Document text
Suppression of FM-to-AM conversion in third-harmonic generation
by tuning the ratio of modulation depth
O
(N
o
Q
.
O ,
"—I ,
> '
q :
o ■
• I— I '■
Oh.
>
o
o
o
X
Yisheng Yang^^^, Yizhou Tan^, Bin Feng^, Fuquan Li^, Wei Han^, and Jichun Tan^
^College of Science, National University of Defense Technology, Changsha 410073, China
^Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China
Issues of Frequency-to-Amplitude modulation (FM-to-AM) conversion occurred in piiase-
modulated third-harmonic generation (THG) process are investigated. An expression about group-
velocity is theoretically derived to suppress the FM-to-AM conversion, which appears to be depen-
dant on the ratio of modulation depth of fundamental to second-harmonic when given the same
modulation frequencies of them. Simulation results indicate that the induced AM in THG process
can be suppressed effectively when the expression about group- velocity is satisfied.
PACS numbers: 42.65.Ky, 42.65. Yj, 42.65.Lm
Third-harmonic generation (THG) is a powerful tech-
nique to produce tunable-wavelength laser pulses, which
have varieties of applications in fields such as inertial
confinement fusion (ICF), photolithography, and biol-
ogy [l], To meet the requirement of suppression
of stimulated Brillouin scattering or beam smoothing,
it is desirable to efficiently frequency-triple laser pulse
that has a broad spectrum imposed by phase modula-
tion Ideally, this phase modulation does not in-
duce any variations in pulse intensity. However, as result
of frequency-dependent effects like group- velocity disper-
sion, frequency modulation (FM) of input fields will be
converted into amplitude modulation (AM) of output
fields, which is named as FM-to-AM conversion [4]. Gen-
erally, the induced AM can lead to some higher-order
nonlinear effects or may cause damages to optical ele-
ments due to instantaneous ultrahigh intensity in process,
and thus needs to be prevented Recently, the sup-
pression of this FM-to-AM conversion has been attract-
ing increasing interests, and many approaches, such as
angular spectral dispersion, dual-tripler scheme, and pre-
compensation with gratings or crystals, have been pro-
posed and demonstrated f7|-[l^.
In a previous paper, we reported that the induced AM
in THG process could be suppressed at the retracing
point of a crystal [l^. Here, the issue of FM-to-AM
conversion in THG process is investigated from the view-
point of modulation properties of laser pulses. An ex-
pression about group- velocity, which reveals the intrinsic
group-velocity-matched relationship in phase-modulated
THG process, is given to guide the suppression of FM-
to-AM conversion.
Considering the effect of group-velocity mismatch,
nonlinear coupling equations describing the THG pro-
cess under the plane wave approximation can be written
as [13:
dAi{z,t) 1 dAi{z,t) iujide
dz
Vgl
dt
nic
dA2{z,t) , 1 dA2{z,t) iL02d,
dz
Vg2
dt
n2C
A3 Ale
iAkoz
(1)
(2)
dA3{z,t) 1 dA^^Zjt) iuj^de
dz
Vg3
dt
AiA2e
— iAkoz
(3)
where subscripts 1, 2, and 3 refer to fundamental (FH),
second-harmonic (SH), and third-harmonic (TH) pulse,
respectively. Parameters Vg, A/cq, and dg represent
group-velocity, original phase-mismatch and nonlinear
coefficient, respectively.
For FH and SH Gaussian pulses with sinusoidal phase
modulation, amplitudes can be written as the form of
exp(— t^/2rj^) exp [cTi sin(27rr2it)] , where a and ft are
modulation depth and modulation frequency, respec-
tively. By transforming the coordinate {z,t) to local co-
ordinates {z,T — t — z/vgi) and {z,T' — t — z/vg2), am-
plitude of the output TH field in frequency domain can
be obtained under the pump undcplction approximation:
A3{Z,UJ)
iUJsde
"3C Jo
^exp{-iAk„^)d^,
where
exp [-a{t + $j/i)2 - b{t + £,V2f
(4)
(5)
• exp icji sin {2'KVli{t + ^J^i)) + i<J2 sin {2'KVl2{t + $^^2))
• exp(ia;i)c?t.
