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Focal shifts on focusing through a plane interface
Optics Communications 282 (2009) 2286–2291
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Focal shifts on focusing through a plane interface Jakob J. Stamnes a, Dhayalan Velauthapillai b,* a b
Department of Physics and Technology, University of Bergen, Allégaten 55, N–5007 Bergen, Norway Department of Engineering, University College of Bergen, Postboks 7030, Nygardstangen, N–5020 Bergen, Norway
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 30 August 2008 Received in revised form 20 February 2009 Accepted 2 March 2009
We derive explicit formulas for the focal shift, the intensity increase, and the axial zeros obtained on focusing at low Fresnel numbers through a plane interface between two isotropic media, and demonstrate their validity through comparisons with numerical plots of axial intensity distributions. Also, we show that for the special case in which the two isotropic media have the same refractive index, these formulas are in complete agreement with previous results presented for focusing in a single medium. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
where
In the 1980s large discrepancies were observed between the results of the Debye and Kirchhoff theories for focusing of scalar waves [1–6]. It was shown experimentally that the Debye theory is not adequate for calculating intensity distributions around the focus when a converging spherical wave is diffracted through a circular aperture at low Fresnel numbers [7]. Focal shift phenomena were observed also for focusing of electromagnetic waves [8–12]. The previous works on scalar waves [1–7,13,14] were concerned with focal shifts obtained on focusing in a single, homogeneous, isotropic medium, whereas here we consider focusing of scalar waves through a plane interface between two different homogeneous, isotropic media. 2. Focusing through a plane interface between two isotropic media Consider the three-dimensional focusing geometry in Fig. 1, where a scalar converging spherical wave is diffracted through a circular aperture of radius a in the plane z ¼ 0 in an isotropic medium with refractive index n1 . The diffracted field is then transmitted through a plane interface at z ¼ z0 separating the isotropic medium of the incident wave from another isotropic medium with refractive index n2 . If the second medium had the same refractive index as the first medium, the focus of the wave would be at ð0; 0; z1 Þ, where z1 > z0 . Within the paraxial approximation the intensity of the focused field is given by [10]
Iðr; zÞ ¼
2 pa2 1 þ nr Zz1 k1
2 Z 2
0
1
2 1 2 t tdt ; J 0 ðv tÞ exp i u 2
* Corresponding author. Tel.: +47 55587711. E-mail address: [email protected] (D. Velauthapillai). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.03.004
1 n2 Z ¼ z0 þ ðz z0 Þ; nr ¼ ; nr n1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p a v ¼ r; r ¼ x2 þ y2 ; k1 Z 2p a ðZ z1 Þ: k1 z1 Z
ð2Þ ð3Þ
2
¼ u
ð4Þ
2.1. Derivation The derivation of (1) in [10] is quite complicated since it involves focusing of an electromagnetic wave into a biaxial medium. Another rather complicated derivation for the focusing of an electromagnetic wave into an isotropic medium is given in section 16.2 in [15]. The complications arise because the derivations are based on exact solutions of the electromagnetic problem. Therefore, we now give a simplified derivation, whose validity is restricted to paraxial geometries and scalar fields, but which provides considerable physical insight. To that end, we start with the first Rayleigh– Sommerfeld diffraction formula for the diffracted field uI obtained upon diffraction through an aperture in the plane z ¼ 0 in a single, isotropic medium:
uI ¼
1 2p
Z Z
ui A
@ expðikRÞ 0 0 dx dy : @z R
ð5Þ
Here, ui is the field in the aperture plane z ¼ 0; A is the aperture area, k is the wave number, and R is the distance from an integration point Pðx0 ; y0 ; 0Þ inside the aperture and the observation point Pðx; y; zÞ, i.e.
ð1Þ R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x0 Þ2 þ ðy y0 Þ2 þ z2 :
ð6Þ
Carrying out the differentiation in (5), and assuming that kR 1, we get
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Substituting from (10)–(12) into (8), we obtain
uI ¼
1 2 1 1 þ nr ik1 Zz
Z Z
0
0
expðik1 ðR1 þ nr R2 R1;0 Þdx dy ;
ð13Þ
A
where 2=ð1 þ nr Þ is the transmission coefficient T at normal incidence. The refraction point P0 ðx0 ; y0 ; z0 Þ at the interface follows from Snell’s law:
n1 sin h1 ¼ n2 sin h2 ;
ð14Þ
where h1 is the angle between the line P0 P0 and the interface normal, and h2 is the angle between the line P 0 P and the interface normal. In the paraxial approximation, Snell’s law becomes Fig. 1. A converging spherical wave in an isotropic medium of refractive index n1 is diffracted through a circular aperture of radius a in the plane z ¼ 0 and then refracted into a second isotropic medium of refractive index n2 P n1 through a plane interface at z ¼ z0 . Here, z ¼ zf is the distance from the aperture plane z ¼ 0 to the geometrical focal point in the second medium. For the special case in which n1 ! n2 , the focus of the wave would be at z ¼ z1 > z0 .
