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

## Water immersion objective details

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

Mag | 60X |

Wavelength | 1064nm |

NA | 1.2 |

medium | water |

Max ray angle | 64.5degrees |

f # of eyepiece | 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 (spherical aberration control) | .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 λ
_{v}= .590μm. - Wavelength of the IR λ
_{IR}= 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:

- [math]\mathrm{\theta} = \sin^{-1}\frac{1.22\lambda_v}{nD}= 8.1e^{-5}rad[/math]

- Spatial resolution in water;

- [math]\mathrm{\Delta l} = \frac{1.22f\lambda_v}{nD}= 122nm[/math]

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:

- [math]\mathrm{\Delta l} = \frac{1.22\lambda_v}{2n}\sqrt{(\frac{n}{NA})^{2}-1}= 127nm[/math]

- Now the minimum spotsize (beam waist ω
_{o}) can be calculated using the same formula where Δl=2ω_{o}. But this time for infrared wavelength- Minimum beam waist;

- [math]\mathrm{\omega_o} = \frac{1.22f\lambda_{IR}}{2nD}= 112nm[/math]

and the beam diameter 224nm.

- In terms of NA:

- [math]\mathrm{\omega_o} = \frac{1.22\lambda_{IR}}{4n}\sqrt{(\frac{n}{NA})^{2}-1}= 116nm[/math]

and the beam diameter 232nm.

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

- With Gaussian approach;

- [math]\mathrm{\omega_o} = \frac{\lambda_{IR}}{\pi n}\sqrt{(\frac{n}{NA})^{2}-1}= 121nm[/math]

and the beam diameter 242nm.

- A LabView code is given in the link to calculate the parameters[3].

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 30^{o}). For high convergent beam like here this approximation (Gaussian paraxial approach) is not valid any more and that's why calculated spotsize here is an orders of magnitude less. A geometrical view of the problem is presented in the link:[4] For better approximation one have to use spotsize equation given by electromagnetic field theory with higher order Gaussian corrections[5][6][7][8]. 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; This value is much closer to the experimental value so i am taking it as the beamwaist inside water for my optical trap.**). This value seems to be closer to the experimental value and gives a Rayleigh range of 567nm. This figure is used to calculate the PSF (point spread function) for the optical trap, which will be used in caliberation for the sensitivity. Also a quick way to estimate the spotsize: **ω _{o}=λ_{0}/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[9][10][11]. 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.