- Email: [email protected]
Pulse compression effects of passive mode locker in ultrafast optical parametric oscillator
Accepted Manuscript Title: Pulse compression effects of passive mode locker in ultrafast optical parametric oscillator Author: Jieyu Wang Li Wang Yi Zhan PII: DOI: Reference:
S0030-4026(15)01304-2 http://dx.doi.org/doi:10.1016/j.ijleo.2015.09.223 IJLEO 56428
To appear in: Received date: Accepted date:
14-11-2014 25-9-2015
Please cite this article as: J. Wang, L. Wang, Y. Zhan, Pulse compression effects of passive mode locker in ultrafast optical parametric oscillator, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.09.223 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Pulse compression effects of passive mode locker in ultrafast optical parametric oscillator
ip t
Jieyu Wang, Li Wang*, Yi Zhan College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
cr
Abstract: The pulse compression effects of passive mode locker, such as graphene and Kerr lens
us
medium, which has been inserted into a synchronously-pumped optical parametric oscillator (SPOPO) pumped by pulse laser, and a continuous-wave (cw) pumped actively mode-locked OPO has been
an
studied theoretically. By solving the nonlinear coupled-wave equations in the plane-wave approximation, we simulate the relations of the pulse width and peak power of the signal pulses to the
M
normalized aperture radius, the nonlinear crystal length and the beam radius on the nonlinear crystal. The numerical results indicate that the Kerr lens medium and graphene are both favorable for the pulse
te
d
compression in the SPOPO, and each of them has different advantages. Whereas in the actively mode locked OPO, the Kerr lens medium may be a better alternative for the pulse compression.
Ac ce p
Keywords: pulse compression, optical parametric oscillator, Kerr lens medium, graphene saturable absorber, acousto-optic modulation
1. Introduction
Since ultrashort pulses have paved a way for many applications including
material processing, nonlinear optics, attosecond science, and metrology due to the advantages of high peak intensity and broad optical spectrum, there has been an ongoing effort in the field to reduce the pulse width and increase the power of lasers
* Corresponding author. E-mail address: [email protected]
Page 1 of 15
[1,2]. Among these, the Kerr lens mode locking (KLM) technique can support the generation of femtosecond pulses due to the non-resonant nature of the optical Kerr effect, lending itself to a wide range of gain media [3,4]. Besides that, the graphene
ip t
saturable absorber is another excellent choice for femtosecond pulses generation,
cr
especially with the application in a wide spectrum [5,6]. However, the tuning range of
us
these kinds of lasers is confined to limited spectral bands in the near-infrared.
Optical parametric oscillators (OPOs) have been extensively investigated for
an
over four decades, expanding the wavelength range of lasers to new spectral coverage [7,8]. Synchronously-pumped optical parametric oscillators (SPOPOs) are established
M
as reliable light sources with stable tunable ultrashort pulses suitable for many applications across the optical spectrum from ultraviolet to the mid-infrared [9,10].
te
d
Nevertheless, the requirement for cavity length synchronization, especially the need for mode-locked ultrafast laser sources, increases the cost and complexity of system
Ac ce p
design for such ultrafast OPOs. In recent years, an approach of active mode-locking methods, as used in conventional lasers, to OPOs under continuous wave (cw) pumping has been realized [11-13]. This kind of OPOs circumvents the need for the sources of picosecond and femtosecond pulses and synchronous pumping, offers a new technique of generating tunable ultrashort pulses using OPOs, whereas the pulse width of oscillation beams is a few hundred picoseconds to one nanosecond. Hybrid mode-locking techniques are an efficient way for the pulse compression and have been demonstrated in lasers. By employing dual-loss-modulation method, stable Q-switched and mode-locked pulses can be generated [14]. It would be
Page 2 of 15
expected, therefore, that similar mode-locking strategies could be available in OPOs due to the analogy between the parametric devices and the lasers, since oscillation pulse width could be compressed efficiently, and the pump sources of longer pulses
ip t
may be applied. In previous work, by using Kerr lens medium, pulse compression in a
cr
SPOPO is demonstrated experimentally [15]. In this work, the pulse compression of
us
inserting passive mode locker, such as graphene and Kerr lens medium, into a SPOPO and cw-pumped actively mode-locked OPO is simulated and analyzed. The simulated
an
results show that in the SPOPO, the Kerr lens medium and graphene are both favorable for the pulse compression, and the Kerr lens medium can support a
M
relatively wide tuning range of pulse width, while the graphene is easier for cavity alignment. In an cw-pumped actively mode locked OPO, however, the situation is
te
d
different that using the Kerr lens medium, the pulse compression will occur only if the normalized aperture radius, pump power and the beam radius on the nonlinear crystal
Ac ce p
can satisfy the certain conditions, whereas the graphene can not be applied for the pulse compression in the actively mode locked OPO.
