Abstract

I thank Canales et al. very much for their attention to the effect of the phase apodizer, especially for their comment on my paper [Appl. Opt. 44, 4870 (2005)]. I reinvestigated the three-dimensional intensity distribution by a pure phase-shifting apodizer, and the main results on focal shift, focal split, and three intensity distributions are correct. However, the changing principle of the Strehl ratio is wrong, as Canales et al. show in their comment. Here, I show the cause leading to the erroneous result and give the correct dependence of the Strehl ratio on the inner radius and relative waist width. I am very sorry for the incorrect results in the paper.

© 2007 Optical Society of America

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Figures (1)

Fig. 1

Dependence of the Strehl ratio on the inner radius b and the relative waist width w for (a) $a = 0.1$ , (b) $a = 0.3$ , (c) $a = 0.5$ , (d) $a = 0.7$ , and (e) $a = 0.9$ .

Equations (6)

$G ( ρ , u ) = 2 ∑ j = 1 3 exp ( i ϕ j ) ∫ r j − 1 r j r J 0 ( ρ r ) exp [ - ( 1 w 2 + i u 2 ) r 2 ] d r ,$
$S = [ G a ≠ b ( 0 , 0 ) / G a = b = 0 ( 0 , 0 ) ] 2 .$
$G a ≠ b ( 0 , 0 ) = ∑ j = 1 3 exp ( i ϕ j ) ∫ r j − 1 r j exp ( - r 2 w 2 ) d ( r 2 ) = ∑ j = 1 3 exp ( i ϕ j ) ( − w 2 ) ∫ r j − 1 r j exp ( - r 2 w 2 ) d ( - r 2 w 2 ) = ( − w 2 ) [ exp ( - b 2 w 2 ) − 1 ] − ( − w 2 ) [ exp ( - a 2 w 2 ) − exp ( - b 2 w 2 ) ] + ( − w 2 ) [ exp ( - 1 w 2 ) − exp ( - a 2 w 2 ) ] = ( − w 2 ) [ 2 exp ( - b 2 w 2 ) − 1 − 2 exp ( - a 2 w 2 ) + exp ( - 1 w 2 ) ] .$
$G a = b = 0 ( 0 , 0 ) = ( − w 2 ) [ exp ( - 1 w 2 ) − 1 ] ,$
$S = [ 2 exp ( - b 2 w 2 ) − 1 − 2 exp ( - a 2 w 2 ) + exp ( - 1 w 2 ) ] 2 [ exp ( - 1 w 2 ) − 1 ] 2 .$
$S = [ G a ≠ b ( 0 , 0 ) / G a = b = 0 ( 0 , 0 ) ] 2 = { 4 exp ( - 2 w 2 b 2 ) + 1 + 4 exp ( - 2 w 2 a 2 ) + exp ( - 2 w 2 ) − 4 exp ( - 1 w 2 b 2 ) + 4 exp ( - 1 w 2 a 2 ) − 2 exp ( - 1 w 2 ) − 8 exp ( - a 2 + b 2 w 2 ) + 4 exp ( - b 2 + 1 w 2 ) − 4 exp ( - a 2 + 1 w 2 ) } × [ 1 − exp ( - 1 w 2 ) ] − 4 .$