n1
x 0 x0 x0 x ¼ n2 ; z z0 z0
uI ¼
1 i
z expðikRÞ 0 0 dx dy : ui R kR A
ð7Þ
Next, we modify (7) in order to account for the fact that we are focusing through an interface from an isotropic medium with refractive index n1 into a second isotropic medium with refractive index n2 . To that end, we consider the contribution from one source point Pðx0 ; y0 ; 0Þ inside the aperture to the field at the observation point Pðx; y; zÞ in the second medium. The source at Pðx0 ; y0 ; 0Þ emits a spherical wave, which we can decompose into an angular spectrum of plane waves, and the main contribution to the field at Pðx; y; zÞ will be that particular plane wave emitted by the source point Pðx0 ; y0 ; 0Þ, which after refraction is directed towards Pðx; y; zÞ. This implies that the main contribution to the field at Pðx; y; zÞ from the point source at Pðx0 ; y0 ; 0Þ is obtained by replacing ½expðikRÞ=ðkRÞ by ½expðik1 R1 þ k2 R2 Þ=ðk1 R1 þ k2 R2 Þ. By adding the contributions from all source points Pðx0 ; y0 ; 0Þ to the total field at Pðx; y; zÞ, we obtain from (7)
1 uI ¼ i
Z Z
z expðik1 R1 þ k2 R2 Þ 0 0 dx dy ; Tu R k 1 R1 þ k 2 R2 A i
ð8Þ
where T is the transmission coefficient, R1 and R2 are the distances from Pðx0 ; y0 ; 0Þ to the refraction point P 0 ðx0 ; y0 ; z0 Þ at the interface and from P0 ðx0 ; y0 ; z0 Þ to Pðx; y; zÞ, respectively, given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx0 x0 Þ2 þ ðy0 y0 Þ2 þ z20 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ ðz z0 Þ2 : R1 ¼
1 k1 R1 þ k2 R2 k1 z0 þ ðz z0 Þ ¼ k1 Z; nr
nr ¼
n 2 k2 ¼ : n 1 k1 ð10Þ
ð16Þ
ðz z0 Þðx0 xÞ ðz z0 Þðy0 yÞ ; y0 x ¼ ; nr Z nr Z z0 ðx0 xÞ z0 ðy0 yÞ ; y0 y0 ¼ : x0 x0 ¼ Z Z
x0 x ¼
ð17Þ ð18Þ
2.1.2. Fresnel approximation We now apply the Fresnel approximation to obtain from (9), (12), (17) and (18)
R1 z0 þ
ðx0 x0 Þ2 þ ðy0 y0 Þ2 z0 ¼ z0 þ 2 ½ðx0 xÞ2 þ ðy0 yÞ2 ; 2z0 2Z
R2 z z0 þ
ð19Þ
ðx0 x0 Þ2 þ ðy0 y0 Þ2 z z0 ¼ z z0 þ 2 2 ½ðx0 xÞ2 þ ðy0 yÞ2 ; 2ðz z0 Þ 2nr Z ð20Þ
x02 þ y02 : R1;0 z1 þ 2z1
ð21Þ
Thus, we get
x2 þ y 2 R1 þ nr R2 R1;0 z0 z1 þ nr ðz z0 Þ þ 2Z x02 þ y02 1 1 xx0 þ yy0 : þ 2 Z z1 Z 0
x ¼ at cos /;
y0 ¼ at sin /;
ð22Þ
x ¼ r cos b;
y ¼ r sin b;
ð23Þ
a2 expðik1 wÞ ik1 z1 Z Z 1 Z 2p 2 1 2 t tdt; expfiv t cosð/ bÞgd/ exp i u 1 þ nr 0 0 2 ð24Þ
where
w ¼ z0 z1 þ nr ðz z0 Þ þ r 2 =2Z;
ð11Þ
where we have used the paraxial approximation in the final step, and where R1;0 is the distance from Pðx0 ; y0 ; 0Þ to Pf ð0; 0; z1 Þ, i.e.
R1;0
z0 y þ ðz z0 Þy0 =nr ; Z
0
Taking into account that each source point Pðx ; y ; 0Þ inside the aperture emits a converging spherical, which would be focused at P f ð0; 0; z1 Þ in the absence of the interface at z ¼ z0 , we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x02 þ y02 þ z21 :
y0 ¼
and hence
uI ¼
exp ðik1 R1;0 Þ expðik1 R1;0 Þ ; R1;0 z1
z0 x þ ðz z0 Þx0 =nr ; Z
to obtain from (13), (22), and (23)
0
ui ¼
ð15Þ
For a circular aperture of radius a, we let
ð9Þ
2.1.1. Paraxial approximation In the paraxial approximation, we have
z 1; R
y 0 x0 y x ¼ n2 0 ; z z0 z0
which give
x0 ¼ Z Z
n1
ð12Þ
v ¼
2p a r; k1 Z
¼ u
2p a2 Z z1 : k1 z1 Z ð25Þ
Finally, using
J 0 ðxÞ ¼
1 2p
Z 2p 0
eix cosð/bÞ d/;
ð26Þ
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we have from (24)
At the focal point ð0; zf Þ, both (35) and (36) give
expðik1 wÞ pa 2 2 i k1 z1 Z 1 þ nr 2
uI ¼
Z
1
0
1 2 t tdt; J 0 ðv tÞ exp i u 2
ð27Þ
Ið0; zf Þ ¼ I0 ¼
2 pa2 1 þ nr k1 z21
2 ð37Þ
:
so that the intensity I ¼ juI j2 becomes equal to the result in (1). 2.1.3. Fresnel number The Fresnel number N 1 for focusing in a single medium with wavelength k1 , aperture radius a, and focal distance z1 is defined by
N1 ¼
ð28Þ
2
a ; k2 ½zf þ z0 ðnr 1Þ
ð29Þ
where zf is the distance from the aperture plane z ¼ 0 to the geometrical focal point in the second medium. From elementary geometrical considerations and the paraxial form of Snell’s law it follows that:
zf ¼ z0 þ nr ðz1 z0 Þ;
ð30Þ
zf z1 ¼ ðnr 1Þðz1 z0 Þ:
ð31Þ
Thus, if nr > 1, then zf z1 > 0, implying that the geometrical focal plane z ¼ zf is displaced away from the interface compared to its position at z ¼ z1 for nr ¼ 1. Conversely, if nr < 1, then zf z1 < 0, and the geometrical focus is displaced towards the interface compared to its position at z ¼ z1 for nr ¼ 1. Upon substitution from (30) in (29), we have
a2 a2 ¼ ¼ N1 ¼ N; k2 nr z1 k1 z1
ð32Þ
Z z1 z1 p N 1 ¼ 2 ; Z Z
z1 u ¼1 ; 2p N Z
ð33Þ
so that (1) may be written
Z 1 2 2 2 u 2 pa 1 2 2 t tdt : Iðr; zÞ ¼ 1 J 0 ðv tÞ exp i u 2 2p N 1 þ nr k 1 z 1 2 0
2
ð34Þ For observation points on the r ¼ 0 axis, (34) yields
2 u 2 =4Þ; Ið0; zÞ ¼ I0 1 sinc ðu 2pN
2 pa2 I0 ¼ 1 þ nr k1 z21
;
ð35Þ
2.1.4. Focal plane intensity f Þ ¼ z1 , so that (4) gives From (2) and (30) it follows that Zðz ðzf Þ ¼ 0. Thus, (34) gives the familiar Airy diffraction pattern for u the intensity distribution in the focal plane, i.e.