2. Theoretical model
We consider the PPKTP- OPO with the passive mode locker of Kerr lens or
graphene medium to be singly resonant at the signal frequency, and assume that the method of type I quasi-phase-matching is adopted in order to improve the conversion efficiency, the parametric interaction based on the nonlinear coupled-wave equations in the plane-wave approximation can be described as [16]
Page 3 of 15
As 1 As Ap Ai*e ikz z vs t Ai 1 Ai Ap As*e ikz z vi t z
1 Ap As Ai eikz v p t
ip t
Ap
(1)
where Ap2, s ,i p p , s ,i dn p , s ,i / dt are the amplitudes of pump, signal and idler waves,
cr
n p , s ,i are the numbers of photons of pump, signal and idler waves, respectively, and
k k p k s ki k m 0
us
v p , s ,i are the group velocities. k is the wave-vector mismatch and satisfied
(2)
an
where k p , s ,i are the wave vectors, and km is the grating vector of the PPKTP. The coupling coefficient is
(2) psi h0 1/ 2 ( ) c 2nrp nrs nri S
M
(3)
d
where nrp , s ,i are the refractive indices, p , s ,i are the frequencies of the waves, S is
te
the effective area in the PPKTP crystal, h is Planck constant, 0 377 .
Ac ce p
The signal beams is amplified by PPKTP and modulated by the mode locker in
each round trip, and finally become stable. In synchronously pumped OPO, the mode locker is the graphene or the Kerr lens medium. The modulation effect of the graphene can be described as [17]
q (t )
q0 A(t ) 1 PA
(4)
2
where q0 is the initial saturable absorption, PA
EA is the saturation power of the A
graphene, A is the relaxation time, E A is the saturation energy of the graphene. The nonlinearity of Kerr lens medium can be described as [18]
Page 4 of 15
2 (l ) 12 [(1
l 2 l 2 P ) ( ) (1 )] 2 R1 Pc n01
(5)
where is the signal wavelength, n0 is the linear refractive index, 1 is the
ip t
initial beam radius, R1 is the initial radius of curvature, l is the length of the Kerr lens medium, (l ) is the beam radius after the beam passing through the Kerr lens
cr
medium, P is the beam power and Pc c 0 2 /(2 n2 ) is the critical power of
us
self-focusing, n2 is the nonlinear refractive index. Combined with a suitable aperture, the Kerr lens medium has shown the capability of modulated loss associated with the
an
power difference. In the actively mode locked OPO, however, the mode lockers
M
contain two parts: a passive mode locker and an active mode locker. Here we take an acousto-optic modulator (AOM) as the active mode locker. The intracavity periodic
d
loss can be written as [11]
(6)
te
T (t ) 1 sin 2 (t / 2)
Ac ce p
where is the modulation depth of the AOM, 2 rf is repetition rate of pulses,
rf is the radio frequency of AOM. Since the round-trip time of the cavity is much shorter than the time-scale for
signal pulses to reach the steady-state condition, to simulate pulse compression in the SRO, our strategy is to use the Runge-Kutta numerical method incorporating with split-step method to solve the nonlinear coupled-wave equations in the presence of the modulated loss of active and passive mode locker. A large amount of the calculation is accomplished in the MATLAB environment.
3. Simulation results and discussion
Page 5 of 15
To facilitate the discussions, we use the normalized aperture radius in the calculation, which means each value is divided by the value corresponding to the lowest power of the pulse. As an example, a KTP crystal is taken as the Kerr lens medium in our
ip t
model. The parameters involved are set as follows. The PPKTP length l NLO is 1cm,
cr
the effective nonlinearity coefficient d eff is 5.2pm/V. The beam spot NLO of pump, signal and idler in the PPKTP crystal is assumed to be overlapped perfectly.
us
The wavelengths are set to be: p = 532nm, s = 989nm and i = 1151nm. The
an
linear loss only comes from the transmittance of output coupler of 1%, neglecting the absorption of PPKTP, the intrinsic loss of the passive mode locker and the loss of
monolayer
and
bilayer
M
coating. The nonlinear refractive index n2 of KTP is 29.4×10-16cm2/W. For graphene,
respectively,
q0 =0.54%
and
1.30%,
d
FA =14.5μJ/cm2 and 22.5μJ/cm2, where FA PA / A is the saturation fluence, and
te
A is the beam spot size in the graphene.
