## Abstract

The operation of a cascaded second-order mode-locked Nd:YVO_{4} laser has been investigated considering it as a soft-aperture Kerr lens type and using complex beam parameters. A self consistent complex beam propagation method is used to incorporate the effect of cascaded Kerr nonlinearity on radially varying gain aperturing. The analysis deduces a stable pulsewidth of ~9.5 ps which agrees well with the experimental value of 10.3 ps.

© 2016 Optical Society of America

## 1. Introduction

Mode-locking in femtosecond regime using Kerr lensing in Ti:sapphire laser has been quite successful in recent years as it is very stable and self-starting [1]. Q-switching is also very stable in the nano-second regime [2]. When operated in pico-second (ps) regime, Ti:Sapphire lasers are again stable, although the pulse is not bandwidth limited [3]. It is quite challenging to achieve bandwidth limited mode-locked laser operation in ps regime especially in other materials such as Nd based gain medium [4] because of low gain and high threshold. Here we point out that cascaded second-order nonlinear optical processes can play an important role in generating self-amplitude modulation and phase variations of light beams propagating in non-centro-symmetric materials [5]. They have been widely studied for applications related to almost all third order effects such as all-optical switching in waveguides, for generation of spatio-temporal solitons, nearly degenerate four-wave mixing, cascaded interaction in optical parametric oscillator, etc. [6–11] in past decades. Earlier we reported experimental realization of nonlinear mirror modelocking based on cascaded second-order interaction [12–16]. The generation of ultra-short pulses by modelocking of solid-state lasers with this approach is comparatively new and has not been analyzed properly to date in a quantitative manner [17–19]. Here we provide such an analysis and compare it with observations in a Nd:YVO_{4} laser.

The second order cascaded process can be briefly stated as: when two second-order nonlinear optical interactions take place simultaneously in a nonlinear crystal (*ω _{f}* +

*ω*=

_{f}*ω*and

_{sh}*ω*=

_{sh}− ω_{f}*ω*), it produces a large effective third-order nonlinearity $({\chi}_{eff}^{(3)})$, which can mimic the effects of intrinsic third order process in a second order nonlinear optical crystal at much lower power [5, 20, 21]. The $({\chi}_{eff}^{(3)})$ can be positive or negative and can be realized by the phase mismatch (Δ

_{f}*k*) between fundamental and second-harmonic beam, conforming to a self-focusing or a self-defocusing medium. This cascaded process can also be useful for mode-locking, namely cascaded second-order mode-locking (CSM) [21], where a second-harmonic generating (SHG) crystal is placed in off-phase-matched condition and uses the real part of the induced $({\chi}_{eff}^{(3)})$ to introduce a nonlinear phase shift in the fundamental beam. It does not provide direct amplitude modulation as in the case of nonlinear mirror modelocking (NLM) [22]; rather forward and return pass through the nonlinear crystal (NLC) tracks a nonlinear phase shift (Φ

*) on the fundamental beam giving nonlinear mode size variation. This in turn can be converted into nonlinear gain modulation at the gain medium. Empirically, this scheme is analogous to Kerr-lens modelocking (KLM) since KLM uses the self-focusing effect in gain medium due to intrinsic third-order nonlinearity $({\chi}_{\mathit{int}}^{(3)})$ of the medium [1,23].*

_{NL}In this report, we present a complete picture of CSM experimentally (section 2) and explain results theoretically. We study the role of radially varying gain in CSM for producing stable ps pulses. We find that this approach favors modelocking by a mechanism analogous to fast saturable absorption as discussed in section 3. In Sections 3 and 4, we derive the characteristic modelocking parameters and study the stability of CSM laser. We use the master-modelocking equation to determine the stable pulsewidth, chirp parameter and stability criteria. The measured pulsewidth is 10.3 ps, whereas, from our theoretical analysis, we get ~9.5 *ps* in excellent agreement with the experimental value.

