1. Introduction

The mode-locking of a laser to obtain short optical pulses has been extensively studied both experimentally^1,2 and theoretically^3,4. Theoretical (and numerical) modelling of the evolution of the optical field in laser cavity can be a formidable task, necessitating the description of both transverse and longitudinal field variations as well as dynamical evolution, perhaps over a huge range of time scales. Previous analysis has tended to concentrate on either spatial or temporal analysis. The former generally assumes that the response time of the nonlinear medium is instantaneous so that the steady-state analysis can be applied to the pulsed regime^5–8, while the latter does not include spatial effects explicitly, but includes an effective saturable absorption to take the self-focusing due to the nonlinear medium into account^9–11. One example of this is the well-known master equation approach of Haus et al.¹⁰ in which a partial differential equation is used to describe evolution on fast and slow time scales, considered as independent variables.

We have recently derived a spatio-temporal master equation (ME) which combines the spatial and temporal approaches above by using the Haus approach of a master equation, but including spatial effects via a transverse propagation operator which is explicitly dependent on the ABCD elements of the cavity¹². Our ME can also handle nonlinearity, although our present requirement of a single nonlinear element limits our model to a ring cavity with one nonlinear element, or a Fabry-Perot cavity with the nonlinear element at an end mirror. In short-pulse (mode-locked) lasers, our partial differential equation (pde) approach treats pulse evolution (dispersion) and transverse effects (diffraction) in mathematically and computationally very similar ways.

In this paper we apply our ME to the problem of mode-locking a laser by an internal phase or amplitude modulation. We then extend the analysis to include nonlinear effects via a Kerr lens and demonstrate the effect this has on the pulse shortening process.

2. Master Equation

Our spatio-temporal master equation consists of spatial and temporal operators, $\widehat{\pi }$ _x and $\widehat{\pi }$ _t respectively, and may also include gain and nonlinear terms:

The spatial operator takes the form¹²:

ABCD are elements of the usual cavity matrix at the chosen reference plane, cos ψ = S ≡ (A + D)/2 is independent of the cavity reference plane, T_R is the round-trip time, based on the group velocity.

We have also included a complex gain or loss g, which we assume to be uniform, linear, and unsaturated, and a nonlinear function N(E) which depends only on the local field E at the reference plane. For a Kerr lens N(E) = ik|E|² E. The reference plane must be set at the nonlinear element in this ME.

This ME has the correct transverse mode and related properties¹², and automatically accounts for the sensitivity of the nonlinear gain to the location of the Kerr medium and the cavity configuration in the low-power limit. Comparison with numerical results from the exact Huygen’s integral method¹³ (in x only) has shown that our results are also remarkably accurate even when a substantial nonlinearity is included¹⁴.

For simplicity (and ease of computing) we consider only one transverse direction here, but extension to two transverse dimensions is simple. Astigmatism can be included in the model if required.

3. Intracavity modulation

We assume the usual form for the temporal propagator¹⁰:

where β is a complex parameter whose real and imaginary parts describe group velocity dispersion and gain bandwidth respectively. Re (β) > 0, as we have chosen in these simulations, corresponds to the case of anomalous dispersion. Other fast-time effects (such as intracavity modulation) can also be incorporated. In this paper we are interested in the effect of an intracavity amplitude or phase modulator. It is well-known³ that pulses form at an extremum of the modulation cycle, and are usually much shorter than the modulation period. We can therefore set t = 0 at an extremum, and replace the modulation cycle by a term in the ME which is quadratic in t, i.e. by αt ², where α is an arbitrary complex parameter, real in the case of amplitude modulation and imaginary for phase modulation.

Our temporal ME operator then becomes

4. Properties of the linear Master Equation

The linear ME for a cavity with an internal modulator is therefore

We know¹² that this has gaussian solutions in x of the form

where R is the phase front curvature of the field and w is its width. We expect a similar solution in t, and so we consider a field of the form

where q_t describes the temporal properties of the field in a similar manner to q_x .

Substituting this into the ME and equating terms in equivalent powers of x and t we obtain the following three equations:

For a matched beam solution, ie. a mode of the cavity, the only change in the field after a cavity round trip is in its amplitude. In other words dq_x /dT = dq_t /dT = 0.

Using this condition in Eq. (7) we find that for the cavity mode

and so the gain/loss in the cavity is

For a confined beam we require that Im(Q_x ) > 0 and Im(Q_t ) > 0. This determines the choice of the signs in (8) and (9).

To examine the stability of the cavity we set Q_x = Q_xm + Δ_x; Q_t = Q_tm + Δ_t, and neglect ${\mathrm{\Delta }}_x^2$, ${\mathrm{\Delta }}_t^2$ in (7). Then

Again the sign is determined by the choice of sign for the confined mode. In order for the cavity to be stable we require that Re(∓2iψ) < 0 and Re(∓4√-iαβ) < 0.

Analysis of (10) shows that phase modulation (arg(α) = ±π/2) is stable for either choice of sign, but amplitude modulation (arg(α) = 0 or π) is only stable if arg(α) = π. Thus our ME reproduces the known properties of linear field evolution in optical cavities containing apertures and/or modulators.

5. Numerical analysis

We simulate our ME numerically using the split-step model¹⁵, with periodic boundary conditions. At first sight the form of the ME seems to make this method unsuitable, since the “(A - D)” term belongs in both real space and k-space.

To get rid of this cross term we transform the physical field E(x,t,T) as follows¹⁴:

where

Since the transformation is a simple multiplication by a function U(x), and need only be done at the start and end of the simulation, the computational penalty is very slight.

