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Focal shift in unapertured Bessel-Gauss beams
15 June 1994
OPTICS COMMUNICATIONS Optics Communications 109 (1994) 43-46
ELSEVIER
Focal shift in unapertured Bessel-Gauss beams Baida Lfi, Wenlong Huang Department of Opto-Electronic Science and Technology, Sichuan University, Chengdu 610064, China Received 31 January 1994
Abstract
The focal shift of unapertured Bessel-Gauss beams is studied by means of the Collins diffraction integral. The results of unapertured Bessel beams and Gaussian beams follow readily as two limiting cases of our study.
1. Introduction In a number of paper dealing with focused beams [ 1-6 ], it was found that under certain conditions the point of m a x i m u m intensity of the output beam is not at the geometrical focus, but somewhat closer to the focused lens. This is the well-known concept of focal shift and it is of importance from both the theoretical and practical viewpoint. The present paper is aimed at investigating the focal shift of a new type of coherent beams - Bessel-Gauss beams. We will examine the focused property ofunapertured BesselGauss beams and derive a rather simple expression determining the point of m a x i m u m intensity, from which the focal shift is calculated numerically. Furthermore, it will be shown that in two limiting cases our results are reduced to those of unapertured Gaussian beams and Bessel beams.
2. The field distribution along the Z-axis Consider a Bessel-Gauss beam of the form [ 7 ]
UB°(r, O) =AoJo( ar) exp[ -- (r/w) ]2
(1)
incident at an unapertured thin lens in the plane Z = 0 as shown in Fig. 1, where Jo is the zeroth-order Bessel
function of the first kind, w is the waist radius of the Gaussian term, Ao is a complex constant and r denotes the radial coordinate. According to the Collins diffraction integral [ 8 ], we find the field at any point Z = 12 along the Z axis, not too close to the lens -boo
UB°(O, 12)=~zkexp(-ikl2) ~ e x p ( - ~ 2 ) 0
×Jo(ar) exp[ ( - i k / 2 ) ( 1/12 - 1 / f ) r 2 ] rdr,
(2) w i t h f being the focal length of the lens ( f > 0). Let
Nw=w:/2f
be the Fresnel number associated with the Gaussian term,
N,~ = 1/2fa 2
(4)
be the Fresnel number associated with the Bessel term (note that the central spot radius of Bessel beams is approximately equal to a - l ) and
nNw AZ
u= - l+AZ '
12- f
AZ= -f
Recalling the integral formula [ 9 ]
0030-4018/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved
SSDI 0030-4018 ( 94 ) 00144-J
(3)
(5,6)
B. Lfi, W.. Huang / Optics Communications 109 (1994) 43-46
44
Point of maximum intensity Thin lens
Geometrical
focus
L
Z
I
Fig. 1. A Bessel-Gauss beam focused by an unapertured thin lens L.
UG(r) =Ao exp ( - rZ/w 2) ,
i
XJo(aX) e x p ( - f i x 2) dx
1
0
= (1/2fl) exp(--O~2/4fl)
( 11 )
(7)
,
for any choice of the (possibly complex) parameters a and fl, and straightforward integral calculations yield US°(0, u) = iAo nNw e x p ( - i k l z ) ( 1 - u/nNw) 1 -iu
Nw × exp(-4N.(1 -iu)) "
(8)
From Eq. (8) we obtain the intensity distribution I Bc (0, u) of the transmitted Bessel-Gauss beam along the Z-axis
IG(O,u)=IGv(O,O) l-~-N-~w l + u 2 ,
(12)
IVY(0, 0) = IAo 12n2N 2, ,
(13)
corresponding to the results of Gaussian beams, (ii) w = or, Nw = ~ . Eqs. ( 1 ) and (9) are reduced to
UB(r) =AoJo( ar) ,
(14)
IB(0, AZ) = IAo 12(1/AZ) 2 ,
(15)
which are the corresponding results of Bessel beams. It should be noted that IB(0, AZ) cannot be expressed in terms of the intensity at the geometrical focus, because Iv~(0, O ) = o o .