## 2. Experiment

The schematic diagram of the soft aperture CSM laser is shown in Fig. 1. The pump radiation at 808 nm from a fiber coupled laser diode array of maximum output power 12 W is focused to a spot size of 240 *µ*m at the centre of the gain medium which is a 4×4×8 mm^{3}, *a*-cut, Nd:YVO_{4} crystal having Nd^{3+} concentration of 0.5%. The rear face of the rear mirror (RM) is anti-reflection coated at 808 nm and has high reflectivity (> 99.5%) at 1064 nm on the other side. The Nd:YVO_{4} crystal is anti-reflection coated on both of its parallel faces for wavelengths 1064 nm and 808 nm and is tilted by an angle of 2° to reduce the effect of undesired satellite cavity. Two concave mirrors *M*_{1} and *M*_{2} of radii of curvature 500 mm and 250 mm respectively are used to focus the beam at the output coupler. The tilting angle of *M*_{1} and *M*_{2} is kept as low as possible to avoid cavity astigmatism and thus providing TEM_{00} mode throughout the cavity which has been found to be necessary for stable cw ML operation. The output coupler is partially reflecting at both 1064 nm and 532 nm (R@1064 nm:92% and R@532 nm:30%). A second harmonic crystal, usually in non-phase-matched condition, is placed near the output coupler. Here, different nonlinear crystals, such as, KTP (positive biaxial crystal, cut for type-II SHG of 1064 nm: *θ* = 90°, *ϕ* = 23.5°, size: 7*×*7*×*9 mm^{3}), BBO (negative uniaxial crystal, cut for type-I SHG of 1064 nm: *θ* = 22.8°, *ϕ* = 90°, size: 6*×*6*×*12 mm^{3}), and LBO (negative biaxial crystal, cut for type-I SHG of 1064 nm: *θ* = 90°, *ϕ* =*×*11*×.*3°, size: 5*×*5*×*15 mm^{3}) are used in combination with the output coupler, to optimize the laser performance with respect to the pulse width and output power. The different arms of the Z-shaped passive cavity of length 81.7 cm are optimized to be 250 mm, 405 mm and 162 mm respectively which gives an appropriate cavity mode size at the center of the gain medium for cw modelocking, as determined by the ABCD matrix formalism. Here, we can manually adjust the separation of different cavity mirrors to get the suitable mode size for stable modelocking. To get the stable mode-locked pulses, tuning of the SHG crystal away from the phase-matched condition and then varying the separation between SHG crystal and the output coupler is necessary. The measured non-collinear SHG intensity auto-correlation trace for KTP crystal is shown in Figs. 2(a) and 2(b) shows the measured line-width of the output pulse. The modelocked pulsewidth was found to be 10.3 ps and the line-width was 0.18 nm. Therefore, the time-bandwidth product was ~0.491 which is close to the value for transform-limited *sech*^{2} pulses (~0.315) and essentially equal to the limit (~0.44) for Gaussian-shaped pulses. The variation of output power with input pump power for different NLC at 1064 nm and 532 nm is shown in Fig. 3. Here raw power indicates the cw output power without any intra-cavity NLC. The oscilloscope trace of stable modelocked pulses is shown in Fig. 4 in milisecond timescale for showing the amplitude stability. The power fluctuation is less than 1% and the output power never drops even if it runs for hours.

## 3. Theoretical analysis of CSM

The chief ingredient for constructing a stable modelocked laser is to introduce a mechanism which could dynamically select certain oscillating modes inside the cavity and amplify them in contrast to other oscillating modes. These so called “special” modes are solely determined by the geometrical configuration of the cavity. For an end pumped Kerr lens modelocked laser this mechanism is attributed to self-focusing inside gain medium (in KLM the gain medium is also a Kerr medium). To understand how self-focusing leads to mode selection and amplification we must note that the gain medium is longitudinally pumped by a Gaussian beam, suggesting a transverse variation in the gain. It may be suitable to approximate the transverse distribution by a spatial normal distribution function with variance equal to the variance of the pump mode. There will be an inherent random distribution of flux [24] across various longitudinal modes. Modes with higher flux will be more focused due to the self-focusing effect (which is proportional to the intensity of the mode) inside the gain medium. Assuming these modes to be concentric with the pump mode we can easily infer that these modes have a better overlap than the low flux modes with the population inverted region. Thus they are likely to have a higher probability for amplification by stimulated emission [25] as they have access to higher number of population inverted sites. After a finite amount of time we expect to find only a few modes out of many oscillating modes with significant amplification. These are the modes which contribute to generate a modelocked pulse. This entire process could be modeled by simply considering a transversely varying radial gain profile inside gain medium. This idea has been developed to study KLM by J. Hermann and others [26–29] and we apply similar idea here to study the CSM process.

