User:Pranav Rathi/Notebook/OT/2010/12/10/Olympus Water Immersion Specs

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|          | The best theoretical value for the spotsize is needed to be multiplied by pi (121 X pi = 380 nm; i do not know why?). This value is seems to be closer to the experimental value. Also a quick way to estimate the spotsize: '''ω<sub>o</sub>=λ<sub>0</sub>/2n=400nm''' (if we also divide it by '''π''' we will get 127nm).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |          | The best theoretical value for the spotsize is needed to be multiplied by pi (121 X pi = 380 nm; i do not know why?). This value is seems to be closer to the experimental value. Also a quick way to estimate the spotsize: '''ω<sub>o</sub>=λ<sub>0</sub>/2n=400nm''' (if we also divide it by '''π''' we will get 127nm).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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| -        | The results of the beam waist can be experimentally verified by directly measuring the spot size, but the process is rather cumbersome and hardly interesting[https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B5jWjFYiQdkfM2QxN2U4NTYtYTNmYi00OWNhLWJhOTEtNjgwMTk3Zjc4ZWVi&hl=en]https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B5jWjFYiQdkfZTI0MGUwYzMtYzcxNi00YTVjLWI1NzgtY2FlNTJlMGI3YTk2&hl=en. The results given here are the best in terms of the theoretical limits of the aberration free optics. Experimentally we suffer on number of bases: one of them is experimental-setup itself; because of aberrations introduces by index mismatch among water, oil and glass interfaces, which also introduce the multiple reflection. And we should also not forget that the focal plane of the objective is not infinitely thin in z-direction, which implies the out of focus rays degrading the overall image reducing the resolution and degrading the spotsize. | +        | The results of the beam waist can be experimentally verified by directly measuring the spot size, but the process is rather cumbersome and hardly interesting[https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B5jWjFYiQdkfM2QxN2U4NTYtYTNmYi00OWNhLWJhOTEtNjgwMTk3Zjc4ZWVi&hl=en][https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B5jWjFYiQdkfZTI0MGUwYzMtYzcxNi00YTVjLWI1NzgtY2FlNTJlMGI3YTk2&hl=en]. The results given here are the best in terms of the theoretical limits of the aberration free optics. Experimentally we suffer on number of bases: one of them is experimental-setup itself; because of aberrations introduces by index mismatch among water, oil and glass interfaces, which also introduce the multiple reflection. And we should also not forget that the focal plane of the objective is not infinitely thin in z-direction, which implies the out of focus rays degrading the overall image reducing the resolution and degrading the spotsize. |
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Water immersion objective details

WE are using Olympus UPLANSAPO (UIS 2) water immersion IR objective for DNA stretching and unzipping. The detail specifications of the objective can be found in the link:[1] Other specification are as follows;

Mag                         60X
Wavelength                  1064
NA                          1.2
medium                      water
Max ray angle               64.5degrees
f #                         26.5
Effective FL in water       1.5 to 1.6mm (distance between the focal spot and the exit aperture surface)
Entrance aperture diameter  8.5mm
Exit aperture diameter      6.6mm
Working distance            .28mm
Cover glass correction      .13 to .21 (we use .15)

Resolution and achievable spot size

The resolution and the spotsize (beam waist) presented here is in the theoretical limits; we cannot achieve better than this. Resolution and spotsize are diffraction limited and to reach these limits our optics has to be perfect; no aberrations and other artifacts. Since our optics is not perfect and very clean we can hardly reach these limits in real life; definitely the resolution and spotsize in real is worse than the numbers presented here. A good way to do a quick estimation of the resolution (diameter of the airy disk) is that its 1/3 of a wavelength λ=.580 μm; λ/3*n= 145 nm. Since we do all our experiments in water we will have take index of water in account (n=1.33). I am ignoring the NA of the condenser in the calculations.

• Wavelength of the visible light λv = .590μm.
• Wavelength of the IR λIR = 1.064μm.
• Diameter of the incident beam at the exit pupil (D=2ω'o.)=6500μm. (ω'o is the incident beam waist)
• Focal length of the objective f=1500μm.
• Angular resolution inside water:
$\mathrm{\theta} = \sin^{-1}\frac{1.22\lambda_v}{nD}= 8.1e^{-5}rad$
• Spatial resolution in water;
$\mathrm{\Delta l} = \frac{1.22f\lambda_v}{nD}= 122nm$

Since we are not too sure of the focal length of the objective, so i derived the resolution formula in terms of the numerical aperture NA (the math can be seen through this link[2]).

• Resolution in terms of NA:
$\mathrm{\Delta l} = \frac{1.22\lambda_v}{2n}\sqrt{(\frac{n}{NA})^{2}-1}= 127nm$
• Now the minimum spotsize (beam waist ωo) can be calculated using the same formula where Δl=2ωo. But this time for infrared wavelength
• Minimum beam waist;
$\mathrm{\omega_o} = \frac{1.22f\lambda_{IR}}{2nD}= 112nm$

and the beam diameter 224nm.

• In terms of NA:
$\mathrm{\omega_o} = \frac{1.22\lambda_{IR}}{4n}\sqrt{(\frac{n}{NA})^{2}-1}= 116nm$

and the beam diameter 232nm.

I also used Gaussian approach to calculate the spot size and the results are not much different, which proves that either approach is right.

• With Gaussian approach;
$\mathrm{\omega_o} = \frac{\lambda_{IR}}{\pi n}\sqrt{(\frac{n}{NA})^{2}-1}= 121nm$

and the beam diameter 242nm.

Results are not much different and either approach is right. BUT zeroth order Gaussian approximation (paraxial) is not correct for high NA (for high convergent beams) objective lens. All above approaches are based on paraxial approximation (when diffraction angle is less than 30o). For high convergent beam like here this approximation is not valid any more and that's why calculated spotsize here is an orders of magnitude less. For better approximation one have to use spotsize equation given by electromagnetic field theory with higher order Gaussian corrections[3][4][5][6]. As we reach higher order corrections we get better and better theoretical results. But still it will be in the range of above calculate spotsize.

The best theoretical value for the spotsize is needed to be multiplied by pi (121 X pi = 380 nm; i do not know why?). This value is seems to be closer to the experimental value. Also a quick way to estimate the spotsize: ωo=λ0/2n=400nm (if we also divide it by π we will get 127nm).

The results of the beam waist can be experimentally verified by directly measuring the spot size, but the process is rather cumbersome and hardly interesting[7][8]. The results given here are the best in terms of the theoretical limits of the aberration free optics. Experimentally we suffer on number of bases: one of them is experimental-setup itself; because of aberrations introduces by index mismatch among water, oil and glass interfaces, which also introduce the multiple reflection. And we should also not forget that the focal plane of the objective is not infinitely thin in z-direction, which implies the out of focus rays degrading the overall image reducing the resolution and degrading the spotsize.
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