Evolution of the transverse response of an optical system with complex filters Silvia Ledesma a, * , Juan C. Escalera b , Juan Campos b , Marı ´a J. Yzuel b a Departamento de Fı ´sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina b Departamento de Fı ´sica, Universidad Auto ´ noma de Barcelona, 08193 Bellaterra, Spain Received 25 October 2004; received in revised form 23 December 2004; accepted 17 January 2005 Abstract In many optical systems a specific axial behaviour is needed. It is very common to search for high focal depth and an almost constant distribution along the axial coordinate. An additional condition to be required could be symmetrical axial response. The transverse response in the best image plane is also of great importance, for instance to produce superresolution. Moreover, the invariance of the transverse response in defocused planes could be a requirement for an optical system. In this paper, we study the transverse response at defocused planes produced by complex filters with high focal depth. In order to analyze the transverse responses we extend the transverse gain factor for defocused planes. Here, we derive several conditions that complex filters may satisfy to produce symmetrical axial response. Numerical examples of transverse responses for some filters that produce high depth of focus are shown. Ó 2005 Elsevier B.V. All rights reserved. PACS: 42.79.Ci; 42.30.Kq; 42.79.Vb Keywords: Phase modulation; Apodization; Depth of focus; Resolution 1. Introduction Optical systems need, in general, a specific axial behaviour [1–4]. In many optical systems a high depth of focus (DOF) is needed [5,6]. The trans- verse response in the best image plane (BIP) is also usually investigated [1–6]. Nevertheless, the trans- verse response in defocused planes has to be stud- ied to ensure that the response is also good in those planes. Amplitude-only filters have been widely studied to obtain, for instance, superresolution or apodi- zation [7]. Sheppard and Hegedus [1] proposed some parameters that describe in a simple way the transverse and axial behaviour of an optical system when amplitude-only filters are introduced. 0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.01.019 * Corresponding author. Tel.: +54 11 45763353; fax: +54 11 45763357. E-mail address: ledesma@df.uba.ar (S. Ledesma). Optics Communications 249 (2005) 183–192 www.elsevier.com/locate/optcom