Our experimental setup explores modelocking stability in the picosecond regime along the lines discussed above but utilizing self-focusing induced by a second order cascaded nonlinearity. The nonlinear interaction gives rise to a nonlinear phase shift which subsequently controls the overlapping of the cavity modes with the population inverted sites inside the gain medium. This nonlinear phase shift can be tuned by varying the phase-mismatch wave vector between the fundamental and second harmonic beam. This provides an additional degree of freedom to explore the modelocking regime. Our system has two non-trivial cavity components: (i) gain medium (ii) nonlinear crystal. Cumulative effect of each of the components could be analyzed temporally by writing a fundamental equation for modelocking [30].

This equation describes the temporal dynamics of a stable modelocked pulse inside the cavity. *A* is the pulse amplitude, other parameters are the dynamic gain (Ω), resonator loss (*l*), self-phase modulation (*δ*), self-amplitude modulation (*γ*) and dispersion (D). We determine resonator losses by the Findlay and Clay method [31] using output couplers with different reflectivities and determining threshold power for lasing for each mirror. The threshold electrical input is given:
$\left[{P}_{th}=\frac{1}{2\eta}\left(l-\mathrm{ln}\phantom{\rule{0.2em}{0ex}}R\right)\rho {I}_{s}\right]$; where *ρ* is the emission cross section, *R* is the reflectivity of the output coupler, *η* is the efficiency factor, *l* is the resonator loss and *I _{s}* is the flux in the gain medium. From this equation, the mirror reflectivity can be written as:
$\left[-\mathrm{ln}\phantom{\rule{0.2em}{0ex}}R=\frac{2\eta}{A{I}_{s}}{P}_{th}-l\right]$. By extrapolation of the straight line plot of

*−*ln

*R*versus

*P*, at

_{th}*P*= 0, the round trip resonator loss (

_{th}*l*) is obtained.

Ω is the modified gain considering the transverse beam overlap factor at the center of the gain medium. An approximate expression showing the dependence of Ω on the cavity and pump mode sizes could be derived by considering the propagation of the cavity mode inside the gain medium following reference [26]. The starting point is to write the wave propagation equation inside the gain medium. For a wave propagating along *z* direction with slow amplitude variation, this reads,

*r*

^{2}=

*x*

^{2}+

*y*

^{2}and

*w*is the pump beam spot size of 1/e amplitude.

_{p}The second term in the left-hand side of Eq. (2) accounts for the transverse variation of the cavity mode and the exponential term in the right hand side takes care of the radially varying gain. In the above equation we neglected the depletion of the pump beam in the z direction as the gain medium length is small. The transverse profile of the circulating cavity mode in the resonator is assumed to be Gaussian. The field amplitude A(z) is also Gaussian and is of the form [26],

where, G(z) is the amplitude amplification and*Q*(

*z*) =

*Q*′(

*z*) +

*iQ*″(

*z*) is the complex beam parameter.

*α*(

*z*) is a secondary beam parameter satisfying the requirement of self-consistency after one complete round trip in the resonator,

*A*

_{0}is the initial electric field amplitude. The lowest-order transverse mode has been considered here. Q(z) is related to the commonly used ’

*q*’ factor by

*w*is the Gaussian cavity beam spot size of 1

_{s}*/e*amplitude and

*R*is the radius of curvature. We are interested in finding the dependence of effective gain, G(z), on beam parameters after integrating over the transverse variables. To do this we substitute the ansatz (Eq. (3)) into Eq. (1) followed by multiplication of the resulting equation by

*A*

^{*}and

*r*

^{2}

*A*

^{*}. This gives us equations for zeroth and second order moments of intensity distribution in transverse plane respectively. Integrating these equations over transverse co-ordinates leads us to Eq. 5(a) and 5(b):

As we had earlier emphasized that the gain medium plays a minimal role in the dynamics of beam propagation in longitudinal direction, therefore we expect *α*(*z*) to still satisfy the real part of the gaussian beam propagation equation in vacuum given by,

Imposing this constraint in Eq. 5(a) and 5(b) followed by division of Eq. 5(b) by 2 and subtracting Eq. 5(a) from that we obtain,

The above expression for gain of the cavity mode captures the effect of the nonlinear crystal through the parameter *Q*″. The cascaded process in the nonlinear crystal gives rise to an effective nonlinear refractive index which modifies the cavity mode size at the gain medium.