(16)
IBm(0, u) = IAo I:n:N: x
1
N2 3. The focal shift
1--ueXp(2Uwu2
( 1~-~-u:)N,~] ' (9)
where 1 ~ ( 0 , 0) = ]Ao] 2-2~T2n ~vw e x p ( - N w / 2 N ~ )
(10)
is the intensity at the geometrical focus F. Two limiting cases are of interest. (i) a = 0, N~ = oo. Eqs. ( 1 ), ( 9 ) and ( 10 ) become
From the condition d/ dAZ
dI du =0, du dAZ
(17)
the equation determining the point of maximum intensity of focused Bessel-Gauss beams along the Zaxis (/=/2max) can be derived in a similar way used by Li and Wolf [2,3]. After some differential calculus, we get
B. Lii, W. Huang I Optics Communications 109 (1994)43-46
u ) 1-,,,,w~
N~u 2 N , ( l + u 2)
1 nmw
R=O .
(18)
The relation focal shift AZ BG is defined as AzBG= (12. . . .
f)/f.
45
Of interest are unapertured Bessel beams. From Eqs. ( 15 ), (16) and ( 19 ), we find immediately that there is no focal shift for unapertured Bessel beams, because in this case the maximum intensity along the Z-axis is reached at the geometrical focus AZ= 0.
(19)
AZ B~ can be obtained from Eqs. (18) and (19) numerically, some numerical calculation results are compiled in Fig. 2, from which it can be seen that AZ BG is determined dominantly by two Fresnel numbers Nw, N~, and lies in the region of ( - 1, 0). Hence, the point of maximum intensity of unapertured Bessel-Gauss beams is not located at the geometrical focus, but somewhat closer to the lens in general, and IAzBG [ decreases with increasing Nw and N~. Letting a = 0 in Eq. ( 18 ) and combining with Eqs. (5) and ( 19 ), we obtain AZ G - [1 + (ZrNw)2] - ' ,
(20)
AZ G denotes the focal shift of unapertured Gaussian beams and is consistent with the result in Ref. [ 4 ].
-0.2
4. Concluding remarks In conclusion, we have derived the expressions determining the intensity distribution along the Z-axis and the focal shift of unapertured and focused Bessel-Gauss beams. It has been shown that the dominant parameters of unapertured Bessel-Gauss beams are two Fresnel numbers Nw, N~, and no focal shift exists for ideal unapertured Bessel beams. In contrast, using the Bessel beam produced at the geometrical focus was found experimentally instead of a point of maximum intensity along the Z-axis [ 10 ]. Hence, it reveals that the focused characteristics of apertured Bessel beams carrying a finite energy are essentially distinguishable from those of unapertured
¢~~
%!'li
~
5~I~; /
-0.4-
_o.
:~'I~i
---
@
.0 " c ,, )
.~.
' 6"
"-.
~il "0~ % {b)
Fig. 2. (a) AZ a° as a function of Nw, parameter is N,~, (b) AZ BG versus Nw and N~,.
'"""~'
46
B. Lfi, W. Huang /Optics Communications 109 (1994) 43-46
Bessel b e a m s , t h e r e l a t e d results a n d d i s c u s s i o n s will b e p u b l i s h e d elsewhere.
Acknowledgement T h e a u t h o r s are t h a n k f u l to the N a t i o n a l S c i e n c e a n d T e c h n o l o g y f o u n d a t i o n for the s u p p o r t o f the work.
References [ 1 ] H. Kogelnik, Bell Syst. Tech. J. 44 ( 1965 ) 455. [ 2 ] Y. Li and E. Wolf, Optics Comm. 39 ( 1981 ) 211. [ 3 ] Y. Li and E. Wolf, Optics Comm. 42 (1982) 151. [4] Y. Li, Optics Comm. 95 (1993) 13. [ 5 ] W.C. Carter, Appl. Optics 21 (1982) 1989. [6] T. Poon, Optics Comm. 65 (1988) 401. [ 7 ] F. Gori, G. Guattari and C. Padovani, Optics Comm. 64 (1987) 491. [8] S.A. Collins, J. Opt. Soc. Am. 60 (1970) 1168. [9]A. Erdelyi, ed., Table of integral transforms, Vol. 2 (McGraw-Hill, 1954) p. 51. [ 10] B. Lii et al., Propagation characteristics of focused Bessel beams with a finite beam width, to be published.