2 pa2 Iðr; zf Þ ¼ 1 þ nr k1 z21
2 2J 1 ðv f Þ2 v ; f
ð39Þ
so that
Z j N ¼ : z1 N 2j
ð40Þ
From (2) and (30) it follows that:
nr ðZ z1 Þ ¼ z zf ;
ð41Þ
so that
zj zf Z j z1 Zj N ¼ nr ¼ nr 1 ¼ nr 1 ; N 2j z1 z1 z1
ð42Þ
where we have used (40) in the last step. Thus, we have
zj zf 2jnr ¼ : z1 N 2j
ð43Þ
2p a v f ¼ r: k1 z1
z z1 2j ¼ ; N 2j z1
ð44Þ
in agreement with equation (12.64a) in [15]. 2.1.6. Focusing in a single medium As a check, we now specialize to the case of focusing in a single medium. Then we have n2 ¼ n1 ¼ n; nr ¼ 1, and hence k1 ¼ k2 ¼ k, ¼ z, while (3), (4), and (37) yield so that (2) gives Z
2p a r; k z
! u0 ¼ 2pN u
z z1 ; z
I0 ¼
pa2 kz21
2 :
ð45Þ
Thus, as expected, for focusing in a single medium (34) and (35) reduce to equations (12.66a) and (12.59d), respectively in [15]. 3. Focal shift and intensity increase on focusing through an interface Letting
u x¼ ; 4
b¼
2
pN
;
ð46Þ
the axial intensity in (35) becomes
2
where sincðxÞ ¼ sinðxÞ=x.
j z1 2j N 2j u ¼1 ; ¼1 ¼ N N 2p N Z j
v ! v 0 ¼
where we have used k2 ¼ k1 =nr . In terms of the Fresnel number we have
¼ 2pN u
ð38Þ
In the special case of focusing in a single medium, so that nr ! 1 and zf ! z1 , (43) gives
which gives
N2 ¼
j ¼ 4jp ðj ¼ 1; 2; . . .Þ: u From (38) and the last expression in (33), we have
a2 : k1 z1
On focusing through a plane interface, as illustrated in Fig. 1, the Fresnel number does not change. This can be explained as follows. Upon refraction through the interface the exit pupil (i.e. the image of the aperture through the interface) will have the same size as the aperture, but will be moved away from the interface by a distance equal to z0 ðnr 1Þ if nr > 1 and moved towards the interface by the distance z0 ð1 nr Þ if nr < 1. Thus, after refraction the Fresnel number becomes
N2 ¼
2.1.5. Axial zeros According to (35), the axial intensity has zeros when
ð36Þ
Ið0; zÞ ¼ I0 ½ð1 þ bxÞsincðxÞ2 ;
ð47Þ
where I0 ¼ Ið0; zf Þ is given in (37). The point of maximum intensity along the axis is determined from the equation
dIð0; zÞ dI dx ¼ ¼ 0: dz dx dz
ð48Þ
From (4) and (46) it follows that:
dx pa2 ; ¼ dz 2nr k1 Z 2
ð49Þ
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which is different from zero, so that the solution of (48) must satisfy dI ¼ 0. From (47) we have dx
pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffi d dI 1 ¼ 2 Ið0; zÞ I0 þ b sin x dx dx x 1 1 ¼ 2I0 2 sin x þ þ b cos x ¼ 0; x x
ð50Þ
which gives
tanðxÞ ¼ xð1 þ bxÞ ¼ 0:
ð51Þ
Thus, the point of maximum axial intensity is found by determining the points of intersection between y ¼ tanðxÞ and y ¼ xð1 þ bxÞ. The principal or greatest maximum occurs for values of x in the range 0 < x < p=2 [3]. 3.1. Focal shift
Table 1 Relative intensity increase DI/I0 from (57) after obtaining Xm numerically from (51) (0 < xm < p/2) and relative focal shift Dz/zf with Dz from (55) and zf = z1 nr (from (2) with z0 = 0). Fresnel number, N
b ¼ 2=pN
xm
DI=I0
Dz=zf
0.6710 0.7592 0.8526 1.0040 1.3010 1.6016 2.0830 2.7427 3.5969 4.5396 6.1518 7.4800 9.5397 13.1648
0.9488 0.8385 0.7466 0.6364 0.4894 0.3975 0.3056 0.2321 0.1770 0.1402 0.1035 0.0851 0.0667 0.0484
1.1986 1.1577 1.1163 1.1055 0.9433 0.8482 0.7241 0.5969 0.4816 0.3945 0.2993 0.2490 0.1971 0.1439
1.7589 1.4308 1.1177 0.8993 0.5733 0.3983 0.2484 0.1495 0.0895 0.0571 0.0315 0.0215 0.0133 0.0070
0.5321 0.4926 0.4546 0.4016 0.3158 0.2521 0.1812 0.1217 0.0785 0.0524 0.0300 0.0208 0.0130 0.0069
Let us denote the root of (51) corresponding to the principal coordinates by maximum by xm , and the corresponding z and Z zm and Z m , respectively. From (33) and (46) we have
m z1 u ¼1 ¼ 1 þ bxm ; 2pN Z m
ð52Þ
which gives
Z m ¼
z1 1 ¼ z0 þ ðzm z0 Þ; nr 1 þ bxm
ð53Þ
where we have used (2) in the final step. Solving (53) for zm , we find the maximum intensity to be located at
zm ¼ z0 ð1 nr Þ þ
z1 nr : 1 þ bxm
ð54Þ
From (30) and (54) we find the focal shift Dz to be
Dz ¼ zm zf ¼ z1 nr
bxm ; 1 þ bxm
b¼
2
pN
ð55Þ
:
As expected, we note from (55) that the focal shift always is negative, i.e. the shift is towards the aperture. 3.2. Relative intensity increase The maximum axial intensity follows from (47) with z ¼ zm , corresponding to x ¼ xm . Thus
Ið0; zm Þ ¼ I0 ½ð1 þ bxm Þsincðxm Þ2 ;
ð56Þ
where I0 is the intensity at the geometrical focal point given in (37). The relative intensity increase becomes
DI I m I 0 2 ¼ ¼ ½ð1 þ bxm Þsincðxm Þ 1; I0 I0
Im ¼ Ið0; zm Þ:
ð57Þ
4. Numerical results Table 1 shows the values xm obtained from (51) ð0 < xm < p=2Þ for different values of the Fresnel number N and the corresponding values of b ¼ p2N. Also shown in Table 1 is the relative intensity increase DI=I0 given by (57) as well as the quantity Dz=z1 nr ¼ bxm =ð1 þ bxm Þ (cf. (55)), which is the focal shift Dz relative to the focal distance zf ¼ z1 nr that is obtained when the aperture lies on the interface so that z0 ¼ 0. Fig. 2 shows the relative focal shift Dz=zf for different Fresnel numbers N. Dz was obtained from (55) after determining xm numerically from (51) ð0 < xm < p=2Þ. We varied the Fresnel number by letting the aperture radius a and the wavelength k1 of the incident wave be fixed ða ¼ 0:5 mm; k1 ¼ 0:633 lmÞ and changing the val-
Fig. 2. Relative focal shift Dz=zf for the focusing geometry in Fig. 1 versus the Fresnel number N ¼ a2 =k1 z1 . Here, k1 is the wavelength in the medium with refractive index n1 and zf is the distances from aperture plane z ¼ 0 to the focal plane. a ¼ 0:5 m; k1 ¼ 0:633 lm; nr ¼ n2 =n1 ¼ 1:5; z0 ¼ 10 mm, and 50 mm 6 z1 6 590 mm.
