J. Eur. Opt. Society-Rapid Publ. 22, 47( 2026) 465
Fig. 1. Two normalized temporal pulses with center frequency x 0 = 2.36 10 15 rad / s( corresponding to a free-space wavelength k 0 = 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.
X << x 0: ð4Þ
Let us next write the pulse as the product of an envelope function A( t) and a carrier signal of frequency x 0:
Eð0; tÞ ¼ At ðÞe �ix0t:
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
At ðÞ¼E 0
¼ E 0
Z 1
�1
Z 1
�1 ð5Þ
e � ð x�x 0Þ 2 = X 2 e �i ð x�x 0 Þt dx ð6Þ
e �x02 = X 2 e �ix0t dx 0; ð7Þ
where x 0 = x�x 0, and with the integration in equation( 1) formally extended to minus infinity. This is a standard integral with solution
At ðÞ¼A 0 e �t2 = T 2; ð8Þ where A 0 ¼ E p
0X ffiffiffi p, and with the pulse duration given by T = 2 / X. Using this expression for the envelope function in equation( 5) gives us
Eð0; tÞ ¼ A 0 e �t2 = T 2 e �ix0t: ð9Þ
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 x 0 = 2.36 10 15 rad / s( corresponding to a free-space wavelength k 0 = 800 nm) and A 0 = 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.
3 Passage through a dispersive medium
In the interferometer sketched in Figure 2, a Wollaston prism is oriented such that it splits an incident linearlypolarized pulse into two fully correlated orthogonally-polarized pulses with equal amplitude, denoted E x( t) andE 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( x), 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.
We take the entrance plane of the dispersive medium to be at z = 0. Thefield in this plane is, according to equation( 9),
E y ð0; tÞ ¼ A 0 e �t2 = T 2 e �ix0t; ð10Þ
and therefore, from equation( 3), the spectral amplitude there is
E yð0; xÞ ¼ E 0 e � ð x�x 0Þ 2 = X 2: ð11Þ
Each frequency component in equation( 11) propagates with wave number k( x) through the dispersive medium. Hence, after a distance L, the spectral amplitude becomes
E yðL; xÞ ¼ E 0 e � ð x�x 0Þ 2 = X 2 e ikðxÞL: ð12Þ
We next use a Taylor expansion of k( x) around the center frequency x 0 and retain only quadratic terms and lower [ 14 ]
E yðL; xÞ ¼ E � x�x0 0 e ð Þ2 = X 2 e i kL þ k0 ðx � x 0 ÞL þ k 00 ðx � x 0 Þ 2 L = 2
: ð13Þ
The values of k, k 0 = dk / dx and k 00 = d 2 k / dx 2 are to be evaluated at the carrier frequency x 0. Substitution from equation( 13) into equation( 1) yields the time domain expression
Z E y ðL; tÞ ¼ E 1
0 e ikL
e i k0 ðx�x 0 ÞLþk 00 ðx�x 0 Þ 2 L = 2
0 e � ð x�x 0Þ 2 = X 2
½ Š e �ixt dx: ð14Þ