OPTICS COMMUNICATIONS Optics Communications 109 (1994) 43-46
ELSEVIER
Focal shift in unapertured Bessel-Gauss beams Baida Lfi, Wenlong Huang Department of Opto-Electronic Science and Technology, Sichuan University, Chengdu 610064, China Received 31 January 1994
Abstract
The focal shift of unapertured Bessel-Gauss beams is studied by means of the Collins diffraction integral. The results of unapertured Bessel beams and Gaussian beams follow readily as two limiting cases of our study.
1. Introduction In a number of paper dealing with focused beams [ 1-6 ], it was found that under certain conditions the point of m a x i m u m intensity of the output beam is not at the geometrical focus, but somewhat closer to the focused lens. This is the well-known concept of focal shift and it is of importance from both the theoretical and practical viewpoint. The present paper is aimed at investigating the focal shift of a new type of coherent beams - Bessel-Gauss beams. We will examine the focused property ofunapertured BesselGauss beams and derive a rather simple expression determining the point of m a x i m u m intensity, from which the focal shift is calculated numerically. Furthermore, it will be shown that in two limiting cases our results are reduced to those of unapertured Gaussian beams and Bessel beams.
2. The field distribution along the Z-axis Consider a Bessel-Gauss beam of the form [ 7 ]
UB°(r, O) =AoJo( ar) exp[ -- (r/w) ]2
(1)
incident at an unapertured thin lens in the plane Z = 0 as shown in Fig. 1, where Jo is the zeroth-order Bessel
function of the first kind, w is the waist radius of the Gaussian term, Ao is a complex constant and r denotes the radial coordinate. According to the Collins diffraction integral [ 8 ], we find the field at any point Z = 12 along the Z axis, not too close to the lens -boo
UB°(O, 12)=~zkexp(-ikl2) ~ e x p ( - ~ 2 ) 0
×Jo(ar) exp[ ( - i k / 2 ) ( 1/12 - 1 / f ) r 2 ] rdr,
(2) w i t h f being the focal length of the lens ( f > 0). Let
Nw=w:/2f
be the Fresnel number associated with the Gaussian term,
N,~ = 1/2fa 2
(4)
be the Fresnel number associated with the Bessel term (note that the central spot radius of Bessel beams is approximately equal to a - l ) and
nNw AZ
u= - l+AZ '
12- f
AZ= -f
Recalling the integral formula [ 9 ]
0030-4018/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved
SSDI 0030-4018 ( 94 ) 00144-J
(3)
(5,6)
B. Lfi, W.. Huang / Optics Communications 109 (1994) 43-46
44
Point of maximum intensity Thin lens
Geometrical
focus
L
Z
I
Fig. 1. A Bessel-Gauss beam focused by an unapertured thin lens L.
UG(r) =Ao exp ( - rZ/w 2) ,
i
XJo(aX) e x p ( - f i x 2) dx
1
0
= (1/2fl) exp(--O~2/4fl)
( 11 )
(7)
,
for any choice of the (possibly complex) parameters a and fl, and straightforward integral calculations yield US°(0, u) = iAo nNw e x p ( - i k l z ) ( 1 - u/nNw) 1 -iu
Nw × exp(-4N.(1 -iu)) "
(8)
From Eq. (8) we obtain the intensity distribution I Bc (0, u) of the transmitted Bessel-Gauss beam along the Z-axis
IG(O,u)=IGv(O,O) l-~-N-~w l + u 2 ,
(12)
IVY(0, 0) = IAo 12n2N 2, ,
(13)
corresponding to the results of Gaussian beams, (ii) w = or, Nw = ~ . Eqs. ( 1 ) and (9) are reduced to
UB(r) =AoJo( ar) ,
(14)
IB(0, AZ) = IAo 12(1/AZ) 2 ,
(15)
which are the corresponding results of Bessel beams. It should be noted that IB(0, AZ) cannot be expressed in terms of the intensity at the geometrical focus, because Iv~(0, O ) = o o .
