44 J. Eur. Opt. Society-Rapid Publ. 21, 5( 2025)
Appendix A Demonstration of equation( 7) via Fourier inversion In this Appendix, we again aim at calculating( see Eq. 7) b m ðÞ¼ t 1 Z 1
Z 1
x
2p ðx~ m � xÞ O mðt 0 Þ exp ðixðt 0 � tÞÞdt 0 dx;
�1 �1 ðA1Þ
with minimum assumption on the incident field, i. e. on O m( t 0). The approach just assumes that O m is in L 1 ðRÞ, the space of functions integrable over the real line. It is based on the well-known Fourier inversion formula. For fixed values of x, we first calculate the integral Z
1 1 x
2p �1 ðx~ m � xÞ O mðt 0 Þ exp ðixðt 0 � tÞÞdt 0, which is equal to x exp �ixt
ðx~ m � xÞ ð Þ O m ðxÞ. Further integrating over R with
Z 1 x respect to x, we aim at calculating exp �ixt
�1 ðx~ m � xÞ ð Þ x O m ðxÞdx. Replacing ðx~ m � xÞ by x~ m
� 1, we first obtain ðx~ m � xÞ Z 1
x~ m b m ðÞ¼ t �1 þ exp ð�ixtÞO ðx~ m � xÞ
m ðxÞdx and thus find
�1
Z 1
1 b m ðÞ¼�O t m ðÞþ t x~ m
�1 ðx~ m � xÞ exp ð�ixtÞO m ðxÞdx: ðA2Þ
�1
Gt t! 1
To evaluate the integral on the right-hand side, we demonstrate that the function G( t) = expði x~ m tÞFt ðÞ, where Ft ðÞ¼
R 1 1 exp �ixt
�1 ðx~ m � xÞ ð Þ O m ðxÞdx satisfies a simple differential equation that we can solve. It can be verified that G is a differentiable function of the variable t 2 R, because the conditions of the well-known differentiation theorem for integrals are met for functions in L 1 ðRÞ. Differentiating G with respect to t yields dG Z dt ¼ 1 i exp ði x~ m tÞ exp ð�ixtÞO m ðxÞdx ¼ i exp ði x~ m tÞO m ðÞ t. Considering that lim ðÞ¼0( the limit and the integral can be interchanged due to the integrand being suitably dominated), we obtain Gt ðÞ¼i R t exp ð i x~ �1 mt 0 ÞO m ðt 0 Þdt 0.
Thus Ft ðÞ¼i R t exp ð i x~ �1 mðt 0 � tÞÞO m ðt 0 Þdt 0 and coming back to equation( A2), wefind
Z t
b m ðÞ¼�O t m ðÞþi t x~ m exp ði x~ m ðt 0 � tÞÞO m ðt 0 Þdt 0: ðA3Þ
�1
This concludes a demonstration of equation( 7).