Ac ce p
3.1 Synchronously pumped OPO
To evaluate the feasibility of pulse compression using the passive mode locker in the SPOPO, we solve the
nonlinear
coupled-wave
equations, Eq. (1), with Eq. (4) or Eq. Fig.1 Relation of the pulse width and peak power to the
(5) simultaneously. The pump pulse
normalized aperture radius at NLO =100μm
energy is about 28nJ, corresponding
to five times the pump threshold. The relations of the pulse width S and the peak
Page 6 of 15
power of the signal pulses to the normalized aperture radius are shown in Fig. 1. As can be seen, compared to the situation without the Kerr medium, the signal pulse width gets compressed after the Kerr medium and the aperture are set in the cavity,
ip t
and the degree of compression can be controlled by adjusting the aperture radius. The
cr
reason of compression by Kerr medium is due to the three order nonlinear effect. The
us
power of the peak section of the pulse is much higher than that of tails, resulting in the different focus spot size after passing through the Kerr medium, thus different losses
an
will be suffered after passing through an aperture. In addition, the nonlinear parametric gain of peak section is greater than that of tails in nonlinear amplifying,
M
due to the characteristics of the synchronously pumping. As a result, the difference of the power between the peak section and the tails becomes larger gradually, leading to
te
d
the pulse compression when the pulses become stable. Furthermore, the difference of the loss of the power increases with the reduction of the aperture radius, resulting in
Ac ce p
further pulse compression at the expense of the peak power. Replacing
the
Kerr
lens
medium with the monolayer and bilayer graphene, the relations of signal pulse width S to the beam radius on the graphene are discussed,
Fig.2 Relation of the pulse width to the beam radius in the monolayer and bilayer graphene
as shown in Fig. 2. As evident from Fig. 2, the pulses get compressed
with the graphene insertion, and pulse width decrease a little with the increase of
Page 7 of 15
radius. At the same spot size, the compression effect using bilayer graphene is more obvious than that of monolayer graphene, implying that increasing the graphene layers will be helpful for the pulse compression in synchronously pumped OPO. The
ip t
reason is obvious that the saturable absorption of graphene is proportional to the
cr
instant power, thus the tails of pulses suffer more loss than the peak section. And the
us
absorption increase with the layers of graphene, leading to a larger difference of power between peak section and tails. What is more, the peak section of pulses gets
an
more parametric gain during the nonlinear interaction, finally resulting in pulse compression when the performance of OPO becomes stable. Compared with the
M
results shown in Fig. 1, we can readily seen that the tuning range of pulse width is wider by using Kerr lens medium, as well as shorter pulse generation. Yet the cavity
te
d
alignment by using graphene is relatively simple, and the requirement for the spot size is relatively low, implies that each of the passive mode lockers has advantages in
Ac ce p
pulse compression.
Because the signal pulse always get compressed when passive mode locker is
Fig.3 Relation of the pulse width (a) and peak power (b) to the beam radius in the PPKTP for different crystal length
Page 8 of 15
inserted into the cavity, in order to optimum the pulse width and peak power, we study the relation of them to the beam radius on the PPKTP crystal at different crystal length without any passive mode locker in the cavity, with the results illustrated in Fig.
ip t
3. As can be seen, as the crystal length is fixed, the signal pulse width decreases with
cr
the increase of the spot radius on PPKTP crystal, and the peak power also increases.
us
When the beam radius on PPKTP crystal is invariant, the shorter the crystal length is, the narrower the pulse width is, and the higher the peak power is. From the pump
an
threshold of singly-resonant OPO formula written as [19]
with
c 0 n p ns ni s i A 8 2 d 2l 2
（7）
（8）
d
M
Pth (Ts Vs )
te
it can be seen that the pump threshold increases with the spot area on the nonlinear
Ac ce p
crystal, and decreases with the growth of crystal length. Here Ts denotes the transmission of the output coupler and Vs the sum of all residual cavity losses, λs,i is the signal and idler wavelengths, d= χ(2)/2 is the nonlinear coefficient, Ip is the pump
intensity, and nj (j=p, s, i) the refractive indices of the interacting waves. As long as the pump power density gets closer to the pump threshold, the signal pulse width becomes narrower, and the peak power becomes stronger, indicating that it will be conducive for the pulse compression.
Page 9 of 15
us
cr
ip t
3.2 Actively mode locked OPO
Fig.4 Relation of the pulse width (a) and peak power (b) to the normalized aperture radius for
NLO =150μm
an
different pump power at
In the actively mode-locked OPO, an AOM is inserted into the cavity as an
M
active mode locker, combined with the passive mode locker to modulate the signal pulses. Thus, we solve the nonlinear coupled-wave equations, Eq. (1), the periodic
te
d
loss equation of the AOM, Eq. (6), with Eq. (4) or Eq. (5) simultaneously to study the pulse compression effects of passive mode locker in the cavity. The pump source is a
Ac ce p
cw laser. The modulation depth of the AOM is 50%, and the repetition rate =
160MHz. The relations of signal pulse width S and peak power to normalized aperture radius are calculated under different pump power Ppump , as shown in Fig. 4. We observed from Fig. 4(a) that at first, the pulse width is compressed with the decrease of the normalized aperture radius to a certain value related to the peak power of signal pulses. After that as the aperture radius continue to decrease, pulse width will increase gradually, and then gradually decreased. Under the same aperture radius, the lower pump power is, the narrower signal pulse is. Compared with the condition of no Kerr lens medium insertion, the normalized aperture radius required for the
Page 10 of 15
pulse compression get smaller with reduction of the pump power after Kerr lens medium and aperture are set in the cavity. The peak power keep constant until the aperture radius decreases to the certain value, as illustrated in Fig. 4(b), and then
ip t
begin to decline. Other than the situations in the SPOPO, however, the front pulse tail
cr
obtain more parametric gain than peak section under the cw pumping until pump
us
power reduce to a certain power, diminish the power difference between the peak section and the tails, resulting in the pulse broadening. After that, the net loss of the
an
tails is greater than that of the peak section, leading to the pulse compression. And the Fig. 4(b) also shows that the power of peak section becomes higher as the pump
M
power get closer to the threshold, indicating that getting closer to the pump threshold is not only benefit for pulse compression, but also for higher peak power of pulse.