We now consider the following cascaded process in the nonlinear crystal. The fundamental beam (FW) is first converted in to the second-harmonic beam (SH), i.e. *ω _{f}* +

*ω*=

_{f}*ω*and that SH is back converted into FW again by difference frequency generation i.e.

_{sh}*ω*−

_{sh}*ω*=

_{f}*ω*. These two processes occur simultaneously to impose a nonlinear phase-shift on the FW. The net phase-shift $\left(\mathrm{\Delta}{\mathrm{\Phi}}_{NL}^{tot}\right)$ can increase or decrease the beam spot size inside cavity depending on the resonator configuration. The nonlinear interaction in the crystal could be modeled by the following two coupled wave equations for FW and SH under slowly varying wave approximation,

_{f}*A*

_{1}and

*A*

_{2}are the complex amplitudes for FW and SH respectively. Δ

*k*=

*k*2

_{sh}−*k*is the phase-mismatch between the FW and SH wave.

_{f}The above coupled equations are numerically integrated to find the nonlinear phase-shift acquired by the forward and backward propagating fundamental beam. As the diffraction length is larger than the crystal length (Rayleigh range is ~17 cm), we can neglect the transverse variation of FW and SH beam in NLC (*L _{g}*~1 cm). Here, the reflectivity of the output coupler at FW and SH (

*r*

_{1}and

*r*

_{2}, respectively) has been considered in the numerical analysis. The total nonlinear phase shift incurred in the pulse as it goes through the NLC, could be thought as giving rise to a nonlinear intensity dependent refractive index $\left({n}_{2}^{eff}\right)$ which modifies the mode size of the cavity beam. This case is very similar to the optical Kerr effect where one writes

*n*=

*n*

_{0}+

*n*

_{2}

*I*. In this case, by analogy, it can be said that the total nonlinear phase-shift is $\mathrm{\Delta}{\mathrm{\Phi}}_{NL}^{tot}=\frac{2\pi L}{\lambda}{n}_{2}^{eff}I$. From here, ${n}_{2}^{eff}$ can be easily calculated when $\mathrm{\Delta}{\mathrm{\Phi}}_{NL}^{tot}$ is known. The variation of ${n}_{2}^{eff}$ with Δ

*kL*is shown in Fig. 5.

The higher the intensity in a particular longitudinal mode, the tighter the mode is focused. As a result, it experiences higher gain since the pump mode has a Gaussian profile in the gain medium. We can consistently treat the nonlinear medium as an effective Kerr medium with refractive index ${n}_{2}^{eff}$ which could be represented as a matrix [32]:

where,*d*is the effective optical length and $\zeta ={\left[1+\frac{1}{4}{\left(\frac{2\pi {w}_{c}^{2}}{\lambda {d}_{e}}-\frac{\lambda {d}_{e}}{2\pi {w}_{0}^{2}}\right)}^{2}\right]}^{-1}\frac{P}{{P}_{c}}$.

_{e}*P*is the critical power for self-focussing and can be determined by the expression, ${P}_{c}=\frac{c{\epsilon}_{0}{\lambda}^{2}}{2\pi {n}_{2}^{eff}}$.

_{c}*w*is the spot-size at the center of the NLC and

_{c}*w*

_{0}is the beam waist of passive cavity at output coupler. Using the ABCD matrix formalism, we can compute

*Q*″ at the center of gain medium as a function of input pump power. The variation of cavity mode spot size at the center of gain medium (

*w*) with input pump power (

_{s}*P*) is shown in Fig. 6. The self-phase modulation parameter (

_{in}*δ*) can be written as, $\delta =k{\epsilon}_{0}{n}_{2}^{eff}$

*L*[33]. When the modelocking is stable, we can derive heuristically Eq. (1) by considering small effect of the gain medium and the nonlinear crystal on the modelocked pulse similar to [30]. The transverse amplitude of the modelocked pulse after n

*and (n+1)*

^{th}*round-trip are related as,*

^{th}In a steady state, for obtaining stable modelocked pulses, it must be ensured that *A _{n}*

_{+1}(

*t*) =

*CA*(

_{n}*t*).

