466
J. Eur. Opt. Society-Rapid Publ. 22, 47( 2026)
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, E x( t) and E y( 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). M 1 and M 2 are mirrors. The upper arm contains a variable free-space time delay.
Because of condition( 4) we can formally extend the lower limit of integration to minus infinity and complete the square, giving us the result
where
E y ðL; t
Þ ¼ E p
0 e ikL e �ix 0t ffiffiffiffiffiffiffiffiffi p = A e �B2 = ð4A
Þ; ð15Þ
A ¼ 1 X 2 � ik00 L 2; ð16Þ
B ¼ k 0 L � t; ð17Þ
and p 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 k 0. Since k( x)= n( x) x / c, with c the speed of light in vacuum, repeated use of the chain rule yields [ 15 ]
k 0 ¼ 1 c
dn n � k 0; ð18Þ dk 0
k 00 ¼ k3 0 d 2 n: ð19Þ
2pc 2 dk 2 The values of dn / dk 0 and d 2 n = dk 2
0 for a specific medium can be obtained from the Sellmeier dispersion model [ 15 ].
As an example we study BK7, a borosilicate optical glass, at k 0 = 800 nm. Using its Sellmeir coefficients [ 16 ], we find that at that particular wavelength n = 1.51078, dn / dk 0 = �0.0198418 lm �1, and d 2 n = dk 2
0 ¼ 0:0492482 lm �2. These values correspond to k 0 = 5089 fs / mm and k 00 = 44.59fs 2 / 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.
0
Fig. 3. Time evolution of the electric field components E x ðÞ t and E y ðL; tÞ of a T ¼ 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
Er ð; t
� Þ ¼ E x ðr; tÞ; E y ðr; tÞ
; ð20Þ
where E x( r, t) andE y( r, t) are complex analytic signal representations of the two Cartesian field components. The instantaneous Stokes parameters are defined as [ 18 ]
S 0 ðr; t
S 1 ðr; t
Þ ¼ jE x ðr; tÞj 2 þ E y ðr; tÞ 2;
Þ ¼ jE x ðr; tÞj 2 � E y ðr; tÞ 2; ð21Þ
ð22Þ
S
2 ðr; tÞ ¼ E x ðr; tÞE yðr; tÞþE y ðr; tÞE xðr; tÞ; ð23Þ
h i S 3 ðr; tÞ ¼ i E y ðr; tÞE xðr; tÞ�E x ðr; tÞE yðr; tÞ; ð24Þ
where * denotes complex conjugation. These parameters are all real-valued and it is easily verified that
S 2 0 ðr; tÞ ¼ S 2
1 ðr; t
ÞþS2 2 r; t
Hence, the three normalized quantities s i ðr; t ð ÞþS 2
3 ðr; tÞ: ð25Þ
Þ ¼ S i ðr; tÞ = S 0 ðr; tÞ with i ¼ 1; 2; 3; ð26Þ constitute an instantaneous Poincaré vector sr ð; tÞ ¼ ðs 1 ðr; tÞ; s 2 ðr; tÞ; s 3 ðr; tÞÞ; ð27Þ
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