te
d
Connected the results with Eqs. (7)-(8), some wisdom can be derived. The pump power should be as high as possible. And shorter PPKTP crystal length, as well as
Ac ce p
expanding the beam radius on PPKTP appropriately is preferred, in order to make the pump power close to OPO threshold. These operations will benefit for the signal pulse compression in a cw-pumped OPO using Kerr lens medium. The relations of the signal pulse width S to the beam radius on monolayer and bilayer graphene are
Fig.5 Relation of the pulse width to the beam radius in the monolayer and bilayer graphene
calculated, as illustrated in Fig. 5. The pump power Ppump is about
Page 11 of 15
316W, corresponding to five times the pump threshold. As can be seen from Fig. 5, in the cw pumped OPO, the signal pulse become broader after inserting the graphene into the cavity, comparing with the situation of no graphene insertion. And the
ip t
increase of the graphene layers makes the signal pulses even broader, indicating that
cr
the graphene is not a suitable device for the signal pulse compression in a cw pumped
us
actively mode-locked OPO.
an
4. Conclusion
In conclusion, we have theoretically discussed the effects of a passive mode locker in
M
a SPOPO or actively mode-locked OPO under cw pumping. The numerical results show that in the SPOPO, the Kerr lens medium and graphene are both available for
te
d
the pulse compression, and the Kerr lens medium is in favor of relatively wide tuning range of pulse width, while the graphene is easier for cavity alignment. Nevertheless,
Ac ce p
the situation in a actively mode locked OPO under cw pumping is different that the pulse compression by using the Kerr lens medium will occur only if the normalized aperture radius, pump power and the beam radius on PPKTP can satisfy the certain conditions, yet the graphene can not take effect on the pulse compression. The discussion here could be helpful not only for the techniques of the pulse compression in an ultrafast OPOs using passive mode locker, but also for the choice of passive mode locker in different types of ultrafast OPOs.
Acknowledgement
Page 12 of 15
This work was supported by the Key Project of Beijing Municipal Education Commission Research Program (Kz201110005010) and special programs for
ip t
advanced talents of the sea poly project.
cr
Reference
us
[1] L. Shah, M. E. Fermann, J. W. Dawson, C. P. J. Barty, Micromachining with a 50 W, 50 μJ, subpicosecond fiber laser system, Opt. Express 14 (2006) 12546–12551.
an
[2] U. Keller, Femtosecond to attosecond optics, IEEE Photon. J. 2 (2010) 3.
[3] A.A. Lagatsky, C.T.A. Brown, W. Sibbett, Highly efficient and low threshold diode-pumped
M
Kerr-lens mode locked Yb:KYW laser, Opt. Express 12 (2004) 3928-3933.
[4] C. G. Durfee, T. Storz, J. Garlick, S. Hill, J. A. Squier, M. Kirchner, G. Taft, K. Shea, H. Kapteyn, M.
(2012) 13677–13683.
te
d
Murnane, S. Backus, Direct diode-pumped Kerr-lens mode-locked Ti:sapphire laser, Opt. Express 20
Ac ce p
[5] E. Ugolotti, A. Schmidt, V. Petrov, J. W. Kim, D. I. Yeom, F. Rotermund, S. Bae, B. H. Hong, A. Agnesi, C. Fiebig, G. Erbert, X. Mateos, M. Aguilo, F. Diaz, U. Griebner, Graphene mode-locked femtosecond Yb:KLuW laser, Appl. Phys. Lett. 101 (2012) 161112. [6] A. A. Lagatsky, Z. Sun, T. S. Kulmala, R. S. Sundaram, S. Milana, F. Torrisi, O. L. Antipov, Y. Lee, J. H. Ahn, C. T. A. Brown, W. Sibbett, A. C. Ferrari, 2μm solid-state laser mode-locked by single-layer
graphene, Appl. Phys. Lett. 102 (2013) 013113. [7] H.Y. Lin, X. H. Huang, Y. C. Xu, X. G. Meng, L. E Cai, Mid-infrared, wide-tunable, continuous-wave Nd:YVO4/PPMgLN intracavity optical parametric oscillator, Optik 125 (2014) 6969–6971
Page 13 of 15
[8] M. Ebrahim-Zadeh, S. C. Kumar, A. E. Martin, G. K. Samanta, Breakthroughs in Photonics 2012: Breakthroughs in Optical Parametric Oscillators, IEEE Photon. J. 5 (2013) 0700105.
ip t
[9] M. Ebrahim-Zadeh, Efficient Ultrafast Frequency Conversion Sources for the Visible and Ultraviolet Based on BiB3O6, IEEE J. Sel. Top. Quantum Electron. 13 (2007) 679–691
cr
[10] S. C. Kumar, M. Ebrahim-Zadeh, Fiber-laser-based green-pumped picosecond MgO:sPPLT
us
optical parametric oscillator, Opt. Lett. 38 (2013) 5349-5352.