In Eq.([ni), a = l/2Tf and b = 1/2T2^ are parameters de-
termined by pulse-duration of FH and SH pulses, while
vi — l/vg3 — 1/vgi and V2 — l/vg3 — l/vg2 are the so-
called group-velocity mismatches. The pulse-duration
term exp[— a(t + ^vif' — h{t + ^^^2)^] , as analyzed in
Ref . [isj , is significantly crucial for ultrashort (e.g., pi-
cosecond, femtosecond, or even shorter) laser pulses.
However, for phase-modulated broadband THG, pulse
duration is generally around nanosecond [3], and the ef-
fect of that pulse-duration term is negligible.
Since the pulse-duration term can be neglected, Eq.([S])
turns out to be the Fourier transformation of
exp
lai sm
1(72 sm
{2TTVl2{t + £,V2))
(6)
2
For simplicity, we initially apply Fourier transforms on
exp
iai sin (27rf2i(t + £,t^i))
, and achieve
Jn{'^i)^{'-^ — 27rrir2i) exp(iw^;^i).
(7)
Substituting ([7]) into (jl}, results show that the intensity
of TH pulse in frequency domain possesses the charac-
teristic of
|A3(z, oc sinc^{uji'iz),
(8)
which implies that the output TH pulse will become
intensity-modulated if vi ^ 0. Since 7^ means no
group-velocity mismatch between FH and TH pulses, we
could conclude that the FM-to-AM conversion in process
is basically caused by group- velocity mismatches between
interacting phase-modulated pulses.
Similarly, for Eq.®, we assume modulation frequen-
cies of FH and SH pulses to be the same {fti =02 — ^1),
since random or unequal relations between fli and D,2
makes analysis complicated. Let x = (T2/C1, and reorga-
nize Eq.® as below
exp lai
sin(27rf2t) [cos(27r51^i^i) -I- a; cos(27ri7^t/2)]
-I- cos(27rrit) [sin(27ril^i^i) + a; sin(27rri^j/2)]
(9)
Obviously, if group- velocity mismatches j/i and 1^2 satisfy
sin(27rri^j/i) -t- a; sin(27rri^:/2) = 0,
(10)
^ could be simplified to exp [ia' sin(27rrit)] , just with
the new modulation depth changing to a' which is equiv-
alent to CTi [cos(27rri^:/i) -I- a; cos(27rri^j/2)] ■ Compared
with analysis ahead, this indicates that no amplitude
modulation will be induced on output TH pulse.
Under pump undepletion 0) and accordingly
sin 9 w 9 approximations, by substituting i>i and 1^2,
Eg. dTOl) transforms to
l+x
— + — {x^^,n2 = n,).
'Vga Vgl Vg2 (Tl
(11)
This expression about group-velocity gives guidance for
suppression of FM-to-AM conversion occurred in phase-
modulated THG process. Coincidentally, Eg. pT]) looks
exactly the same with Eq.(19) in Ref.[l^, the only differ-
ence between them is the physical meaning of parameter
X. X here represents the ratio of modulation depth of
SH to FH pulse, while x in Ref.[l5| represents the ra-
tio of pulse duration of FH to SH pulse. Meanwhile,
on the other hand, the two equations are consistent with
each other, as frequency bandwidth of sinusoidally phase-
modulated pulse is determined by 2(Tf7, while bandwidth
of transform-limited ultrashort pulse is determined by
the reciprocal of pulse duration.
Based on the split-step Fourier transform and the
fourth-order Runge-Kutta algorithm, the THG process
TABLE I. Basic Parameters of Input luj and 2u) pulses
Wavelength Pulse Duration Modulation Frequency Group-velocity Pea'
To[ns\ Q.[GHz\ in KDP [m/s] /o[(
lcj/1.053/im
2a;/0.527^m
10
10
2.02x10**
1.94x10*
in Type H potassium dihydrogen phosphate (KDP) crys-
tal is numerically simulated to verify Eg. dlip . In simula-
tion, we change value of Vg-^ to have different x according
Ea. ([TT|) . while values of Vgi and Vg2 are calculated accord-
ing the Sellmeier equations of KDP crystal [l^ , as shown
in Table I, where values of other basic input parameters
of pulses are also presented.