ues of z1 from 50 mm to 590 mm. The distance between the aperture plane and the interface was z0 ¼ 10 mm, and the relative refractive index was nr ¼ 1:5. It follows from Fig. 2 that the focal shift is less than 3% for Fresnel numbers higher than N ¼ 6, and that it increases rapidly as N decreases below about N ¼ 2:5. When N ¼ 2, the focal shift is nearly 18%, and it is larger than 40% for N < 1. For the same focusing geometries as in Fig. 2, the relative increase in intensity DI=I0 is shown in Fig. 3 as a function of the Fresnel number N. DI=I0 was obtained from (57) after determining xm numerically from (51) (0 < xm < p=2). It follows from Fig. 3 that the ratio DI=I0 increases as N decreases, that it is less than 3% for Fresnel numbers higher than N ¼ 6, and that it increases rapidly as N decreases below about N ¼ 2:5. When N ¼ 2, the relative intensity increase is nearly 24%, and it is larger than 100% for N < 1. Figs. 4–7 show normalised axial intensities Ið0; zÞ=I0 (from (35)) against z zf [with zf from (30)] for Fresnel numbers of N ¼ 0:6694; N ¼ 0:9873; N ¼ 3:949, and N ¼ 7:899, respectively. The focusing geometries were similar to those in Figs. 2 and 3 (a ¼ 0:5 mm; k1 ¼ 0:633 lm; z0 ¼ 10 mm, and nr ¼ 1:5), and the values of z1 were 590, 400, 100, and 50 mm in Figs. 4–7, respectively. From the plots in Figs. 4–7 we find the maxima to be at zm ¼ 408:4 mm for N ¼ 0:6694 (Fig. 4), zm ¼ 351:5 mm for
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Fig. 3. Relative intensity increase DI=I0 ¼ ðIm I0 Þ=I0 for the focusing geometry in Figs. 1 and 2 versus Fresnel number N ¼ a2 =k1 z1 . Here, Im ¼ Ið0; zm Þ is the maximum axial intensity and I0 ¼ Ið0; zf Þ is the intensity at the focus. Fig. 6. Axial intensity distribution for the focusing geometry in Fig. 1 with the same parameters as in Fig. 4, except for z1 ¼ 100 mm, giving N ¼ a2 =k1 z1 ¼ 3:949.
Fig. 4. Axial intensity distribution for the focusing geometry in Fig. 1 with a ¼ 0:5 mm; nr ¼ n2 =n1 ¼ 1:5; z0 ¼ 10 mm; k1 ¼ 0:633 lm, and z1 ¼ 590 mm, giving N ¼ a2 =k1 z1 ¼ 0:6694. Fig. 7. Axial intensity distribution for the focusing geometry in Fig. 1 with the same parameters as in Fig. 4, except for z1 ¼ 50 mm, giving N ¼ a2 =k1 z1 ¼ 7:899.
Fig. 5. Axial intensity distribution for the focusing geometry in Fig. 1 with the same parameters as in Fig. 4, except for z1 ¼ 400 mm, giving N ¼ a2 =k1 z1 ¼ 0:9873.
N ¼ 0:9873 (Fig. 5), zm ¼ 135:0 mm for N ¼ 3:949 (Fig. 6), and zm ¼ 68:6 mm for N ¼ 7:899 (Fig. 7). The corresponding values for zm obtained from (54) after determining xm numerically from (51) ð0 < xm < p=2Þ were zm ¼ 408:4 mm for N ¼ 0:6694; zm ¼ 351:4 mm for N ¼ 0:9873; zm ¼ 134:9 mm for N ¼ 3:949, and zm ¼ 68:6 mm for N ¼ 7:899. Thus, there is very good agreement between the maximum positions determined from the plots in Figs. 4–7 and the corresponding maximum positions obtained from (54) after determining xm numerically from (51) ð0 < xm < p=2Þ. From the plots in Figs. 4–7 we find the first two axial zeros to be at z ¼ 217 mm and z ¼ 122 mm for N ¼ 0:6694 (Fig. 4), z ¼ 193:4 mm and z ¼ 113; 8 mm for N ¼ 0:9873 (Fig. 5), first three axial zeros z ¼ 94; 6 mm; z ¼ 69; 6 mm and z ¼ 54; 6 mm for N ¼ 3:949 (Fig. 6) and z ¼ 54:9 mm; z ¼ 44:8 mm and z ¼ 37:7 mm for N ¼ 7:899 (Fig. 7). The corresponding values for zj obtained from (42) were precisely the same to three significant figures. Thus, there is very good agreement between the axial zeros determined from the plots in Figs. 4–7 and the corresponding axial positions obtained from (42).
J.J. Stamnes, D. Velauthapillai / Optics Communications 282 (2009) 2286–2291
5. Conclusions We have derived explicit formulas for the focal shift and the intensity increase obtained on focusing at low Fresnel numbers through a plane interface between two isotropic media, and shown that these formulas give good agreement with numerical results obtained from plots of the axial intensity distributions. Also, we have shown that for the special case in which the two isotropic media have the same refractive index, the derived formulas are in complete agreement with previous results presented for focusing in a single medium. Finally, we have derived explicit formulas for the zeroes of the axial intensity distribution, and shown that they give good agreement with numerical results obtained from plots of the axial intensity distributions. Our results may be of importance in optical trapping experiments, where focused light interaction takes place through one or more surfaces. Also, they may be of importance in second-harmonic generation of light. Although this is a non-linear effect, focused light is used to obtain high intensities, and a linear theory
2291
of focusing may provide a good first-order estimate of the intensity of the focused field.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J.H. Erkkila, M.E. Rogers, J. Opt. Soc. Am. 71 (1981) 904. J.J. Stamnes, B. Spjelkavik, Opt. Commun. 40 (1981) 81. Y. Li, E. Wolf, Opt. Commun. 39 (1981) 211. E. Wolf, Y. Li, Opt. Commun. 39 (1981) 205. Y. Li, Optik 64 (1983) 207. Y. Li, J. Opt. Soc. Am. 72 (1982) 770. Y. Li, H. Platzer, Opt. Acta 30 (1983) 1621. H. Ling, S.W. Lee, J. Opt. Soc. Am. A 1 (1984) 965. V. Dhayalan, J.J. Stamnes, Pure Appl. Opt. 6 (1997) 347. J.J. Stamnes, G.S. Sithambaranathan, M. Jain, J.K. Lotsberg, V. Dhayalan, Opt. Commun. 226 (2003) 107. Y. Li, J. Opt. Soc. Am. 22 (2005) 68. Y. Li, J. Opt. Soc. Am. 22 (2005) 77. C.J.R. Sheppard, P. Török, J. Opt. Soc. Am. A 20 (2003) 2156. Y. Li, J. Opt. Soc. Am. 20 (2003) 234. J.J. Stamnes, Waves in Focal Regions, Adam Hilger, Bristol and Boston, 1986.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Focal shifts on focusing through a plane interface Jakob J. Stamnes a, Dhayalan Velauthapillai b,* a b