[11] N. Forget, S. Bahbah, C. Drag, F. Bretenaker, L. Michel, E. Rosencher, Actively mode-locked
an
optical parametric oscillator, Opt. Lett. 31 (2006) 972-974
[12] K. Devi, S. C. Kumar, M. Ebrahim-Zadeh, Mode-locked, continuous-wave, singly resonant optical
M
parametric oscillator, Opt. Lett. 37 (2012) 3909-3911.
[13] K. Devi, S. C. Kumar, M. Ebrahim-Zadeh, Directly phase-modulation-mode-locked doubly
te
d
resonant optical parametric oscillator, Opt. Express 21 (2013) 23365-23375. [14] H. J. Zhang, S. Z. Zhao, K. J. Yang, G. Q. Li, D. C. Li , J. Zhao, Theoretical and experimental
Ac ce p
study of single mode-locking pulse generation with low repetition rate in a dual-loss-modulated QML YVO4/Nd:YVO4 laser with EO and GaAs, Appl. Phys. B 112 (2013) 185–192.
[15] R. Laenen, C. Rauscher, A. Laubereau, Kerr lens mode-locking of a sub-picosecond optical parametric oscillator, Opt. Commun. 115 (1995) 533-538. [16] J. Khurgin, J. M. Melkonian, A. Godard, M. Lefebvre, E. Rosencher, Passive mode locking of optical parametric oscillators: an efficient technique for generating sub-picosecond pulses, Opt. Express 16 (2008) 4804-4818. [17] F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, U. Keller, Control of solid state laser dynamics by semiconductor devices, Opt. Engine. 34 (1995) 2024-2036.
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[18] V. Magni, G. Cerullo, S. De Silvestri, ABCD matrix analysis of propagation of gaussian beams through Kerr media, Opt. Commun. 96 (1993) 348-355.
ip t
[19] S. Guha, Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,
Ac ce p
te
d
M
an
us
cr
Appl. Phys. B 66 (1998) 663–675.
Page 15 of 15
S0030-4026(15)01304-2 http://dx.doi.org/doi:10.1016/j.ijleo.2015.09.223 IJLEO 56428
To appear in: Received date: Accepted date:
14-11-2014 25-9-2015
Please cite this article as: J. Wang, L. Wang, Y. Zhan, Pulse compression effects of passive mode locker in ultrafast optical parametric oscillator, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.09.223 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Pulse compression effects of passive mode locker in ultrafast optical parametric oscillator
ip t
Jieyu Wang, Li Wang*, Yi Zhan College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
cr
Abstract: The pulse compression effects of passive mode locker, such as graphene and Kerr lens
us
medium, which has been inserted into a synchronously-pumped optical parametric oscillator (SPOPO) pumped by pulse laser, and a continuous-wave (cw) pumped actively mode-locked OPO has been
an
studied theoretically. By solving the nonlinear coupled-wave equations in the plane-wave approximation, we simulate the relations of the pulse width and peak power of the signal pulses to the
M
normalized aperture radius, the nonlinear crystal length and the beam radius on the nonlinear crystal. The numerical results indicate that the Kerr lens medium and graphene are both favorable for the pulse
te
d
compression in the SPOPO, and each of them has different advantages. Whereas in the actively mode locked OPO, the Kerr lens medium may be a better alternative for the pulse compression.
Ac ce p
Keywords: pulse compression, optical parametric oscillator, Kerr lens medium, graphene saturable absorber, acousto-optic modulation
1. Introduction
Since ultrashort pulses have paved a way for many applications including
material processing, nonlinear optics, attosecond science, and metrology due to the advantages of high peak intensity and broad optical spectrum, there has been an ongoing effort in the field to reduce the pulse width and increase the power of lasers
* Corresponding author. E-mail address: [email protected]
Page 1 of 15
[1,2]. Among these, the Kerr lens mode locking (KLM) technique can support the generation of femtosecond pulses due to the non-resonant nature of the optical Kerr effect, lending itself to a wide range of gain media [3,4]. Besides that, the graphene
ip t
saturable absorber is another excellent choice for femtosecond pulses generation,
cr
especially with the application in a wide spectrum [5,6]. However, the tuning range of
us
these kinds of lasers is confined to limited spectral bands in the near-infrared.
