Supplementary material
To quantify the improvement produced by the addition of the plano-concave lens,
we performed theoretical analyses.
Figure S1A is a schematic drawing of our optics
for detecting small intensity changes in the rat cerebral cortex stained with a voltage
sensitive dye (RH-414) using the photodiode array described above. The shape of the
photoactive area in this photodiode array is a circle with a radius of 25.5 mm (for details
see [1]).
To obtain a high intensity real image, we used a tandem lens assembly consisting
of two photographic lens complexes, both designed for a 35 mm single-lens reflex
camera. One of them is a standard lens (focal length, 50 mm) used as the objective
lens, and the other is a telescopic lens (focal length, 300 mm) used for the imaging lens.
In Fig. S1A, when the object, OA, was placed at the focal plane of the objective lens,
rays from any point on the object form a parallel beam, and the angle of the beam with
respect to the optical axis is uniquely defined by the distance from the focal point, O, on
the object.
The imaging lens forms an enlarged real image of the object at the focal
plane of the imaging lens, the magnifying power of which is determined by the ratio of
focal lengths of the objective to the imaging lenses.
In the present optics, the focal
lengths are 50 mm and 300 mm, respectively, and the magnifying power is x6.
Using Fig. S1B, we calculated the position of the real image of the object, which
lies ℓ lower and parallel to the focal plane of the objective lens.
In this figure, we
simplified the optics by assuming that the distance between the two lens complexes (D
in Fig. S1A) and the thickness of each lens complex were zero.
We used the thin lens
formula for the objective lens:
1
1
1
+
=
,
a1
b1
50
(1)
where a1 is the distance from the object to the objective lens, b1 is the distance from the
objective lens to its virtual image and 50 (mm) is the focal length of the objective lens.
From Fig. S1B, a1 is given by:
a1 = 50 + ℓ ,
then Eq. (1) is rewritten:
(2)
1
1
1
+
=
(50 + ℓ)
b1
50
.
(3)
Similarly, it follows from the thin lens formula for the imaging lens:
1
1
1
+
=
a2
b2
300 ,
(4)
where a2 is the distance from the imaging lens to the virtual image of the objective lens,
b2 is the distance from the imaging lens to its real image, and 300 (mm) is the focal
length of the imaging lens.
From Fig. S1B, a2 and b2 are given by:
a2 = -b1 ,
(5)
b2 = 300 - ℓ' ,
(6)
then Eq. (4) is rewritten:
-
1
1
1
+
=
.
b1
(300 - ℓ') 300
(7)
By combining Eqs. (3) and (7), we obtain:
1
1
7
.
+
=
(50 + ℓ) (300 - ℓ')
300
(8)
Hereafter, we evaluated the effect of the plano-concave lens inserted between the
focal plane and the surface of the rat cerebral cortex (Figs. S1C and S1D).
We
assumed that the rat cortical surface was a sphere with a radius of 20 mm [2].
Therefore, we used a plano-concave lens with a radius of curvature of 20 mm so that it
aligned with the curved surface of the cerebral cortex.
As described above, the
photoactive area of the detector is a circle with a radius of 25.5 mm, and the quotient of
25.5 mm divided by the magnifying power, 6, is 4.25 mm. Thus, only points within
4.25 mm from the optical axis on the object can make a real image in the photoactive
area.
Using the assumptions mentioned above, we focused the tandem lens system so
as to have a real image of O in the image plane. P1 in Fig. S1C, which lies 4.25 mm
away from the optical axis, is projected at the marginal circumference in the active
window of the photodetector.
unfocused site on the object.
Hence, P1 is considered to be situated at the most
In Fig. S1C, P1 lies on the edge of the arc formed by a
radius of 20 mm and the central angle Ψ.
Thus:
4.25 = 20 sinΨ .
(9)
Rearrangement of Eq. (9) gives:
sinΨ =
4.25
= 0.2125 ,
20
(10)
so that:
cosΨ ≈ 0.9772 .
(11)
First, we calculated ℓ’ in Fig. S1B under the condition without the plano-concave
lens inserted in the light path.
In this condition, the tandem lens system is focused so
that O, which lies at the focal point of the objective lens, is projected at O’ on the image
plane, ℓ1 (the length of P1P2 in Fig. S1C), which corresponds to ℓ in Fig. S1B. We
designate ℓ’ calculated in this manner as ℓ1’. From Fig. S1C, ℓ1 is given by:
ℓ1 = 20 (1 – cosΨ) .
(12)
By inserting cosΨ calculated from Eq. (11) into Eq. (12), we obtain:
ℓ1 ≈ 0.457 (mm) .
(13)
Then, from Eqs. (8) and (13):
ℓ1' ≈ 15.46 (mm) .
(14)
Second, we calculated ℓ’ in Fig. S1B when the plano-concave lens was inserted into
the light path.
The plano-concave lens is made of BK-7 glass, its radius of curvature is
20 mm and its center thickness is 3.0 mm.
