© The Author(s), published by EDP Sciences, 2026

1 Introduction

Optical pulses are used in a wide range of applications such as control of chemical reactions [1], observation of electron dynamics [2], time-resolved spectroscopy [3], nonlinear microscopy [4], the generation of entangled photons [5], and quantum key distribution [6], to name but a few. The state of polarization of an optical field is a fundamental property that determines how it scatters [7] and propagates through a material medium [8]. Pulse shaping is concerned with spatio-temporal control of intensity and frequency. In addition, the ability to control the instantaneous polarization state provides another degree of freedom that can be exploited in all of the above-mentioned applications. This was examined by Gerber et al. by using an LCD inside a zero-dispersion compressor [9–11]. Here we present a theoretical analysis of an alternative method, based on optical dispersion, to manipulate the evolution of the instantaneous polarization state over the duration of the pulse.

We analyze an interferometric setup in which a polarizing beam splitter, such as a Wollaston prism, creates two orthogonally-polarized and fully correlated pulsed fields with equal amplitude, E_x(t) and E_y(t). Whereas one field, E_x(t) say, is left unchanged, E_y(t) is modified by letting it travel a distance L through a deterministic, linear, and non-absorbing dispersive medium. The two fields are then recombined to form an output pulse whose polarization evolution is found to be strongly dependent on the thickness L of the dispersive medium. As an example we study a glass-like medium. We discuss two regimes. When the thickness L ≈ 10λ, the polarization evolution of the output pulse is mainly governed by the time delay between the two pulses. When L ≈ 1000λ, the polarization evolution is determined by both the time delay and the stretching of E_y (t). We find that in both regimes it is possible for the instantaneous Poincaré vector to reach all points of the Poincaré sphere.

2 Plane-wave pulses

Consider a linearly-polarized pulse which is represented by an analytic signal E (z, t) ([12], Sect. 3.1). Such a pulse can be expressed in terms of its spectral components by using the Fourier transform$$E(z,t)={\int }_0^{\infty }‍\tilde{E}(z,\omega ){\mathrm{e}}^{-\mathrm{i}\mathit{\omega t}}\mathit{d\omega },$$(1)with inverse relation$$\tilde{E}(z,\omega )=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }‍E(z,t){\mathrm{e}}^{\mathrm{i}\mathit{\omega t}}\mathit{dt}.$$(2)Since E (z, t) is an analytic signal with Ẽ (z, ω) = 0 if ω < 0, the lower limit of integration in equation (1) is zero rather than −∞. We assume that in a plane z = 0 the spectrum has a Gaussian shape around a center frequency ω₀, i.e.,$$\tilde{E}(0,\omega )={\tilde{E}}_0{\mathrm{e}}^{-{\left(\omega -{\omega }_0\right)}^2/{\mathrm{\Omega }}^2},$$(3)where Ẽ₀ is a constant and Ω denotes the effective width. The spectrum must be zero for negative frequencies, and also only contain optical frequencies. Together this leads to the constraint$$\mathrm{\Omega }\ll {\omega }_0.$$(4)Let us next write the pulse as the product of an envelope function A(t) and a carrier signal of frequency ω₀:$$E(0,t)=A\left(t\right){\mathrm{e}}^{-\mathrm{i}{\omega }_{0}t}.$$(5)

We note that writing the pulse as an enveloped carrier signal is meaningful even at the single-cycle level [13]. To determine the envelope function we substitute from equations (3) and (5) into equation (1) to obtain$$A\left(t\right)={\tilde{E}}_0{\int }_{-\infty }^{\infty }‍{\mathrm{e}}^{-{\left(\omega -{\omega }_0\right)}^2/{\mathrm{\Omega }}^2}{\mathrm{e}}^{-\mathrm{i}\left(\omega -{\omega }_0\right)t}\mathit{d\omega }$$(6) $$={\tilde{E}}_0{\int }_{-\infty }^{\infty }‍{\mathrm{e}}^{-\omega {\prime }^2/{\mathrm{\Omega }}^2}{\mathrm{e}}^{-\mathrm{i}\omega \prime t}{\mathit{d\omega }}^{\prime },$$(7)where ω′ = ω − ω₀, and with the integration in equation (1) formally extended to minus infinity. This is a standard integral with solution$$A\left(t\right)=A_0{\mathrm{e}}^{-t^2/T^2},$$(8)where $A_0={\tilde{E}}_0\mathrm{\Omega }\sqrt{\pi }$, and with the pulse duration given by T = 2/Ω. Using this expression for the envelope function in equation (5) gives us$$E(0,t)=A_0{\mathrm{e}}^{-t^2/T^2}{\mathrm{e}}^{-\mathrm{i}{\omega }_{0}t}.$$(9)