Optical parametric oscillators (OPOs) have been extensively investigated for
an
over four decades, expanding the wavelength range of lasers to new spectral coverage [7,8]. Synchronously-pumped optical parametric oscillators (SPOPOs) are established
M
as reliable light sources with stable tunable ultrashort pulses suitable for many applications across the optical spectrum from ultraviolet to the mid-infrared [9,10].
te
d
Nevertheless, the requirement for cavity length synchronization, especially the need for mode-locked ultrafast laser sources, increases the cost and complexity of system
Ac ce p
design for such ultrafast OPOs. In recent years, an approach of active mode-locking methods, as used in conventional lasers, to OPOs under continuous wave (cw) pumping has been realized [11-13]. This kind of OPOs circumvents the need for the sources of picosecond and femtosecond pulses and synchronous pumping, offers a new technique of generating tunable ultrashort pulses using OPOs, whereas the pulse width of oscillation beams is a few hundred picoseconds to one nanosecond. Hybrid mode-locking techniques are an efficient way for the pulse compression and have been demonstrated in lasers. By employing dual-loss-modulation method, stable Q-switched and mode-locked pulses can be generated [14]. It would be
Page 2 of 15
expected, therefore, that similar mode-locking strategies could be available in OPOs due to the analogy between the parametric devices and the lasers, since oscillation pulse width could be compressed efficiently, and the pump sources of longer pulses
ip t
may be applied. In previous work, by using Kerr lens medium, pulse compression in a
cr
SPOPO is demonstrated experimentally [15]. In this work, the pulse compression of
us
inserting passive mode locker, such as graphene and Kerr lens medium, into a SPOPO and cw-pumped actively mode-locked OPO is simulated and analyzed. The simulated
an
results show that in the SPOPO, the Kerr lens medium and graphene are both favorable for the pulse compression, and the Kerr lens medium can support a
M
relatively wide tuning range of pulse width, while the graphene is easier for cavity alignment. In an cw-pumped actively mode locked OPO, however, the situation is
te
d
different that using the Kerr lens medium, the pulse compression will occur only if the normalized aperture radius, pump power and the beam radius on the nonlinear crystal
Ac ce p
can satisfy the certain conditions, whereas the graphene can not be applied for the pulse compression in the actively mode locked OPO.
2. Theoretical model
We consider the PPKTP- OPO with the passive mode locker of Kerr lens or
graphene medium to be singly resonant at the signal frequency, and assume that the method of type I quasi-phase-matching is adopted in order to improve the conversion efficiency, the parametric interaction based on the nonlinear coupled-wave equations in the plane-wave approximation can be described as [16]
Page 3 of 15
As 1 As Ap Ai*e ikz z vs t Ai 1 Ai Ap As*e ikz z vi t z
1 Ap As Ai eikz v p t
ip t
Ap
(1)
where Ap2, s ,i p p , s ,i dn p , s ,i / dt are the amplitudes of pump, signal and idler waves,
cr
n p , s ,i are the numbers of photons of pump, signal and idler waves, respectively, and
k k p k s ki k m 0
us
v p , s ,i are the group velocities. k is the wave-vector mismatch and satisfied
(2)
an
where k p , s ,i are the wave vectors, and km is the grating vector of the PPKTP. The coupling coefficient is
(2) psi h0 1/ 2 ( ) c 2nrp nrs nri S
M
(3)
d
where nrp , s ,i are the refractive indices, p , s ,i are the frequencies of the waves, S is
te
the effective area in the PPKTP crystal, h is Planck constant, 0 377 .
Ac ce p
The signal beams is amplified by PPKTP and modulated by the mode locker in
each round trip, and finally become stable. In synchronously pumped OPO, the mode locker is the graphene or the Kerr lens medium. The modulation effect of the graphene can be described as [17]
q (t )
q0 A(t ) 1 PA
(4)
2
where q0 is the initial saturable absorption, PA
EA is the saturation power of the A
graphene, A is the relaxation time, E A is the saturation energy of the graphene. The nonlinearity of Kerr lens medium can be described as [18]
Page 4 of 15
2 (l ) 12 [(1
l 2 l 2 P ) ( ) (1 )] 2 R1 Pc n01
(5)
where is the signal wavelength, n0 is the linear refractive index, 1 is the
ip t
initial beam radius, R1 is the initial radius of curvature, l is the length of the Kerr lens medium, (l ) is the beam radius after the beam passing through the Kerr lens
cr
medium, P is the beam power and Pc c 0 2 /(2 n2 ) is the critical power of
us
self-focusing, n2 is the nonlinear refractive index. Combined with a suitable aperture, the Kerr lens medium has shown the capability of modulated loss associated with the
an
power difference. In the actively mode locked OPO, however, the mode lockers
M
contain two parts: a passive mode locker and an active mode locker. Here we take an acousto-optic modulator (AOM) as the active mode locker. The intracavity periodic
d
loss can be written as [11]
(6)
te
T (t ) 1 sin 2 (t / 2)
Ac ce p
where is the modulation depth of the AOM, 2 rf is repetition rate of pulses,
rf is the radio frequency of AOM. Since the round-trip time of the cavity is much shorter than the time-scale for
signal pulses to reach the steady-state condition, to simulate pulse compression in the SRO, our strategy is to use the Runge-Kutta numerical method incorporating with split-step method to solve the nonlinear coupled-wave equations in the presence of the modulated loss of active and passive mode locker. A large amount of the calculation is accomplished in the MATLAB environment.