The master equation used for the numerical integration is therefore that obeyed by Ẽ(x, t, T)

6. Active mode-locking

In our numerical analysis we once more consider a ring cavity containing a gaussian aperture and two thin lenses¹⁴. We also include an internal modulator, and a gain bandwidth (to limit pulse shortening). At this stage we do not include a Kerr medium, and so we set κ = 0 in (13), which is then linear. The total cavity length is 2m and the strength of the thin lenses is chosen such that the beam focuses close to the slit and to the Kerr lens. The linear power loss in the cavity is ≈ 8%. We include a simple, unsaturable linear gain, which we assume can be large enough to overcome the linear and/or nonlinear losses. The initial field was gaussian in x and flat in t, with an amplitude corresponding to a nonlinear phase shift of 0.1 radians when the Kerr medium is included. We use a 128 × 128 grid, and scale x and t to the width of the gaussian mode.

Only if the gain and loss are in exact balance does (13) stabilise (as determined from the “energy” ∫ ∫ E dx dt), but by rescaling we can investigate the evolution of the beam profile. We find the stable field profiles correspond to the mode profiles calculated from (6) with 1/q_x = Q_xm and 1/q_t = Q_tm . Note that although we have considered anomalous dispersion here pulse formation is also seen with normal dispersion. However, in order to get pulse compression it would be necessary to use a self-defocusing lens and to change the location of the Kerr medium.

Both amplitude and phase modulator produce broadly similar results, the main difference being the time taken for the field to stabilise. We find that the phase modulator takes slightly longer to stabilise the field for the same value of |α|. As expected³, we find that we can get stable solutions with arg(α) = ±π/2 or π, but not with arg(α) = 0. As figure 1 shows, the phase modulated pulses have different temporal widths depending on the sign of arg(α). Note that we use arbitrary units throughout, since our main interest is the relative width, and change in width, in each case. All pulse parameters agree with the analytic predictions (8), (9).

 

Figure 1. Temporal profiles. |α| = 0.09, β = (6.0 - 9.0i) × 10^-3, κ = 0.0, zero net gain. Red, green and blue lines are for arg(α) = π (amplitude modulation), π/2 - π/2 (phase modulation) respectively.

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7. Active mode-locking plus a Kerr medium

Many of the problems associated with active mode-locking can be overcome if pulse compression is provided by an internal nonlinear element. This technique generally leads to significantly shorter pulses than active mode-locking. Therefore we add a nonlinear (Kerr) medium to the cavity model. In terms of the (transformed) ME we do this by adding the term iκ|U(x)^-1 Ẽ|² E (i.e. κ ≠ 0 in (13)). Note that although the function U(x) appears explicitly in the nonlinear term there is no problem with the split-step method as nonlinear terms are handled in real space. We expect that the effect of the Kerr medium will result in some soliton-like pulse shaping, and so the stable field will be some kind of combination of a gaussian and a sech.

We start from a cw field and then turn on the modulator. With the nonlinearity included we find that stable pulses can be produced with both the amplitude and the phase modulator, even when the net gain is non-zero, and even without gain saturation. In comparison with a phase modulator, an amplitude modulator stabilises the field more quickly, as can be seen from figure 2, and results in slightly narrower pulse widths.

In contrast with the linear results, we find that phase modulation results in stable pulses only when arg(α) = -π/2. A similar result was seen in an analysis of soliton dynamics in the presence of phase modulators¹⁶, which suggests that soliton shaping has become the dominant effect in our system. Indeed, when we examine the stable solutions we find that the addition of the Kerr lens has produced a compression in both space and time. The temporal profiles are shown in figures 3 (amplitude modulation) and 4 (phase modulation). Clearly these are now sech-like, becoming more so as the gain is increased. A similar result is also found for the spatial profiles.

 

Figure 2. Dynamics of field in amplitude modulated and phase modulated cavities with a Kerr medium. |α| = 0.09, β = (6.0 - 9.0i) × 10^-3, κ = 0.1. Red and blue lines are for arg(α) = π(AM) and arg(α) = -π/2 (PM) respectively, with no net gain, green and magenta lines are AM and PM respectively, with a net gain of 0.025.

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Figure 3. Temporal profiles. Comparison of linear and nonlinear results using amplitude modulation. Profiles have been scaled to equal heights. Red and blue lines are linear and nonlinear results, respectively, with zero gain, green line is nonlinear result with a net gain of 0.025. A similar result is also found for the spatial profiles.

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Figure 4. Temporal profiles. Comparison of linear and nonlinear results using phase modulation. Profiles have been scaled to equal heights. Red and blue lines are linear and nonlinear results, respectively, with zero gain, green line is nonlinear result with a net gain of 0.025. A similar result is also found for the spatial profiles.

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8. Conclusion

We have shown that the spatio-temporal master equation describing a cavity containing an internal amplitude or phase modulator produces gaussian pulses in space and time, in agreement with the Kuizenga and Siegman theory of active mode-locking³. Stable pulses are produced for α = ±i|α| and α = -|α|.

When a nonlinear (Kerr) medium is added to the cavity the field is compressed in both space and time and the profiles become more soliton-like. In this case stable pulses are only produced for α = -i|α| and α = -|α|.

By analogy with the spatial effects of a gaussian aperture, we can expect that the pulse shapes and stability in the KAML will depend on the location of the modulator, and thus that (4) is only a simple, special case of a more general class of operators π̂_t.