The real-valued temporal pulse equals 2 Re{E(0, t)}, where Re denotes the real part [12].

In the optical regime the pulse duration can be on the order of femtoseconds. Examples of such temporal pulses are shown in Figure 1 for a center frequency ω₀ = 2.36 × 10¹⁵ rad/s (corresponding to a free-space wavelength λ₀ = 800 nm) and A₀ = 1/2, for two different pulse durations. On the left is shown a few-cycle pulse and on the right a many-cycle pulse. Throughout this manuscript we will consider the latter.

Fig. 1 Two normalized temporal pulses with center frequency ω0 = 2.36 × 1015 rad/s (corresponding to a free-space wavelength λ0 = 800 nm) with different pulse durations T. Left: T = 5 fs, right: T = 20 fs. The envelope function ±A(t) is indicated in red. In both examples the constraint (4) is satisfied.

3 Passage through a dispersive medium

In the interferometer sketched in Figure 2, a Wollaston prism is oriented such that it splits an incident linearly-polarized pulse into two fully correlated orthogonally-polarized pulses with equal amplitude, denoted E_x(t) and E_y(t). The latter is modified by traveling a distance L through a deterministic non-absorbing, linear dispersive medium with a known refractive index profile n(ω), and is then recombined with the unchanged signal E_x(t) to produce an output pulse. Fixed phase differences introduced by optical elements are not important in the present context. The path length in the upper arm can be varied, but is initially set to be the same as that in the lower arm. We study the polarization dynamics of the resulting output pulse as a function of the length L of the dispersive medium.

Fig. 2 Sketch of an interferometric setup to generate pulses with adjustable temporal polarization. The linearly polarized output of a pulsed laser is split into two orthogonally-polarized beams, Ex(t) and Ey(t), by a Wollaston prism (WP). The y-polarized beam travels a distance L through a dispersive medium (DM), and is then recombined with the pulse in the upper arm by a 50:50 beam splitter (BS) to produce the output pulse E(t). M1 and M2 are mirrors. The upper arm contains a variable free-space time delay.

We take the entrance plane of the dispersive medium to be at z = 0. The field in this plane is, according to equation (9),$$E_y(0,t)=A_0{\mathrm{e}}^{-t^2/T^2}{\mathrm{e}}^{-\mathrm{i}{\omega }_{0}t},$$(10)and therefore, from equation (3), the spectral amplitude there is$${\tilde{E}}_y(0,\omega )={\tilde{E}}_0{\mathrm{e}}^{-{\left(\omega -{\omega }_0\right)}^2/{\mathrm{\Omega }}^2}.$$(11)

Each frequency component in equation (11) propagates with wave number k(ω) through the dispersive medium. Hence, after a distance L, the spectral amplitude becomes$${\tilde{E}}_y(L,\omega )={\tilde{E}}_0{\mathrm{e}}^{-{\left(\omega -{\omega }_0\right)}^2/{\mathrm{\Omega }}^2}{\mathrm{e}}^{\mathrm{i}k\left(\omega \right)L}.$$(12)

We next use a Taylor expansion of k(ω) around the center frequency ω₀ and retain only quadratic terms and lower [14]$${\tilde{E}}_y(L,\omega )={\tilde{E}}_0{\mathrm{e}}^{-{\left(\omega -{\omega }_0\right)}^2/{\mathrm{\Omega }}^2}{\mathrm{e}}^{\mathrm{i}\left[\mathit{kL}+k\prime \left(\omega -{\omega }_0\right)L+k\prime \prime {\left(\omega -{\omega }_0\right)}^{2}L/2\right]}.$$(13)