3. Simulation results and discussion
Page 5 of 15
To facilitate the discussions, we use the normalized aperture radius in the calculation, which means each value is divided by the value corresponding to the lowest power of the pulse. As an example, a KTP crystal is taken as the Kerr lens medium in our
ip t
model. The parameters involved are set as follows. The PPKTP length l NLO is 1cm,
cr
the effective nonlinearity coefficient d eff is 5.2pm/V. The beam spot NLO of pump, signal and idler in the PPKTP crystal is assumed to be overlapped perfectly.
us
The wavelengths are set to be: p = 532nm, s = 989nm and i = 1151nm. The
an
linear loss only comes from the transmittance of output coupler of 1%, neglecting the absorption of PPKTP, the intrinsic loss of the passive mode locker and the loss of
monolayer
and
bilayer
M
coating. The nonlinear refractive index n2 of KTP is 29.4×10-16cm2/W. For graphene,
respectively,
q0 =0.54%
and
1.30%,
d
FA =14.5μJ/cm2 and 22.5μJ/cm2, where FA PA / A is the saturation fluence, and
te
A is the beam spot size in the graphene.
Ac ce p
3.1 Synchronously pumped OPO
To evaluate the feasibility of pulse compression using the passive mode locker in the SPOPO, we solve the
nonlinear
coupled-wave
equations, Eq. (1), with Eq. (4) or Eq. Fig.1 Relation of the pulse width and peak power to the
(5) simultaneously. The pump pulse
normalized aperture radius at NLO =100μm
energy is about 28nJ, corresponding
to five times the pump threshold. The relations of the pulse width S and the peak
Page 6 of 15
power of the signal pulses to the normalized aperture radius are shown in Fig. 1. As can be seen, compared to the situation without the Kerr medium, the signal pulse width gets compressed after the Kerr medium and the aperture are set in the cavity,
ip t
and the degree of compression can be controlled by adjusting the aperture radius. The
cr
reason of compression by Kerr medium is due to the three order nonlinear effect. The
us
power of the peak section of the pulse is much higher than that of tails, resulting in the different focus spot size after passing through the Kerr medium, thus different losses
an
will be suffered after passing through an aperture. In addition, the nonlinear parametric gain of peak section is greater than that of tails in nonlinear amplifying,
M
due to the characteristics of the synchronously pumping. As a result, the difference of the power between the peak section and the tails becomes larger gradually, leading to
te
d
the pulse compression when the pulses become stable. Furthermore, the difference of the loss of the power increases with the reduction of the aperture radius, resulting in
Ac ce p
further pulse compression at the expense of the peak power. Replacing
the
Kerr
lens
medium with the monolayer and bilayer graphene, the relations of signal pulse width S to the beam radius on the graphene are discussed,
Fig.2 Relation of the pulse width to the beam radius in the monolayer and bilayer graphene
as shown in Fig. 2. As evident from Fig. 2, the pulses get compressed
with the graphene insertion, and pulse width decrease a little with the increase of
Page 7 of 15
radius. At the same spot size, the compression effect using bilayer graphene is more obvious than that of monolayer graphene, implying that increasing the graphene layers will be helpful for the pulse compression in synchronously pumped OPO. The
ip t
reason is obvious that the saturable absorption of graphene is proportional to the
cr
instant power, thus the tails of pulses suffer more loss than the peak section. And the
us
absorption increase with the layers of graphene, leading to a larger difference of power between peak section and tails. What is more, the peak section of pulses gets
an
more parametric gain during the nonlinear interaction, finally resulting in pulse compression when the performance of OPO becomes stable. Compared with the
M
results shown in Fig. 1, we can readily seen that the tuning range of pulse width is wider by using Kerr lens medium, as well as shorter pulse generation. Yet the cavity
te
d
alignment by using graphene is relatively simple, and the requirement for the spot size is relatively low, implies that each of the passive mode lockers has advantages in
Ac ce p
pulse compression.
