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J. Eur. Opt. Society-Rapid Publ. 22, 47( 2026)
Fig. 7. Left: the E x ðÞand t E y ðÞpulses t when the former undergoes a time delay Dt ¼ k 0 L 32568:64 fs. Right: trajectory on the Poincaré sphere with axes ðs 1; s 2; s 3 Þ, for a length L ¼ 6:4 mm of the BK7 medium. Time ranges from t ¼ 32500 fs to t ¼ 32780 fs with 2 fs time steps( Visualization 2).
the instantaneous intensity of the output pulse is plotted for different values of L. We note that the entire Poincaré sphere can be covered for values of L for which the field in the two arms still produces a single output pulse( see Visualization 1).
We next study the case of the same T = 20fspulse traversing a much thicker slab of the dispersive medium with L = 6.4 mm. The E y( t) pulse is now significantly dispersed and has stretched to about two times its original width. The E x( t) signal must be delayed by a time Dt k 0 L in order to overlap with the E y( t) signal. This can be achieved with the variable free-space delay line indicated in Figure 2. An example is shown in Figure 7. Herethe delay is such that the two pulse peaks coincide( left panel). The result is a time-symmetric evolution of the instantaneous state of polarization. It starts out as y-polarized( state F), moves up over the upper half of the Poincaré sphere and then retraces its path back to end in state F again. In Visualization 2 both pulses and the resulting polarization trajectory of the output pulse are shown in their dependence on the time delay Dt. IfDt = 32490 fs, the E x signal precedes E y and the path starts at I =( 1,0,0) and ends at F =( �1,0,0). When Dt is increased the path gradually twists and at around Dt = 32515 fs detaches from I. The initial state moves upward and, near Dt = 32560 fs, connects with F, forming a closed loop. Next, the path gets distorted, briefly closes( as in Fig. 7), re-opens, and gets longer and longer. Around Dt = 32616 fs the end of the path is no longer at F and begins to spiral back to state I, illustrating the richness of polarization evolutions that can be achieved with this method.
Finally, it is worth noting that these non-trivial polarization changes take place when the fields in the two arms overlap and the pulse intensity reaches its highest value. This is evidenced from Figure 8 where the instantaneous pulse intensity is shown for three different values of the time delay. The blue curve is for Dt = 32568 fs, which corresponds to the left-hand panel of Figure 7 in which the peaks
Fig. 8. The instantaneous intensity of the output pulse for three selected values of the time delay Dt of the E x field.
of the pulses coincide. This gives rise to an output pulse with a symmetric profile. When Dt is decreased by 20 fs( red curve), or increased( green curve), the intensity profile becomes somewhat asymmetric and has a lower peak value.
6 Conclusions
We have analyzed theoretically and numerically an interferometric setup to control the instantaneous state of polarization of optical pulses. Our description is suitable for media for which third-order and higher dispersion may be neglected. Implementation-dependent issues such as phase stability and the possible use of a wedge-shaped medium to continuously vary the path length L, are not addressed.
In the proposed setup a linearly polarized pulse is split into two orthogonal and fully coherent beams. This can be achieved by a Wollaston prism or another type of polarizing beam splitter. One beam travels through free space, the other is made to propagate a distance L through a linear, non-absorbing, deterministic dispersive glass medium. The two pulses are then recombined to produce an output