(16)
IBm(0, u) = IAo I:n:N: x
1
N2 3. The focal shift
1--ueXp(2Uwu2
( 1~-~-u:)N,~] ' (9)
where 1 ~ ( 0 , 0) = ]Ao] 2-2~T2n ~vw e x p ( - N w / 2 N ~ )
(10)
is the intensity at the geometrical focus F. Two limiting cases are of interest. (i) a = 0, N~ = oo. Eqs. ( 1 ), ( 9 ) and ( 10 ) become
From the condition d/ dAZ
dI du =0, du dAZ
(17)
the equation determining the point of maximum intensity of focused Bessel-Gauss beams along the Zaxis (/=/2max) can be derived in a similar way used by Li and Wolf [2,3]. After some differential calculus, we get
B. Lii, W. Huang I Optics Communications 109 (1994)43-46
u ) 1-,,,,w~
N~u 2 N , ( l + u 2)
1 nmw
R=O .
(18)
The relation focal shift AZ BG is defined as AzBG= (12. . . .
f)/f.
45
Of interest are unapertured Bessel beams. From Eqs. ( 15 ), (16) and ( 19 ), we find immediately that there is no focal shift for unapertured Bessel beams, because in this case the maximum intensity along the Z-axis is reached at the geometrical focus AZ= 0.
(19)
AZ B~ can be obtained from Eqs. (18) and (19) numerically, some numerical calculation results are compiled in Fig. 2, from which it can be seen that AZ BG is determined dominantly by two Fresnel numbers Nw, N~, and lies in the region of ( - 1, 0). Hence, the point of maximum intensity of unapertured Bessel-Gauss beams is not located at the geometrical focus, but somewhat closer to the lens in general, and IAzBG [ decreases with increasing Nw and N~. Letting a = 0 in Eq. ( 18 ) and combining with Eqs. (5) and ( 19 ), we obtain AZ G - [1 + (ZrNw)2] - ' ,
(20)
AZ G denotes the focal shift of unapertured Gaussian beams and is consistent with the result in Ref. [ 4 ].
-0.2
4. Concluding remarks In conclusion, we have derived the expressions determining the intensity distribution along the Z-axis and the focal shift of unapertured and focused Bessel-Gauss beams. It has been shown that the dominant parameters of unapertured Bessel-Gauss beams are two Fresnel numbers Nw, N~, and no focal shift exists for ideal unapertured Bessel beams. In contrast, using the Bessel beam produced at the geometrical focus was found experimentally instead of a point of maximum intensity along the Z-axis [ 10 ]. Hence, it reveals that the focused characteristics of apertured Bessel beams carrying a finite energy are essentially distinguishable from those of unapertured
¢~~
%!'li
~
5~I~; /
-0.4-
_o.
:~'I~i
---
@
.0 " c ,, )
.~.
' 6"
"-.
~il "0~ % {b)
Fig. 2. (a) AZ a° as a function of Nw, parameter is N,~, (b) AZ BG versus Nw and N~,.
'"""~'
46
B. Lfi, W. Huang /Optics Communications 109 (1994) 43-46
Bessel b e a m s , t h e r e l a t e d results a n d d i s c u s s i o n s will b e p u b l i s h e d elsewhere.
Acknowledgement T h e a u t h o r s are t h a n k f u l to the N a t i o n a l S c i e n c e a n d T e c h n o l o g y f o u n d a t i o n for the s u p p o r t o f the work.
References [ 1 ] H. Kogelnik, Bell Syst. Tech. J. 44 ( 1965 ) 455. [ 2 ] Y. Li and E. Wolf, Optics Comm. 39 ( 1981 ) 211. [ 3 ] Y. Li and E. Wolf, Optics Comm. 42 (1982) 151. [4] Y. Li, Optics Comm. 95 (1993) 13. [ 5 ] W.C. Carter, Appl. Optics 21 (1982) 1989. [6] T. Poon, Optics Comm. 65 (1988) 401. [ 7 ] F. Gori, G. Guattari and C. Padovani, Optics Comm. 64 (1987) 491. [8] S.A. Collins, J. Opt. Soc. Am. 60 (1970) 1168. [9]A. Erdelyi, ed., Table of integral transforms, Vol. 2 (McGraw-Hill, 1954) p. 51. [ 10] B. Lii et al., Propagation characteristics of focused Bessel beams with a finite beam width, to be published.