Because the signal pulse always get compressed when passive mode locker is
Fig.3 Relation of the pulse width (a) and peak power (b) to the beam radius in the PPKTP for different crystal length
Page 8 of 15
inserted into the cavity, in order to optimum the pulse width and peak power, we study the relation of them to the beam radius on the PPKTP crystal at different crystal length without any passive mode locker in the cavity, with the results illustrated in Fig.
ip t
3. As can be seen, as the crystal length is fixed, the signal pulse width decreases with
cr
the increase of the spot radius on PPKTP crystal, and the peak power also increases.
us
When the beam radius on PPKTP crystal is invariant, the shorter the crystal length is, the narrower the pulse width is, and the higher the peak power is. From the pump
an
threshold of singly-resonant OPO formula written as [19]
with
c 0 n p ns ni s i A 8 2 d 2l 2
（7）
（8）
d
M
Pth (Ts Vs )
te
it can be seen that the pump threshold increases with the spot area on the nonlinear
Ac ce p
crystal, and decreases with the growth of crystal length. Here Ts denotes the transmission of the output coupler and Vs the sum of all residual cavity losses, λs,i is the signal and idler wavelengths, d= χ(2)/2 is the nonlinear coefficient, Ip is the pump
intensity, and nj (j=p, s, i) the refractive indices of the interacting waves. As long as the pump power density gets closer to the pump threshold, the signal pulse width becomes narrower, and the peak power becomes stronger, indicating that it will be conducive for the pulse compression.
Page 9 of 15
us
cr
ip t
3.2 Actively mode locked OPO
Fig.4 Relation of the pulse width (a) and peak power (b) to the normalized aperture radius for
NLO =150μm
an
different pump power at
In the actively mode-locked OPO, an AOM is inserted into the cavity as an
M
active mode locker, combined with the passive mode locker to modulate the signal pulses. Thus, we solve the nonlinear coupled-wave equations, Eq. (1), the periodic
te
d
loss equation of the AOM, Eq. (6), with Eq. (4) or Eq. (5) simultaneously to study the pulse compression effects of passive mode locker in the cavity. The pump source is a
Ac ce p
cw laser. The modulation depth of the AOM is 50%, and the repetition rate =
160MHz. The relations of signal pulse width S and peak power to normalized aperture radius are calculated under different pump power Ppump , as shown in Fig. 4. We observed from Fig. 4(a) that at first, the pulse width is compressed with the decrease of the normalized aperture radius to a certain value related to the peak power of signal pulses. After that as the aperture radius continue to decrease, pulse width will increase gradually, and then gradually decreased. Under the same aperture radius, the lower pump power is, the narrower signal pulse is. Compared with the condition of no Kerr lens medium insertion, the normalized aperture radius required for the
Page 10 of 15
pulse compression get smaller with reduction of the pump power after Kerr lens medium and aperture are set in the cavity. The peak power keep constant until the aperture radius decreases to the certain value, as illustrated in Fig. 4(b), and then
ip t
begin to decline. Other than the situations in the SPOPO, however, the front pulse tail
cr
obtain more parametric gain than peak section under the cw pumping until pump
us
power reduce to a certain power, diminish the power difference between the peak section and the tails, resulting in the pulse broadening. After that, the net loss of the
an
tails is greater than that of the peak section, leading to the pulse compression. And the Fig. 4(b) also shows that the power of peak section becomes higher as the pump
M
power get closer to the threshold, indicating that getting closer to the pump threshold is not only benefit for pulse compression, but also for higher peak power of pulse.
te
d
Connected the results with Eqs. (7)-(8), some wisdom can be derived. The pump power should be as high as possible. And shorter PPKTP crystal length, as well as
Ac ce p
expanding the beam radius on PPKTP appropriately is preferred, in order to make the pump power close to OPO threshold. These operations will benefit for the signal pulse compression in a cw-pumped OPO using Kerr lens medium. The relations of the signal pulse width S to the beam radius on monolayer and bilayer graphene are
Fig.5 Relation of the pulse width to the beam radius in the monolayer and bilayer graphene
calculated, as illustrated in Fig. 5. The pump power Ppump is about
Page 11 of 15
316W, corresponding to five times the pump threshold. As can be seen from Fig. 5, in the cw pumped OPO, the signal pulse become broader after inserting the graphene into the cavity, comparing with the situation of no graphene insertion. And the
ip t
increase of the graphene layers makes the signal pulses even broader, indicating that
cr
the graphene is not a suitable device for the signal pulse compression in a cw pumped
us
actively mode-locked OPO.
an
4. Conclusion
In conclusion, we have theoretically discussed the effects of a passive mode locker in
M
a SPOPO or actively mode-locked OPO under cw pumping. The numerical results show that in the SPOPO, the Kerr lens medium and graphene are both available for
te
d
the pulse compression, and the Kerr lens medium is in favor of relatively wide tuning range of pulse width, while the graphene is easier for cavity alignment. Nevertheless,
Ac ce p
the situation in a actively mode locked OPO under cw pumping is different that the pulse compression by using the Kerr lens medium will occur only if the normalized aperture radius, pump power and the beam radius on PPKTP can satisfy the certain conditions, yet the graphene can not take effect on the pulse compression. The discussion here could be helpful not only for the techniques of the pulse compression in an ultrafast OPOs using passive mode locker, but also for the choice of passive mode locker in different types of ultrafast OPOs.
Acknowledgement
Page 12 of 15
This work was supported by the Key Project of Beijing Municipal Education Commission Research Program (Kz201110005010) and special programs for
ip t
advanced talents of the sea poly project.
cr
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us
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ip t
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cr
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Ac ce p
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ip t
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