Department of Physics and Technology, University of Bergen, Allégaten 55, N–5007 Bergen, Norway Department of Engineering, University College of Bergen, Postboks 7030, Nygardstangen, N–5020 Bergen, Norway
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 30 August 2008 Received in revised form 20 February 2009 Accepted 2 March 2009
We derive explicit formulas for the focal shift, the intensity increase, and the axial zeros obtained on focusing at low Fresnel numbers through a plane interface between two isotropic media, and demonstrate their validity through comparisons with numerical plots of axial intensity distributions. Also, we show that for the special case in which the two isotropic media have the same refractive index, these formulas are in complete agreement with previous results presented for focusing in a single medium. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
where
In the 1980s large discrepancies were observed between the results of the Debye and Kirchhoff theories for focusing of scalar waves [1–6]. It was shown experimentally that the Debye theory is not adequate for calculating intensity distributions around the focus when a converging spherical wave is diffracted through a circular aperture at low Fresnel numbers [7]. Focal shift phenomena were observed also for focusing of electromagnetic waves [8–12]. The previous works on scalar waves [1–7,13,14] were concerned with focal shifts obtained on focusing in a single, homogeneous, isotropic medium, whereas here we consider focusing of scalar waves through a plane interface between two different homogeneous, isotropic media. 2. Focusing through a plane interface between two isotropic media Consider the three-dimensional focusing geometry in Fig. 1, where a scalar converging spherical wave is diffracted through a circular aperture of radius a in the plane z ¼ 0 in an isotropic medium with refractive index n1 . The diffracted field is then transmitted through a plane interface at z ¼ z0 separating the isotropic medium of the incident wave from another isotropic medium with refractive index n2 . If the second medium had the same refractive index as the first medium, the focus of the wave would be at ð0; 0; z1 Þ, where z1 > z0 . Within the paraxial approximation the intensity of the focused field is given by [10]
Iðr; zÞ ¼
2 pa2 1 þ nr Zz1 k1
2 Z 2
0
1
2 1 2 t tdt ; J 0 ðv tÞ exp i u 2
* Corresponding author. Tel.: +47 55587711. E-mail address: [email protected] (D. Velauthapillai). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.03.004
1 n2 Z ¼ z0 þ ðz z0 Þ; nr ¼ ; nr n1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p a v ¼ r; r ¼ x2 þ y2 ; k1 Z 2p a ðZ z1 Þ: k1 z1 Z
ð2Þ ð3Þ
2
¼ u
ð4Þ
2.1. Derivation The derivation of (1) in [10] is quite complicated since it involves focusing of an electromagnetic wave into a biaxial medium. Another rather complicated derivation for the focusing of an electromagnetic wave into an isotropic medium is given in section 16.2 in [15]. The complications arise because the derivations are based on exact solutions of the electromagnetic problem. Therefore, we now give a simplified derivation, whose validity is restricted to paraxial geometries and scalar fields, but which provides considerable physical insight. To that end, we start with the first Rayleigh– Sommerfeld diffraction formula for the diffracted field uI obtained upon diffraction through an aperture in the plane z ¼ 0 in a single, isotropic medium:
uI ¼
1 2p
Z Z
ui A
@ expðikRÞ 0 0 dx dy : @z R
ð5Þ
Here, ui is the field in the aperture plane z ¼ 0; A is the aperture area, k is the wave number, and R is the distance from an integration point Pðx0 ; y0 ; 0Þ inside the aperture and the observation point Pðx; y; zÞ, i.e.
ð1Þ R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x0 Þ2 þ ðy y0 Þ2 þ z2 :
ð6Þ
Carrying out the differentiation in (5), and assuming that kR 1, we get
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Substituting from (10)–(12) into (8), we obtain
uI ¼
1 2 1 1 þ nr ik1 Zz
Z Z
0
0
expðik1 ðR1 þ nr R2 R1;0 Þdx dy ;
ð13Þ
A
where 2=ð1 þ nr Þ is the transmission coefficient T at normal incidence. The refraction point P0 ðx0 ; y0 ; z0 Þ at the interface follows from Snell’s law:
n1 sin h1 ¼ n2 sin h2 ;
ð14Þ
where h1 is the angle between the line P0 P0 and the interface normal, and h2 is the angle between the line P 0 P and the interface normal. In the paraxial approximation, Snell’s law becomes Fig. 1. A converging spherical wave in an isotropic medium of refractive index n1 is diffracted through a circular aperture of radius a in the plane z ¼ 0 and then refracted into a second isotropic medium of refractive index n2 P n1 through a plane interface at z ¼ z0 . Here, z ¼ zf is the distance from the aperture plane z ¼ 0 to the geometrical focal point in the second medium. For the special case in which n1 ! n2 , the focus of the wave would be at z ¼ z1 > z0 .
