J. Eur. Opt. Society-Rapid Publ. 22, 47( 2026) 467
of instantaneous polarization is discussed in [ 19 ]. Rather than using Stokes parameters, as in the present study, there the analysis is done in terms of polarization ellipses. As explained in [ 17 ], these two approaches are equivalent. An experimental procedure to observe instantaneous polarization is described in [ 18 ].
5 Numerical results
We first analyze the case for which the path length through the dispersive medium L 10 lm. The output pulse EðÞ¼E t x ðÞ^x t þ E y ðÞ^y t with, from equations( 9) and( 15),
E x ðÞ¼A t 0 e �ix0t e ik0L e �t2 = T 2;
E y ðÞ¼ t A 0 p X ffiffiffiffi
A ð28Þ e �ix0t e ikL e �B2 = ð4A Þ: ð29Þ
The expression for E x( t) is supplemented with a phase factor exp( ik 0 L), with k 0 the free-space wave number, to ensure that the path lengths in both arms of the interferometer are identical; and with the z-dependence omitted for brevity. The instantaneous polarization of the output pulse E( t) changes during the duration of the pulse chiefly because of the delay of E y( t). As seen from Figure 3, the dispersive effect of the medium does not lead to any appreciable pulse stretching in this case. Because of the delay experienced by E y( t), the output pulse is initially x-polarized( s 1 = 1) and ends up being y-polarized( s 1 = �1). This is illustrated in Figure 4 where the evolution of the instantaneous Stokes parameters as well as that of the intensity are plotted for the case L = 4lm.
During the pulse the instantaneous Poincaré vector traces out a continuous path on the Poincaré sphere between the initial state I =( 1,0,0) and the final state F =( �1,0,0). The orientation of the path is determined by the thickness L of the dispersive medium. This is illustrated in Figure 5 for four selected values of the length L of the dispersive medium. The cyan curve( L = 5lm) passes over the spherical quadrant( s 2 > 0, s 3 > 0); thebluecurve( L = 10 lm) over the quadrant( s 2 < 0, s 3 > 0); theredcurve( L = 15 lm) over the quadrant( s 2 < 0, s 3 < 0); and the purple curve( L = 17lm) over the quadrant( s 2 > 0, s 3 < 0). Each trajectory consists of 40 steps, indicated by dots with a color gradient transitioning from blue( state I) to yellow( state F) and with a time step of 2 fs between dots( see color bar). The depicted time range is from t = �20 fs to t = 60 fs. Notice that the dot spacing is highly irregular. As illustrated in Visualization 1), as the length L is smoothly varied, so is the orientation of the path connecting states I and F, eventually covering the entire Poincaré sphere. In particular, L can be chosen such that at some moment during the pulse s 3 attains the value ± 1.
When the thickness L of the medium exceeds 10 lm, the two fields essentially no longer overlap, as is seen from Figure 3. In that case, the output field consists of two separate pulses, the first one x-polarized, and the second one y-polarized. The transition to this case is shown in Figure 6, where
Fig
. 4. Evolution of the three instantaneous Stokes parameters of the output pulse for a dispersive BK7 medium with length L ¼ 4 lm. The input pulse duration T ¼ 20 fs. The intensity( in [ a. u.]) of the output pulse is represented by the dashed curve.
Fig. 5. Trajectories on the Poincaré sphere with axes ðs 1; s 2; s 3 Þ connecting states I( initial) and F( final), for different lengths L of the dispersive BK7 medium. Time ranges from t ¼�20 fs to t ¼ 60 fs( Visualization 1).
Fig. 6. The instantaneous intensity of the output pulse for four different values of the thickness of the dispersive medium.