*G*is the sum of the gain accumulated by the beam in forward and backward direction.

_{tot}*C*is an arbitrary complex number with |

*C*| = 1. The effect of gain guiding is to make the gain function a power dependent function as cavity mode size varies with pump as well as intra-cavity power. Taylor expanding

*G*about the stable modelocked intra-cavity power

_{tot}*P*

_{0}, we find,

*γ*is usually termed as soft-aperture Kerr lens modelocking parameter, which is known for our case from the derivative of radially varying gain. This parameter contains the information on the change of gain with power (or flux). Using Eq. (7), we can express average gain at a transverse profile as,

*L*is the length of the gain medium.

_{g}*g*is the saturated gain and is defined as, where,

*I*is the intra-cavity circulating intensity and

_{cir}*I*is the saturation intensity. Here, circulating intensity is defined as

_{sat}*I*= |

_{cir}*A*|

^{2}.

*I*can be calculated from the measured values ${I}_{cir}=\frac{{P}_{out}^{cw}}{{A}_{oc}{T}_{oc}}$ and

_{cir}*I*=

_{sat}*ħ*ω

*/*

_{f}*σ*. Here

_{L}*A*and

_{oc}*T*are the beam spot area and transmission coefficient of the output coupler respectively. The Lorentzian

_{oc}*ω*

_{0}and

*ω*is the gain bandwidth at 1064 nm of Nd:YVO

_{L}_{4}medium.

The Lorentzian line shape function can be promoted to a temporal operator in Eq. 13(b) by replacing the *n ^{th}* power of

*iω*by

*d*and one can rewrite Eq. (10) into Eq. (1) with $\mathrm{\Omega}=\tilde{G}\left({P}_{0}\right)$, where, $\tilde{G}\left({P}_{0}\right)$ defined as,

^{n}/dt^{n}We have assumed gain per pass and deviation from the central frequency *ω*_{0} to be small. The small signal gain (*g*_{0}) is given by,

*σ*is the stimulated emission cross section,

_{L}*τ*is the fluorescence lifetime,

_{L}*η*is the quantum efficiency,

_{Q}*η*is the beam overlap efficiency,

_{B}*η*is the Stokes factor,

_{S}*η*is the system efficiency,

_{sys}*P*is the input pump power,

_{in}*h*is the Planck’s constant,

*ν*is the pump photon frequency,

_{p}*w*and ${w}_{{s}_{in}}$ are the pump beam spot size and initial cavity beam spot size respectively at the gain medium.

_{p}## 4. Results and discussion

Equation (1) is a simplified form of the complex Ginzburg Landau Equation (CGLE) which has been studied extensively in the context of stable spatiotemporal pattern formation in various physical systems. Investigating the stability of solutions obtained from CGLE has been a subject of intense research [34–37]. In this section, we perform a self consistency check for the existence of pulse like solutions and find the minimal conditions on the pulse parameters that must be satisfied. We vary the tuning parameters (phase mismatch and input intensity) and obtain the corresponding variation of the pulse parameters still satisfying the constraint equations. We compare this variation for the two cases i) including radial variation of gain ii) excluding such a variation in gain medium. A qualitative change in the behavior is observed and accounted for in the discussion below. We consider the solution to Eq. (1) to be of the form *A* = *A*_{0}[*sech*(*bt*)]^{(1+}^{iβ}^{)}. The pulse is characterized by pulse width *τ _{p}* = 1

*/b*and amplitude

*A*

_{0}. Substituting the ansatz in Eq. (1), we obtain an expression in

*sech*

^{0}(

*bt*) and

*sech*

^{2}(

*bt*). Taking both the real and imaginary parts of the coefficients of

*sech*

^{0}(

*bt*) and

*sech*

^{2}(

*bt*) to be equal to zero, we get four equations as follows: Real part of

*sech*

^{0}(

*bt*)

Imaginary part of *sech*^{0}(*bt*)

Real part of *sech*^{2}(*bt*)

Imaginary part of *sech*^{2}(*bt*)

Our prime objective is to deduce the chirp parameter *β* and pulsewidth *τ _{p}* as the self-amplitude modulation parameter