n1
x 0 x0 x0 x ¼ n2 ; z z0 z0
uI ¼
1 i
z expðikRÞ 0 0 dx dy : ui R kR A
ð7Þ
Next, we modify (7) in order to account for the fact that we are focusing through an interface from an isotropic medium with refractive index n1 into a second isotropic medium with refractive index n2 . To that end, we consider the contribution from one source point Pðx0 ; y0 ; 0Þ inside the aperture to the field at the observation point Pðx; y; zÞ in the second medium. The source at Pðx0 ; y0 ; 0Þ emits a spherical wave, which we can decompose into an angular spectrum of plane waves, and the main contribution to the field at Pðx; y; zÞ will be that particular plane wave emitted by the source point Pðx0 ; y0 ; 0Þ, which after refraction is directed towards Pðx; y; zÞ. This implies that the main contribution to the field at Pðx; y; zÞ from the point source at Pðx0 ; y0 ; 0Þ is obtained by replacing ½expðikRÞ=ðkRÞ by ½expðik1 R1 þ k2 R2 Þ=ðk1 R1 þ k2 R2 Þ. By adding the contributions from all source points Pðx0 ; y0 ; 0Þ to the total field at Pðx; y; zÞ, we obtain from (7)
1 uI ¼ i
Z Z
z expðik1 R1 þ k2 R2 Þ 0 0 dx dy ; Tu R k 1 R1 þ k 2 R2 A i
ð8Þ
where T is the transmission coefficient, R1 and R2 are the distances from Pðx0 ; y0 ; 0Þ to the refraction point P 0 ðx0 ; y0 ; z0 Þ at the interface and from P0 ðx0 ; y0 ; z0 Þ to Pðx; y; zÞ, respectively, given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx0 x0 Þ2 þ ðy0 y0 Þ2 þ z20 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ ðz z0 Þ2 : R1 ¼
1 k1 R1 þ k2 R2 k1 z0 þ ðz z0 Þ ¼ k1 Z; nr
nr ¼
n 2 k2 ¼ : n 1 k1 ð10Þ
ð16Þ
ðz z0 Þðx0 xÞ ðz z0 Þðy0 yÞ ; y0 x ¼ ; nr Z nr Z z0 ðx0 xÞ z0 ðy0 yÞ ; y0 y0 ¼ : x0 x0 ¼ Z Z
x0 x ¼
ð17Þ ð18Þ
2.1.2. Fresnel approximation We now apply the Fresnel approximation to obtain from (9), (12), (17) and (18)
R1 z0 þ
ðx0 x0 Þ2 þ ðy0 y0 Þ2 z0 ¼ z0 þ 2 ½ðx0 xÞ2 þ ðy0 yÞ2 ; 2z0 2Z
R2 z z0 þ
ð19Þ
ðx0 x0 Þ2 þ ðy0 y0 Þ2 z z0 ¼ z z0 þ 2 2 ½ðx0 xÞ2 þ ðy0 yÞ2 ; 2ðz z0 Þ 2nr Z ð20Þ
x02 þ y02 : R1;0 z1 þ 2z1
ð21Þ
Thus, we get
x2 þ y 2 R1 þ nr R2 R1;0 z0 z1 þ nr ðz z0 Þ þ 2Z x02 þ y02 1 1 xx0 þ yy0 : þ 2 Z z1 Z 0
x ¼ at cos /;
y0 ¼ at sin /;
ð22Þ
x ¼ r cos b;
y ¼ r sin b;
ð23Þ
a2 expðik1 wÞ ik1 z1 Z Z 1 Z 2p 2 1 2 t tdt; expfiv t cosð/ bÞgd/ exp i u 1 þ nr 0 0 2 ð24Þ
where
w ¼ z0 z1 þ nr ðz z0 Þ þ r 2 =2Z;
ð11Þ
where we have used the paraxial approximation in the final step, and where R1;0 is the distance from Pðx0 ; y0 ; 0Þ to Pf ð0; 0; z1 Þ, i.e.
R1;0
z0 y þ ðz z0 Þy0 =nr ; Z
0
Taking into account that each source point Pðx ; y ; 0Þ inside the aperture emits a converging spherical, which would be focused at P f ð0; 0; z1 Þ in the absence of the interface at z ¼ z0 , we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x02 þ y02 þ z21 :
y0 ¼
and hence
uI ¼
exp ðik1 R1;0 Þ expðik1 R1;0 Þ ; R1;0 z1
z0 x þ ðz z0 Þx0 =nr ; Z
to obtain from (13), (22), and (23)
0
ui ¼
ð15Þ
For a circular aperture of radius a, we let
ð9Þ
2.1.1. Paraxial approximation In the paraxial approximation, we have
z 1; R
y 0 x0 y x ¼ n2 0 ; z z0 z0
which give
x0 ¼ Z Z
n1
ð12Þ
v ¼
2p a r; k1 Z
¼ u
2p a2 Z z1 : k1 z1 Z ð25Þ
Finally, using
J 0 ðxÞ ¼
1 2p
Z 2p 0
eix cosð/bÞ d/;
ð26Þ
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we have from (24)
At the focal point ð0; zf Þ, both (35) and (36) give
expðik1 wÞ pa 2 2 i k1 z1 Z 1 þ nr 2
uI ¼
Z
1
0
1 2 t tdt; J 0 ðv tÞ exp i u 2
ð27Þ
Ið0; zf Þ ¼ I0 ¼
2 pa2 1 þ nr k1 z21
2 ð37Þ
:
so that the intensity I ¼ juI j2 becomes equal to the result in (1). 2.1.3. Fresnel number The Fresnel number N 1 for focusing in a single medium with wavelength k1 , aperture radius a, and focal distance z1 is defined by
N1 ¼
ð28Þ
2
a ; k2 ½zf þ z0 ðnr 1Þ
ð29Þ
where zf is the distance from the aperture plane z ¼ 0 to the geometrical focal point in the second medium. From elementary geometrical considerations and the paraxial form of Snell’s law it follows that:
zf ¼ z0 þ nr ðz1 z0 Þ;
ð30Þ
zf z1 ¼ ðnr 1Þðz1 z0 Þ:
ð31Þ
Thus, if nr > 1, then zf z1 > 0, implying that the geometrical focal plane z ¼ zf is displaced away from the interface compared to its position at z ¼ z1 for nr ¼ 1. Conversely, if nr < 1, then zf z1 < 0, and the geometrical focus is displaced towards the interface compared to its position at z ¼ z1 for nr ¼ 1. Upon substitution from (30) in (29), we have
a2 a2 ¼ ¼ N1 ¼ N; k2 nr z1 k1 z1
ð32Þ
Z z1 z1 p N 1 ¼ 2 ; Z Z
z1 u ¼1 ; 2p N Z
ð33Þ
so that (1) may be written
Z 1 2 2 2 u 2 pa 1 2 2 t tdt : Iðr; zÞ ¼ 1 J 0 ðv tÞ exp i u 2 2p N 1 þ nr k 1 z 1 2 0
2
ð34Þ For observation points on the r ¼ 0 axis, (34) yields
2 u 2 =4Þ; Ið0; zÞ ¼ I0 1 sinc ðu 2pN
2 pa2 I0 ¼ 1 þ nr k1 z21
;
ð35Þ
2.1.4. Focal plane intensity f Þ ¼ z1 , so that (4) gives From (2) and (30) it follows that Zðz ðzf Þ ¼ 0. Thus, (34) gives the familiar Airy diffraction pattern for u the intensity distribution in the focal plane, i.e.
2 pa2 Iðr; zf Þ ¼ 1 þ nr k1 z21
2 2J 1 ðv f Þ2 v ; f
ð39Þ
so that
Z j N ¼ : z1 N 2j
ð40Þ
From (2) and (30) it follows that:
nr ðZ z1 Þ ¼ z zf ;
ð41Þ
so that
zj zf Z j z1 Zj N ¼ nr ¼ nr 1 ¼ nr 1 ; N 2j z1 z1 z1
ð42Þ
where we have used (40) in the last step. Thus, we have
zj zf 2jnr ¼ : z1 N 2j
ð43Þ
2p a v f ¼ r: k1 z1
z z1 2j ¼ ; N 2j z1
ð44Þ
in agreement with equation (12.64a) in [15]. 2.1.6. Focusing in a single medium As a check, we now specialize to the case of focusing in a single medium. Then we have n2 ¼ n1 ¼ n; nr ¼ 1, and hence k1 ¼ k2 ¼ k, ¼ z, while (3), (4), and (37) yield so that (2) gives Z
2p a r; k z
! u0 ¼ 2pN u
z z1 ; z
I0 ¼
pa2 kz21
2 :
ð45Þ
Thus, as expected, for focusing in a single medium (34) and (35) reduce to equations (12.66a) and (12.59d), respectively in [15]. 3. Focal shift and intensity increase on focusing through an interface Letting
u x¼ ; 4
b¼
2
pN
;
ð46Þ
the axial intensity in (35) becomes
2
where sincðxÞ ¼ sinðxÞ=x.