The values of k, k′ = dk/dω and k″ = d²k/dω² are to be evaluated at the carrier frequency ω₀. Substitution from equation (13) into equation (1) yields the time domain expression$$E_y(L,t)={\tilde{E}}_0{\mathrm{e}}^{\mathrm{i}\mathit{kL}}{\int }_0^{\infty }‍{\mathrm{e}}^{-{\left(\omega -{\omega }_0\right)}^2/{\mathrm{\Omega }}^2}{\mathrm{e}}^{\mathrm{i}\left[k^{\prime }\left(\omega -{\omega }_0\right)L+k^{″}{\left(\omega -{\omega }_0\right)}^{2}L/2\right]}{\mathrm{e}}^{-\mathrm{i}\mathit{\omega t}}\mathit{d\omega }.$$(14)Because of condition (4) we can formally extend the lower limit of integration to minus infinity and complete the square, giving us the result$$E_y(L,t)={\tilde{E}}_0{\mathrm{e}}^{\mathrm{i}\mathit{kL}}{\mathrm{e}}^{-\mathrm{i}{\omega }_{0}t}\sqrt{\pi /A}{\mathrm{e}}^{-B^2/\left(4A\right)},$$(15)where$$A=\frac{1}{{\mathrm{\Omega }}^2}-\frac{\mathrm{i}{k}^{\prime \prime }L}{2},$$(16)$$B=k^{\prime }L-t,$$(17)and √ indicates the principal root. It is convenient to express the variation of the wave number k in terms of the variation of the refractive index n with the free-space wavelength λ₀. Since k(ω) = n(ω)ω/c, with c the speed of light in vacuum, repeated use of the chain rule yields [15]$$k^{\prime }=\frac{1}{c}\left(n-{\lambda }_0\frac{\mathit{dn}}{d{\lambda }_0}\right),$$(18)$$k^{\prime \prime }=\frac{{\lambda }_0^3}{2\pi c^2}\frac{d^{2}n}{d{\lambda }_0^2}.$$(19)

The values of dn/dλ₀ and $d^{2}n/d{\lambda }_0^2$ for a specific medium can be obtained from the Sellmeier dispersion model [15].

As an example we study BK7, a borosilicate optical glass, at λ₀ = 800 nm. Using its Sellmeir coefficients [16], we find that at that particular wavelength n = 1.51078, dn/dλ₀ = −0.0198418 μm⁻¹, and $d^{2}n/d{\lambda }_0^2=0.0492482\mathrm{\mu }{\mathrm{m}}^{-2}$. These values correspond to k′ = 5089 fs/mm and k″ = 44.59 fs²/mm. Setting the pulse width T = 20 fs we get, for different propagation lengths L, the electric fields shown in Figure 3. Note that for these values of L the effect of the medium is mainly to delay E_y(t) with respect to E_x(t). Essentially no pulse reshaping takes place.

Fig. 3 Time evolution of the electric field components Ex(t) and Ey(L,t) of a T = 20 fs pulse for three selected values of the length L of the dispersive medium.