*γ*, dispersion D and self-phase modulation parameter

*δ*is known for our case. The chirp parameter,

*β*can be determined from Eq. 16(c) and 16(d) by considering normalized pulsewidth ( ${\tau}_{n}=\frac{E{\omega}_{L}^{2}}{2\mathrm{\Omega}}{\tau}_{p}$, E is the pulse energy) and normalized dispersion $\left({D}_{n}=\frac{{\omega}_{L}^{2}}{2\mathrm{\Omega}}D\right)$. Here we may rewrite Eq. 16(c) and 16(d) in a single equation [38]:

We can get a quadratic equation of *β* from Eq. (17) as:

*β*has been taken to determine the pulsewidth. The variation of

*β*with respect to Δ

*kL*and intracavity intensity is plotted in Fig. 7(a) under radially varying gain consideration. Using the value of

*β*in Eq. 16(a), we can find the pulsewidth

*τ*,

_{p}From *β* and *τ _{n}*, normalized bandwidth can be writen as:

To describe the stability of CSM laser, we can derive the stability conditions directly from the master modelocking equation. This equation is reminiscent of the ’particle in a well’ problem which relates forces, potentials and amplitudes:
$F=-\frac{dU}{dA}=m\frac{{d}^{2}A}{d{t}^{2}}$, *U* is the energy and *m* is the mass. In our analysis,
$m=\frac{2\mathrm{\Omega}}{{\omega}_{L}^{2}}+iD$ and the potential well is:
$\left(2\mathrm{\Omega}-l\right)\frac{A}{2}+\left(\gamma -i\delta \right)\frac{{A}^{3}}{3}$. Now, we know that when the laser is stable,
$\frac{{\partial}^{2}U}{\partial {A}^{2}}$ should be greater than 0 at the point where
$\frac{\partial U}{\partial A}=0$. Using this condition we obtain the stability condition to be 2Ω *< l* (where maximum value of Ω is 0.0022) which is satisfied in our case. An additional stability criteria can be found using the constraint relation of Eq. 16(a). We define a stability parameter S which depends on the gain overlap factor, chirp parameter and dispersion [38],

The laser loss saturation characteristic in CSM laser is equivalent to a fast saturable absorber. The parameters of CSM that correspond directly to those of a fast saturable absorber are readily derived. The nonlinear loss (*l _{nl}*) can be realized as [12],

*l*=

_{nl}*l*−

*γ*|

*A*|

^{2}. We have plotted intra- cavity intensity versus

*l*at different value of Δ

_{nl}*kL*in Fig. 8. It is observed interestingly that both saturation (at low power) and inverse saturation (at high power) signatures appear in the

*l*curve. From Fig. 8, it is clear that inverse saturation take place in the gain medium when in-tracavity intensity is higher than ~ 10

_{nl}*MW/m*

^{2}. The nonlinear depth of modulation or change of absorption with power (Δ

*R*) for our case is ~ 4%. The lower limit of acceptable value of ΔR for mode-locking was observed to be 3% corresponding to phase-mismatch of −

*π*, beyond which mode-locking stability was not acceptable. Required intra-cavity pulse energy for stable modelocking depends on Δ

*R*and saturation power of the absorber

*P*( $=\frac{{I}_{s}}{A}$,

_{s}*I*, saturation intensity and

_{s}*A*, area of the beam). The inverse saturation effectively reduces these parameters by orders of magnitude allowing stable mode-locking at a very low pump power. Inverse saturation may be identified as the negative feedback responsible for stability in mode-locking.

There are many known methods of mode-locking other than cascaded Second-Order Modelocking. For example, modelocking based on saturable absorption is widely used in the laser community. Recently two-dimensional materials like graphene [39], topological insulator [40] or *MoS*_{2} [41], have been found to be effective saturable absorbers for stable mode-locking operation. Then, what are the potential advantages of cascaded Second-Order Modelocking in comparison with these materials? The oscillating critical intra-cavity pulse energy (*E _{c}*) required for stable cw mode-locking with fast saturable absorber derived by T. R. Schibli et al [42] is given by
$2\sqrt{\mathrm{\Delta}R{P}_{s}{\tau}_{p}{E}_{L}}$. Here,

*E*is the saturation energy of the gain medium and

_{L}*τ*is the pulse-width. The mode-locking will be very stable if the laser is operated having intra-cavity pulse energy more than