j z1 2j N 2j u ¼1 ; ¼1 ¼ N N 2p N Z j
v ! v 0 ¼
where we have used k2 ¼ k1 =nr . In terms of the Fresnel number we have
¼ 2pN u
ð38Þ
In the special case of focusing in a single medium, so that nr ! 1 and zf ! z1 , (43) gives
which gives
N2 ¼
j ¼ 4jp ðj ¼ 1; 2; . . .Þ: u From (38) and the last expression in (33), we have
a2 : k1 z1
On focusing through a plane interface, as illustrated in Fig. 1, the Fresnel number does not change. This can be explained as follows. Upon refraction through the interface the exit pupil (i.e. the image of the aperture through the interface) will have the same size as the aperture, but will be moved away from the interface by a distance equal to z0 ðnr 1Þ if nr > 1 and moved towards the interface by the distance z0 ð1 nr Þ if nr < 1. Thus, after refraction the Fresnel number becomes
N2 ¼
2.1.5. Axial zeros According to (35), the axial intensity has zeros when
ð36Þ
Ið0; zÞ ¼ I0 ½ð1 þ bxÞsincðxÞ2 ;
ð47Þ
where I0 ¼ Ið0; zf Þ is given in (37). The point of maximum intensity along the axis is determined from the equation
dIð0; zÞ dI dx ¼ ¼ 0: dz dx dz
ð48Þ
From (4) and (46) it follows that:
dx pa2 ; ¼ dz 2nr k1 Z 2
ð49Þ
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which is different from zero, so that the solution of (48) must satisfy dI ¼ 0. From (47) we have dx
pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffi d dI 1 ¼ 2 Ið0; zÞ I0 þ b sin x dx dx x 1 1 ¼ 2I0 2 sin x þ þ b cos x ¼ 0; x x
ð50Þ
which gives
tanðxÞ ¼ xð1 þ bxÞ ¼ 0:
ð51Þ
Thus, the point of maximum axial intensity is found by determining the points of intersection between y ¼ tanðxÞ and y ¼ xð1 þ bxÞ. The principal or greatest maximum occurs for values of x in the range 0 < x < p=2 [3]. 3.1. Focal shift
Table 1 Relative intensity increase DI/I0 from (57) after obtaining Xm numerically from (51) (0 < xm < p/2) and relative focal shift Dz/zf with Dz from (55) and zf = z1 nr (from (2) with z0 = 0). Fresnel number, N
b ¼ 2=pN
xm
DI=I0
Dz=zf
0.6710 0.7592 0.8526 1.0040 1.3010 1.6016 2.0830 2.7427 3.5969 4.5396 6.1518 7.4800 9.5397 13.1648
0.9488 0.8385 0.7466 0.6364 0.4894 0.3975 0.3056 0.2321 0.1770 0.1402 0.1035 0.0851 0.0667 0.0484
1.1986 1.1577 1.1163 1.1055 0.9433 0.8482 0.7241 0.5969 0.4816 0.3945 0.2993 0.2490 0.1971 0.1439
1.7589 1.4308 1.1177 0.8993 0.5733 0.3983 0.2484 0.1495 0.0895 0.0571 0.0315 0.0215 0.0133 0.0070
0.5321 0.4926 0.4546 0.4016 0.3158 0.2521 0.1812 0.1217 0.0785 0.0524 0.0300 0.0208 0.0130 0.0069
Let us denote the root of (51) corresponding to the principal coordinates by maximum by xm , and the corresponding z and Z zm and Z m , respectively. From (33) and (46) we have
m z1 u ¼1 ¼ 1 þ bxm ; 2pN Z m
ð52Þ
which gives
Z m ¼
z1 1 ¼ z0 þ ðzm z0 Þ; nr 1 þ bxm
ð53Þ
where we have used (2) in the final step. Solving (53) for zm , we find the maximum intensity to be located at
zm ¼ z0 ð1 nr Þ þ
z1 nr : 1 þ bxm
ð54Þ
From (30) and (54) we find the focal shift Dz to be
Dz ¼ zm zf ¼ z1 nr
bxm ; 1 þ bxm
b¼
2
pN
ð55Þ
:
As expected, we note from (55) that the focal shift always is negative, i.e. the shift is towards the aperture. 3.2. Relative intensity increase The maximum axial intensity follows from (47) with z ¼ zm , corresponding to x ¼ xm . Thus
Ið0; zm Þ ¼ I0 ½ð1 þ bxm Þsincðxm Þ2 ;
ð56Þ
where I0 is the intensity at the geometrical focal point given in (37). The relative intensity increase becomes
DI I m I 0 2 ¼ ¼ ½ð1 þ bxm Þsincðxm Þ 1; I0 I0
Im ¼ Ið0; zm Þ:
ð57Þ
4. Numerical results Table 1 shows the values xm obtained from (51) ð0 < xm < p=2Þ for different values of the Fresnel number N and the corresponding values of b ¼ p2N. Also shown in Table 1 is the relative intensity increase DI=I0 given by (57) as well as the quantity Dz=z1 nr ¼ bxm =ð1 þ bxm Þ (cf. (55)), which is the focal shift Dz relative to the focal distance zf ¼ z1 nr that is obtained when the aperture lies on the interface so that z0 ¼ 0. Fig. 2 shows the relative focal shift Dz=zf for different Fresnel numbers N. Dz was obtained from (55) after determining xm numerically from (51) ð0 < xm < p=2Þ. We varied the Fresnel number by letting the aperture radius a and the wavelength k1 of the incident wave be fixed ða ¼ 0:5 mm; k1 ¼ 0:633 lmÞ and changing the val-
Fig. 2. Relative focal shift Dz=zf for the focusing geometry in Fig. 1 versus the Fresnel number N ¼ a2 =k1 z1 . Here, k1 is the wavelength in the medium with refractive index n1 and zf is the distances from aperture plane z ¼ 0 to the focal plane. a ¼ 0:5 m; k1 ¼ 0:633 lm; nr ¼ n2 =n1 ¼ 1:5; z0 ¼ 10 mm, and 50 mm 6 z1 6 590 mm.