4 Instantaneous state of polarization

The conventional view of the state of polarization takes into account many cycles of the field [17], implying that the usual Stokes parameters are time-averaged quantities, and thus provide no insight into intrapulse polarization variations. Here however, we are interested in precisely such dynamics. To that end we consider a beam-like optical pulse that propagates along the z-axis. The electric field vector at time t at a point r is$$\mathbf{E}(\mathbf{r},t)=(E_x(\mathbf{r},t),E_y(\mathbf{r},t)),$$(20)where E_x(r,t) and E_y(r,t) are complex analytic signal representations of the two Cartesian field components. The instantaneous Stokes parameters are defined as [18]$$S_0(\mathbf{r},t)={|E_x(\mathbf{r},t)|}^2+{|E_y(\mathbf{r},t)|}^2,$$(21)$$S_1(\mathbf{r},t)={|E_x(\mathbf{r},t)|}^2-{|E_y(\mathbf{r},t)|}^2,$$(22)$$S_2(\mathbf{r},t)=E_x^{\ast }(\mathbf{r},t)E_y(\mathbf{r},t)+E_y^{\ast }(\mathbf{r},t)E_x(\mathbf{r},t),$$(23)$$S_3(\mathbf{r},t)=\mathrm{i}[E_y^{\ast }(\mathbf{r},t)E_x(\mathbf{r},t)-E_x^{\ast }(\mathbf{r},t\left)E_y\right(\mathbf{r},t)],$$(24)where ^* denotes complex conjugation. These parameters are all real-valued and it is easily verified that$$S_0^2(\mathbf{r},t)=S_1^2(\mathbf{r},t)+S_2^2(\mathbf{r},t)+S_3^2(\mathbf{r},t).$$(25)Hence, the three normalized quantities$$s_i(\mathbf{r},t)=S_i(\mathbf{r},t)/S_0(\mathbf{r},t)\text{with}i=1,2,3,$$(26)constitute an instantaneous Poincaré vector$$s(\mathbf{r},t)=(s_1(\mathbf{r},t),s_2(\mathbf{r},t),s_3(\mathbf{r},t)),$$(27)

whose tip is a point on the Poincaré sphere of unit radius. As the polarization state changes, this point will trace out a continuous path on the sphere. The physical significance 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 μm. The output pulse $\mathbf{E}\left(t\right)=E_x\left(t\right)\widehat{\mathbf{x}}+E_y\left(t\right)\widehat{\mathbf{y}}$ with, from equations (9) and (15),$$E_x\left(t\right)=A_0{\mathrm{e}}^{-\mathrm{i}{\omega }_{0}t}{\mathrm{e}}^{\mathrm{i}{k}_{0}L}{\mathrm{e}}^{-t^2/T^2},$$(28)$$E_y\left(t\right)=\frac{A_0}{\mathrm{\Omega }\sqrt{A}}{\mathrm{e}}^{-\mathrm{i}{\omega }_{0}t}{\mathrm{e}}^{\mathrm{i}\mathit{kL}}{\mathrm{e}}^{-B^2/\left(4A\right)}.$$(29)

The expression for E_x (t) is supplemented with a phase factor exp(ik₀L), with k₀ 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 (s₁ = 1) and ends up being y-polarized (s₁ = −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 = 4 μm.

Fig. 4 Evolution of the three instantaneous Stokes parameters of the output pulse for a dispersive BK7 medium with length L = 4 μm. The input pulse duration T = 20 fs. The intensity (in [a.u.]) of the output pulse is represented by the dashed curve.

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.

Fig. 5 Trajectories on the Poincaré sphere with axes (s1, s2, s3) connecting states I (initial) and F (final), for different lengths L of the dispersive BK7 medium. Time ranges from t = −20 fs to t = 60 fs (Visualization 1).

The cyan curve (L = 5 μm) passes over the spherical quadrant (s₂ > 0, s₃ > 0); the blue curve (L = 10 μm) over the quadrant (s₂ < 0, s₃ > 0); the red curve (L = 15 μm) over the quadrant (s₂ < 0, s₃ < 0); and the purple curve (L = 17 μm) over the quadrant (s₂ > 0, s₃ < 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 s₃ attains the value ±1.

When the thickness L of the medium exceeds 10 μm, 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 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).

Fig. 6 The instantaneous intensity of the output pulse for four different values of the thickness of the dispersive medium.

We next study the case of the same T = 20 fs pulse 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 Δt ≈ k′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. Here the 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 Δt. If Δt = 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 Δt is increased the path gradually twists and at around Δt = 32515 fs detaches from I. The initial state moves upward and, near Δt = 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 Δt = 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.

Fig. 7 Left: the Ex(t) and Ey(t) pulses when the former undergoes a time delay Δt = k′L ≈ 32568.64 fs. Right: trajectory on the Poincaré sphere with axes (s1, s2, s3), for a length L = 6.4 mm of the BK7 medium. Time ranges from t = 32500 fs to t = 32780 fs with 2 fs time steps (Visualization 2).