_{p}*E*. Therefore, the question arise which saturable absorber makes

_{c}*E*minimum. The saturable absorber for which Δ

_{c}*R*and

*I*are minimum, will have a definite advantage. Having a low value for critical pulse energy is always advantageous in any modelocking scheme. We obtained the least modulation in ΔR for the case with phase-matched condition. This is due to the fact that there is no significant phase shift in the fundamental beam and hence no feedback to the gain aperture. The other limit, having a large phase-mismatch is also not favourable since now the interaction between the second harmonic and the fundamental beam would be low. Thus, there exists an optimum value for phase mismatch which results in the highest modulation of the nonlinear loss but not necessarily will it correspond to the least value for the critical intra-cavity pulse energy.

_{s}We find that the values of Δ*R* and *I _{s}* for the cascaded second order process (Δ

*R*=0.039 and

*I*=8 MW/

_{s}*m*

^{2}) compare favorably with values for Graphene (Δ

*R*=0.005 and

*I*=103.2 GW/

_{s}*m*

^{2}) and

*MoS*

_{2}(Δ

*R*=0.166 and

*I*=1 GW/

_{s}*m*

^{2}). It is clearly evident that cascaded second order processes not only provide the minimum

*E*, but also can be varied with the phase-mismatch (Δ

_{c}*kL*) which is the potential advantage over any other saturable absorber.

Substituting the values for *l* = 0.63, *ω _{L}* = 0.257 THz,

*D*= 1595 × 10

^{−}^{27}

*s*

^{2}

*/m*,

*β*and Ω in Eq. (19), we estimate a pulse width of ~9.5 ps which is very close to the observed experimental pulse width of 10.3 ps shown in from Fig. 2. The variation of

*β*,

*τ*,

_{p}*w*and

_{n}*S*with phase-mismatch and intra-cavity intensity corresponding to radially varying gain and normal gain is shown in Figs. 7 and 9, respectively. For our case, GVD is positive. Therefore only a large negative value of

*β*is possible [38]. The bandwidth limited pulse duration for Nd:YVO

_{4}gain medium is ~1.7 ps. But we obtain ~ 9.5 ps numerically (experimentally 10.3 ps). This result can be justified from Figs. 7(a) and 9(a) where

*β*has a high negative value. The nonlinear phase-shift is necessary for spatial gain filtering and produces saturable absorption (

*γ*~ 3

*×*10

^{−}^{6}

*W*

^{−}^{1}) which helps to generate the stable pulses, similar to Kerr lens modelocking. From the simulation curves of Fig. 7, it can be observed that the

*β*value is high where

*τ*value is low but bandwidth is high causing lower stability. Now, if we compare the pulse parameters from Figs. 7 and 9, it can be observed that for radially varying gain, the

_{p}*S*and

*w*value is more for a particular intracavity intensity than normal gain $({\mathrm{\Omega}}_{1}=\frac{{g}_{0}}{1+\frac{2{I}_{cir}}{{I}_{sat}}}{L}_{g})$. If the utilized gain bandwidth is high, then pulses are shortened, a well-known phenomenon reflected in our results. The gain rises as the pulse shortens upon each round-trip ensuring a better match with the distribution of population inversion and higher probability for stimulated emission of this mode. The gain increases to the point where the density of population inverted states matches the density of incoming photon flux in the region. Beyond this there is gain saturation or loss depending on the relative area of overlap between the cavity mode photons and the population inversion region.

_{Ln}## 5. Conclusion

Cascaded second order modelocking is experimentally as well as theoretically reported in this paper. Formalism of cascaded second order processes utilizes complex beam parameters of the resonator cavity to account the measured pulse-width and stability. It is found that Kerr lens modelocking and cascaded second order modelocking are similar in terms of self-gain guiding process. Theoretical analysis clearly identifies the saturable absorption nature of CSM and proves the presence of forward saturation and inverse saturation. Due to inverse saturation loss, the modelocked pulse train becomes stabilized. This type of modelocking is better than recently developed two-dimensional materials in high power solid-state lasers in terms of critical intra-cavity pulse energy, low saturation intensity, small modulation depth and inverse saturable absorption. This type of stable picosecond laser is very useful for probing the dynamics of fast chemical and biological processes.

## Acknowledgments

Authors acknowledge ’SGDRI’ project for sharing some component used in this work.

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