ues of z1 from 50 mm to 590 mm. The distance between the aperture plane and the interface was z0 ¼ 10 mm, and the relative refractive index was nr ¼ 1:5. It follows from Fig. 2 that the focal shift is less than 3% for Fresnel numbers higher than N ¼ 6, and that it increases rapidly as N decreases below about N ¼ 2:5. When N ¼ 2, the focal shift is nearly 18%, and it is larger than 40% for N < 1. For the same focusing geometries as in Fig. 2, the relative increase in intensity DI=I0 is shown in Fig. 3 as a function of the Fresnel number N. DI=I0 was obtained from (57) after determining xm numerically from (51) (0 < xm < p=2). It follows from Fig. 3 that the ratio DI=I0 increases as N decreases, that it is less than 3% for Fresnel numbers higher than N ¼ 6, and that it increases rapidly as N decreases below about N ¼ 2:5. When N ¼ 2, the relative intensity increase is nearly 24%, and it is larger than 100% for N < 1. Figs. 4–7 show normalised axial intensities Ið0; zÞ=I0 (from (35)) against z zf [with zf from (30)] for Fresnel numbers of N ¼ 0:6694; N ¼ 0:9873; N ¼ 3:949, and N ¼ 7:899, respectively. The focusing geometries were similar to those in Figs. 2 and 3 (a ¼ 0:5 mm; k1 ¼ 0:633 lm; z0 ¼ 10 mm, and nr ¼ 1:5), and the values of z1 were 590, 400, 100, and 50 mm in Figs. 4–7, respectively. From the plots in Figs. 4–7 we find the maxima to be at zm ¼ 408:4 mm for N ¼ 0:6694 (Fig. 4), zm ¼ 351:5 mm for
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Fig. 3. Relative intensity increase DI=I0 ¼ ðIm I0 Þ=I0 for the focusing geometry in Figs. 1 and 2 versus Fresnel number N ¼ a2 =k1 z1 . Here, Im ¼ Ið0; zm Þ is the maximum axial intensity and I0 ¼ Ið0; zf Þ is the intensity at the focus. Fig. 6. Axial intensity distribution for the focusing geometry in Fig. 1 with the same parameters as in Fig. 4, except for z1 ¼ 100 mm, giving N ¼ a2 =k1 z1 ¼ 3:949.
Fig. 4. Axial intensity distribution for the focusing geometry in Fig. 1 with a ¼ 0:5 mm; nr ¼ n2 =n1 ¼ 1:5; z0 ¼ 10 mm; k1 ¼ 0:633 lm, and z1 ¼ 590 mm, giving N ¼ a2 =k1 z1 ¼ 0:6694. Fig. 7. Axial intensity distribution for the focusing geometry in Fig. 1 with the same parameters as in Fig. 4, except for z1 ¼ 50 mm, giving N ¼ a2 =k1 z1 ¼ 7:899.
Fig. 5. Axial intensity distribution for the focusing geometry in Fig. 1 with the same parameters as in Fig. 4, except for z1 ¼ 400 mm, giving N ¼ a2 =k1 z1 ¼ 0:9873.
N ¼ 0:9873 (Fig. 5), zm ¼ 135:0 mm for N ¼ 3:949 (Fig. 6), and zm ¼ 68:6 mm for N ¼ 7:899 (Fig. 7). The corresponding values for zm obtained from (54) after determining xm numerically from (51) ð0 < xm < p=2Þ were zm ¼ 408:4 mm for N ¼ 0:6694; zm ¼ 351:4 mm for N ¼ 0:9873; zm ¼ 134:9 mm for N ¼ 3:949, and zm ¼ 68:6 mm for N ¼ 7:899. Thus, there is very good agreement between the maximum positions determined from the plots in Figs. 4–7 and the corresponding maximum positions obtained from (54) after determining xm numerically from (51) ð0 < xm < p=2Þ. From the plots in Figs. 4–7 we find the first two axial zeros to be at z ¼ 217 mm and z ¼ 122 mm for N ¼ 0:6694 (Fig. 4), z ¼ 193:4 mm and z ¼ 113; 8 mm for N ¼ 0:9873 (Fig. 5), first three axial zeros z ¼ 94; 6 mm; z ¼ 69; 6 mm and z ¼ 54; 6 mm for N ¼ 3:949 (Fig. 6) and z ¼ 54:9 mm; z ¼ 44:8 mm and z ¼ 37:7 mm for N ¼ 7:899 (Fig. 7). The corresponding values for zj obtained from (42) were precisely the same to three significant figures. Thus, there is very good agreement between the axial zeros determined from the plots in Figs. 4–7 and the corresponding axial positions obtained from (42).
J.J. Stamnes, D. Velauthapillai / Optics Communications 282 (2009) 2286–2291
5. Conclusions We have derived explicit formulas for the focal shift and the intensity increase obtained on focusing at low Fresnel numbers through a plane interface between two isotropic media, and shown that these formulas give good agreement with numerical results obtained from plots of the axial intensity distributions. Also, we have shown that for the special case in which the two isotropic media have the same refractive index, the derived formulas are in complete agreement with previous results presented for focusing in a single medium. Finally, we have derived explicit formulas for the zeroes of the axial intensity distribution, and shown that they give good agreement with numerical results obtained from plots of the axial intensity distributions. Our results may be of importance in optical trapping experiments, where focused light interaction takes place through one or more surfaces. Also, they may be of importance in second-harmonic generation of light. Although this is a non-linear effect, focused light is used to obtain high intensities, and a linear theory
2291
of focusing may provide a good first-order estimate of the intensity of the focused field.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J.H. Erkkila, M.E. Rogers, J. Opt. Soc. Am. 71 (1981) 904. J.J. Stamnes, B. Spjelkavik, Opt. Commun. 40 (1981) 81. Y. Li, E. Wolf, Opt. Commun. 39 (1981) 211. E. Wolf, Y. Li, Opt. Commun. 39 (1981) 205. Y. Li, Optik 64 (1983) 207. Y. Li, J. Opt. Soc. Am. 72 (1982) 770. Y. Li, H. Platzer, Opt. Acta 30 (1983) 1621. H. Ling, S.W. Lee, J. Opt. Soc. Am. A 1 (1984) 965. V. Dhayalan, J.J. Stamnes, Pure Appl. Opt. 6 (1997) 347. J.J. Stamnes, G.S. Sithambaranathan, M. Jain, J.K. Lotsberg, V. Dhayalan, Opt. Commun. 226 (2003) 107. Y. Li, J. Opt. Soc. Am. 22 (2005) 68. Y. Li, J. Opt. Soc. Am. 22 (2005) 77. C.J.R. Sheppard, P. Török, J. Opt. Soc. Am. A 20 (2003) 2156. Y. Li, J. Opt. Soc. Am. 20 (2003) 234. J.J. Stamnes, Waves in Focal Regions, Adam Hilger, Bristol and Boston, 1986.