Fig. 1 plot temporal profiles of output three pulses in
case of different x, with values of cti and (T2 are fixed
to be 15 and 30, respectively. Fig. 2 gives practi-
cal temporal profiles of output pulses in KDP crystal
{vg'i = 1.92 X 10^to/s). Compared with the severe inten-
sity modulations in Fig. 2, intensity modulation of Fig. 1
are much smaller, especially when x = 02! o\ is satisfied
(Fig. lb). When x is away from the ratio 0-2 /ci, inten-
sity modulations gets more severe. All these properties
reveal that there will be no AM induced on output pulses
when X matches with the ratio of cr2/o'i, which confirms
the validity of Eq. (fTT|) .
However, see Fig. lb closely, there are still some resid-
ual intensity modulations on temporal profiles. This is
because the approximation of sin 6* « 6* is assumed for
Eq. (flO|) . If we set cti = 172, then we can still achieve
Eg. pT)) exactly without the assumption of sin0 Q. Fig.
3 presents temporal profiles of output three pulses with
(Tl = (72 = 15 and a: = 1, which shows no intensity mod-
ulation.
Now we could explain the phenomenon that the FM-
to-AM conversion can be suppressed at the retracing
point of a crystal from another aspect. As is known,
the THG retracing point of a crystal corresponds to
a special relationship of — = — 4- — , which is
identical with Eo. (fTT|) when 02 = 2(7i is chosen [13j .
Since finding a crystal with appropriate retracing point
is somewhat difficult, according Eq. ([TT|) . suppression of
FM-to-AM can be achieved easily just by tuning the
ratio of modulation depth of FH to SH pulse, which we
think is more feasible in practical.
This work was partially supported by the Na-
tional Natural Science Foundation of China (Grant
No. 60708007), and the Science and Technology
Foundation of Chinese State Key Laboratory of High
Temperature and Density Plasma Physics (Grant No.
9140C6803010802).
3
-4 -2 2 4
Time (ns)
(c) x=3
Time (ns)
Time (ns)
:xu j-jx i. „
4
FIG. 2. Practical temporal profiles in KDP crystal
Time (ns)
FIG. 3. Temporal profiles witli x — 1 and ai — a2 = 15
[1] P. J. Wegner, M. A. Henesian, D. R. Specie, C. Bibeau,
R. B. Elirlich, C. W. Laumann, J. K. Lawson, and T. L.
Weiland, Appl. Opt. 31, 6414 (1992)
[2] M. Clien, Y. Clien, W. Hsiao, and Z. Gu, Tliin Solid
Films 515, 8515 (2007).
[3] D. Eimerl, J. M. Auerbach, C. E. Barker, D. Milam, and
P. W. Milonni, Opt. Lett. 22, 1208 (1997).
[4] S. Hocquet, D. Penninckx, E. Bordenave, C. Gouedard,
and Y. Jaouen, Appl. Opt. 47, 3338 (2008).
[5] J. A. Marozas, J. Opt. Soc. Am. B 19, 75 (2002).
[6] S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S.
Letzring, and J. M. Soures, J. Appl. Phys. 66, 3434926
(1989).
[7] S. Hocquet, G. Lacroix, and D. Penninckx, Appl. Opt.
48, 2515 (2009).
[8] S. Vidal, J. Luce, and D. Penninckx, Opt. Lett. 36, 3494
(2011).
[9] S. Vidal, J. Luce, and D. Penninckx, Opt. Lett. 36, 88
(2011).
[10] H. Cao, X. Lu, L. Li, X. Yin, W. Ma, J. Zhu, and D.
Fan, Appl. Opt. 50, 3609 (2011)
[11] Chen Y., Qian L., Zhu H., Fan D., Chin. Phys. Lett. 28,
044209 (2011)
[12] W. Wang, W. Han, F. Wang, J. Wang, L. Zhou, H. Jia,
Y. Xiang, K. Li, F. Li, L. Wang, W. Zhong, X. Zhang, S.
Zhao, and B. Feng, J. Opt. Soc. Am. B 28, 475 (2011)
[13] Y. Yang, B. Feng, W. Han, W. Zheng, F. Li, and J. Tan,
Opt. Lett. 34, 3848 (2009)
[14] J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S.
Pershan, Phys. Rev. 127, 1918 (1962).
[15] Y. Yang, W. Han, W. Zheng, J. Tan, F. Li, F. Wang,
Y. Xiang, K. Li, B. Feng, H. Jia, D. Cao, and J. Dong,
Phys. Rev. A 78, 053801 (2008).
[16] K. W. Kirby and L. G. DeShazer, J. Opt. Soc. Am. B 4,
1072 (1987).