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 Δt = 32568 fs, which corresponds to the left-hand panel of Figure 7 in which the peaks of the pulses coincide. This gives rise to an output pulse with a symmetric profile. When Δt is decreased by 20 fs (red curve), or increased (green curve), the intensity profile becomes somewhat asymmetric and has a lower peak value.

Fig. 8 The instantaneous intensity of the output pulse for three selected values of the time delay Δt of the Ex field.

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 pulse of which the polarization evolution is strongly influenced by the path length L. Two regimes can be distinguished. When L is just several wavelenghts, the main effect of the dispersive medium is a delay between the two 20 fs pulses. In that case, the path on the Poincaré sphere, which connects an x-polarized state to a y-polarized state, has an orientation that is governed by the particular value of L. It was found that it is possible to reach any point on the Poincaré sphere in this manner. When L is on the order of 10,000 wavelengths, a time-delay for E_x(t) has to be introduced in order for the pulses to overlap. Also, E_y(t) becomes significantly stretched. This results in a much more complex polarization evolution of the output pulse.

One possible application of polarization-modulated pulses may be the investigation of structural changes in pump-probe experiments.

Funding

This work was supported by the Dutch Research Council (NWO), project P19-13, and the National Natural Science Foundation of China (grant W2441005).

Conflicts of interest

The authors have nothing to disclose.

Data availability statement

No data were generated in the study. Visualizations 1 and 2 are available at https://doi.org/10.6084/m9.figshare.29828546 and https://doi.org/10.6084/m9.figshare.30052795.

Author contribution statement

Conceptualization: T.D.V. and A.T.F.; Methodology: J.X., T.D.V. and A.T.F.; Software: J.X. and T.D.V.; Writing – Original Draft Preparation: J.X. and T.D.V.; Writing – Review & Editing: J.X., T.D.V. and A.T.F.; Project administration: T.D.V.; Funding acquisition: T.D.V. All authors have read and agreed to the published version of the manuscript.

References

All Figures

	Fig. 1 Two normalized temporal pulses with center frequency ω0 = 2.36 × 1015 rad/s (corresponding to a free-space wavelength λ0 = 800 nm) with different pulse durations T. Left: T = 5 fs, right: T = 20 fs. The envelope function ±A(t) is indicated in red. In both examples the constraint (4) is satisfied.
In the text

Fig. 2 Sketch of an interferometric setup to generate pulses with adjustable temporal polarization. The linearly polarized output of a pulsed laser is split into two orthogonally-polarized beams, Ex(t) and Ey(t), by a Wollaston prism (WP). The y-polarized beam travels a distance L through a dispersive medium (DM), and is then recombined with the pulse in the upper arm by a 50:50 beam splitter (BS) to produce the output pulse E(t). M1 and M2 are mirrors. The upper arm contains a variable free-space time delay.

In the text

	Fig. 3 Time evolution of the electric field components Ex(t) and Ey(L,t) of a T = 20 fs pulse for three selected values of the length L of the dispersive medium.
In the text

	Fig. 4 Evolution of the three instantaneous Stokes parameters of the output pulse for a dispersive BK7 medium with length L = 4 μm. The input pulse duration T = 20 fs. The intensity (in [a.u.]) of the output pulse is represented by the dashed curve.
In the text

	Fig. 5 Trajectories on the Poincaré sphere with axes (s1, s2, s3) connecting states I (initial) and F (final), for different lengths L of the dispersive BK7 medium. Time ranges from t = −20 fs to t = 60 fs (Visualization 1).
In the text

	Fig. 6 The instantaneous intensity of the output pulse for four different values of the thickness of the dispersive medium.
In the text

	Fig. 7 Left: the Ex(t) and Ey(t) pulses when the former undergoes a time delay Δt = k′L ≈ 32568.64 fs. Right: trajectory on the Poincaré sphere with axes (s1, s2, s3), for a length L = 6.4 mm of the BK7 medium. Time ranges from t = 32500 fs to t = 32780 fs with 2 fs time steps (Visualization 2).
In the text

	Fig. 8 The instantaneous intensity of the output pulse for three selected values of the time delay Δt of the Ex field.
In the text