n m ng

uan t re foca cific e F ich me n of o pr irradiance distribution corresponding to focusing of a 2760 J. Opt. Soc. Am. A/Vol. 18, No. 11 /November 2001 J. Pu and B. Luthe studies are carried out in the scalar approach. In other words, the polarization of the incident light beam has been neglected in most of the papers concerned with focal shifts. To our knowledge, until now only one paper has investigated the focal shift of vector beams.9 In re-cent years, there has been increasing interest in the po-larization of the light beam.10–16 It has been shown that as the high-power laser beam of linear polarization propa-gates through laser amplifiers, the thermally induced bi-refringence in the laser rods imposes different polariza-tion modulations in different parts of the laser beam, so that the light beam exhibits different polarization states in different parts of the laser beam. This kind of light beam was termed the nonuniformly polarized (NUP) beam.10–12 The NUP beam is also generated from the light beam passing through nonlinear optical materials.17 Although focal shifts have been studied extensively, the influence of the nonuniform polarization of the incident light wave on the focal shifts has not been investigated. How its polarization distribution affects the focusing properties and the polarization distribution in the focused field is unclear. The major purpose of this paper is to in-vestigate the focal shifts of the NUP light wave focused by an apertured lens. A simple analytical formula, which is NUP beam is derived, which allows us to introduce the ef-fective Fresnel number to evaluate the relative focal shifts in Section 3. In Section 4, some examples are pre-sented to illustrate the influence of the polarization dis-tribution of the NUP beam on the effective Fresnel num-ber and the relative focal shifts. It is demonstrated that the effective Fresnel number is the dominant factor in de-termining the relative focal shifts. The effect of the po-larization distribution of the incident NUP light wave across the lens aperture on the polarization distribution in the focused field is also presented in Section 5, and in Section 6 a brief conclusion is given