In this condition (Fig. S1D), after focusing
the tandem lens system so as to have the real image, O, on the image plane, it appeared
as if light comes from O2, which lies on the optical axis 50 mm lower than the objective
lens. This physical phenomenon is based on the fact that light from O passes through
the glass, which has a relatively high refractive index (RI) in the length of O1OS. OSO2
is the apparent depth and it is well known that the length of an apparent depth (OSO2) is
equal to the RI as follows:
OSO2 =
where n is the RI.
OSO
,
n
(15)
In Fig. S1D, the length of OSO is equal to the center thickness (3
mm) of the plano-concave lens, and the RI (n) of BK-7 glass is approximately 1.5 in the
wavelength range that we used. Hence, from these values and Eq. (15), we obtain:
OSO2 = 2 (mm) .
(16)
Here, we can calculate the location of P3, where it appears as if light from P1 comes
from P1 and ℓ1 in Fig. S1D, which are the same as those in Fig. S1C.
In Fig. S1D,
according to Snell’s law:
sinθ1
=n ,
sinθ2
(17)
where n is the RI, θ1 is the angle P3SQ1 and θ2 is the angle P1SQ1. Since SQ1 is
parallel to the optical axis, the angle SCOS is a corresponding angle to the angle P3SQ1.
Hence, the angle SCOS is equal to θ1. For the angle SCOS in the triangle SCOS:
SOS
.
COS
(18)
COS = CO2 - OSO2 = 48 ,
(19)
SOS = QO1 = 4.25 - W ,
(20)
tanθ1 =
From Fig. S1D:
where W is the length of P1Q1.
tanθ1 =
Eq. (18) then becomes:
4.25 - W
.
48
(21)
Similarly, for the angle P1SQ1 in the triangle P1SQ1:
tanθ2 =
P1Q
W
,
=
SQ1
SQ1
(22)
SQ1 = OSO1 = 3 + ℓ1 .
(23)
Eq. (22) is then rewritten as:
tanθ2 =
W
.
3 + ℓ1
(24)
In Fig. S1D, when angles θ1 and θ2 are small, we can consider:
sinθ1 ≈ tanθ1, sinθ2 ≈ tanθ2 ,
(25)
Substituting Eq. (25) into Eq. (17), we obtain:
tanθ1
≈n.
tanθ2
(26)
When θ1 and θ2 are small angles, we can also consider that the line P1P3 is parallel to the
optical axis.
Therefore, using Eq. (21) and Eq. (24), and n ≈ 1.5 (as described above),
Eq. (26) becomes:
4.25 - W
48
≈ 1.5 ,
W
3 + ℓ1
(27)
Substituting ℓ1 from Eq. (13) in Eq. (27), W can be calculated as follows:
W ≈ 0.195 (mm) .
(28)
In Fig. S1D, for the angle P3SQ2 in the triangle P3SQ2:
tanθ1 =
W
.
2 + ℓ2
(29)
where ℓ2 is the length of P3P4. Combining Eqs. (21), (28) and (29), this can be written
as:
ℓ2 ≈ 0.305 (mm) .
(30)
In Fig. S1B, ℓ2 corresponds to ℓ. Here, in the same way as above, we designate ℓ’
calculated in this manner as ℓ2'.
Then, from Eq. (8) and Eq. (30):
ℓ2' ≈ 10.53 (mm) .
(31)
Hence, it is concluded that the length between the image plane and the real
image of P1, which is considered to be situated at the most unfocused site on the object,
is decreased from ℓ1' (15.46 mm) to ℓ2' (10.53 mm) by placement of the plano-concave
lens. Finally, this effect is evaluated to be an approximately 32% improvement in
comparison with the case where the space between the rat cerebral cortex and the
objective lens is air.
Since changing the focus by 300 μm led to a 50% reduction of
signal size [3], it is evident that the increase in the signal size that is yielded by
improving the focus contributes significantly to the increase in S/N.
References
1. Hama N, Ito S, Hirota A (2010) An improved multiple-site optical membrane
potential-recording system to obtain high-quality single sweep signals in intact rat
cerebral cortex. J Neurosci Methods 194:73-80
2. Paxinos
G, Watson C, Carrive P, Kirkcaldie M,
Ashwell
K
(2009)
Chemoarchitectonic Atlas of the Rat Brain, Second Edition. Academic Press,
Amsterdam, Tokyo
3. Salzberg BM, Grinvald A, Cohen LB, Davila HV, Ross WN (1977) Optical
recording of neuronal activity in an invertebrate central nervous system:
simultaneous monitoring of several neurons. J Neurophysiol 40:1281-1291
Figure S1
A: Schematic drawing of our currently used optics composed of a tandem lens system.
B: Simplified ray diagram of the relationship between two objects and their real image
in our optics. L indicates the position of both the objective and imaging lenses.
C:
Auxiliary figure for calculating ℓ1 value. D: Ray diagram with the plano-concave lens
(shaded) on the surface of the rat cerebral cortex that we assumed to be a sphere with a
radius of 20 mm. See details in Supplementary material.