glass laser [18].

5. Approximate analytical description

The following estimates may be used for an approximate analytical determination o f the energy density o f a picosecond pulse.

The absorption o f the dye is assumed to be isotropic ( ro r = 0 ) . The fluorescence lifetime should be long compared to the pulse duration (rF/AtL -> °°). F o r dyes without excited state absorption the energy density transmission is Te = exp [— oL(N0l — eAhs/hvL)] = T0 exp [ aLe ( l — T)/hvL]. I is the sample length, eA b s = e ( l — Te) is the absorbed pulse energy density. Solving the above equation gives

l n ( r

e

) - l n ( r

0

) _

e = (13)

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APPARENT INPUT ENERGY DENSITY E ' / d t r2) IJ cm"2]

Figures Calibration curves for energy density determination. Temporal pulse shape is Gaussian; spatial pulse shape is rectangular in (a) and Gaussian in (b).

Dashed curves: (1') fundamental, (2') second harmonic of ruby laser. Solid curves: (1) second, (2) first, (3) third, (4) fourth harmonic of Nd-glass laser.

Dotted curves: analytical solution of Equation 15 for fundamental ruby laser frequency. Small signal dye transmission 7"n = 0.01. Other data listed in Table I.

The abbreviation es = hvLjoL is called the saturation energy density. F o r finite values o f A fL/ rF

E q u a t i o n 13 may be extended to

C

, i n ( r ) - M r

0

)

l - T , \ K F * F /

E x c i t e d state absorption may be included for fast higher excited state relaxation ( re x = 0 ) i n an approxi- mate manner b y replacing the denominator 1 — T b y exp (~N0oexl) T = Toex/°L — Te leading to

KFKF J

The dotted curves i n Figs. 6a, b are calculated using E q u a t i o n 15 for D D I dissolved i n glycerol (funda- mental ruby laser). They deviate from the corresponding dashed curves l ' b y a factor o f approximately

1.5 (rectangular spatial distribution shown i n F i g . 6a) and 0.8 (Gaussian spatial distribution i n F i g . 6b).

The deviation from Curve 1' o f F i g . 6a results m a i n l y from the fact that anisotropic absorption is not included i n E q u a t i o n 15 ( ro r = 0 ) and slightly from the assumption o f infinitely fast relaxation o f higher excited states ( re x = 0). In F i g . 6b the dotted curve deviates from Curve l ' since it is not integrated spatially.

The analytical solutions may also be used for the determination o f pulse durations b y using the fixed dependence o f energy o n intensity, duration and beam diameter for a specific temporal and spatial pulse shape.

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6. Conclusions

F o r pulse durations short compared to the ground state recovery time, absorbing dyes act as p h o t o n counters (absorbed photons = excited photons) and the i n p u t pulase energy may be determined from energy transmission measurements. This technique o f energy measurement allows one to calibrate p h o t o - detectors to absolute picosecond pulse energy. Simultaneous measurements o f energy transmission and i n p u t peak intensity provides a simple and inexpensive technique to measure the duration o f picosecond pulses i f they are shorter than the fluorescence lifetime (AtL < rF/ 2 ) .

Acknowledgements

The authors are grateful to D r W . B l a u for m a n y helpful discussions and to T h . Ascherl for technical assistance. T h e y thank the Rechenzentrum o f the University for disposal o f computer t i m e .

References

1. A . P E N Z K O F E R and W. B L A U , Opt. Quantum Electron. 15 (1983) 325.

2. A . P E N Z K O F E R , W. F A L K E N S T E I N and W. KAISER, Chem. Phys. Lett. 44 (1976) 82.

3. D. REISER and A . L A U B E R E A U , Appl. Phys. B27 (1982) 115.

4. W. F A L K E N S T E I N , A . P E N Z K O F E R and W. KAISER, Opt. Commun. 27 (1978) 151.

5. A . PENZKOFER and W. F A L K E N S T E I N , Opt. Quantum Electron. 10 (1978) 399.

6. C. V. SHANK, E . P. IPPEN and O. T E S C H K E , Chem. Phys. Lett. 45 (1977) 291.

7. A . P E N Z K O F E R and J. WIEDMANN, Opt. Commun. 35 (1980) 81.

8. I. B. B E R L M A N , 'Handbook of Fluorescence Spectra of Aromatic Molecules' (Academic Press, New York, 1971).

9. R. J. SEYMOUR, P. Y. L E E and R. R. A L F A N O , in 'Picosecond Phenomena IF: (edited by R. M . Hochstrasser, W. Kaiser and C. V . Shank) Springer Series in Chemical Physics Vol. 14 (Springer, Berlin, 1980) pp. 111-4.

10. K. H . D R E X H A G E and U. T. M U L L E R - W E S T E R H O F F , IEEE J. Quantum Electron. QE-8 (1972) 759.

11. P. D E B Y E , 'Polar Molecules' (Dover Publications, London, 1929) p. 84.

12. T. J. C H U A N G a n d K. B. EISENTHAL, Chem. Phys. Lett. 11 (1971) 368.

13. H. E . LESSING and A. VON JENA, in 'Laser Handbook', Vol. 3 (edited by M. L . Stitch) (North Holland, Amsterdam, 1979) Ch. B6, pp. 753-846.

14. J. A . GIORDMAINE, P. M. RENTZEPIS, S. L . SHAPIRO and K. W. WECHT, Appl. Phys. Lett. 11 (1967) 216.

15. W. B L A U and A. P E N Z K O F E R , Opt. Commun. 36 (1981) 419.

16. A . P E N Z K O F E R and W. F A L K E N S T E I N , ibid. 17 (1976) 1.

17. P. LIU, W. L. SMITH, H. L O T E M , J. H. B E C H T E L , N. B L O E M B E R G E N and R. S. A D H A V , Phys. Rev. B17 (1978) 4620.

18. A . P E N Z K O F E R , D. V O N DER LINDE and A. L A U B E R E A U , Opt. Commun. 4 (1972) 377.

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