# Trapping stiffness of optical tweezers

Sciencestd
I read in some articles that the force in optical tweezers can be written as: F=kx, with no minus because the force will increase as the distance increased and the particle moves to the source..., This I can understand, but what I can not understand if I make integral (it is conservative force) the potential will be: -0.5kx^2, which then is a not harmonic potential well... but in the same time I read in some articles that they took the potential as: 0.5kx^2 ...

Can anyone give me hints please..?!

Haborix
If you want us to help you resolve an apparent contradiction, then we need to have the sources that you think are conflicting.

• Sciencestd
Haborix
The first article is not being particularly careful as far as I can tell. The article says that it is a restoring force that acts to bring the particle back towards the center of the waveguide. Also, they only give the magnitude of the forces in their initial list, so the lack of a minus sign may just reflect that. In their schematic of the forces they have the y-coordinate increasing in a direction different from the arrow they give for the relevant force. If that's their convention for what constitutes positive y and a positive force, then there wouldn't be a minus sign. The second article uses what I would consider to be the standard form for a restoring force. Maybe the standard conventions are different in this field.

Summary: Article one is either just working with magnitudes or using a funny convention. Article two is, in my view, standard.

• Sciencestd
I read in some articles that the force in optical tweezers can be written as: F=kx, with no minus because the force will increase as the distance increased and the particle moves to the source..., <snip>

That is not what is written in the second source you provided- check the text just above and Eqn #2.

Edit- one reason for the minus sign is that the trap exerts a restoring force- the direction of applied force is opposite the direction of the trapped particle's displacement.

Sciencestd
That is not what is written in the second source you provided- check the text just above and Eqn #2.

Edit- one reason for the minus sign is that the trap exerts a restoring force- the direction of applied force is opposite the direction of the trapped particle's displacement.

Yes this is the question... in the first article they mentioned that it is F=kx, and in the second article they mentioned F=-kx. and thus the potential will have different sign in every case... which is not compatible!
I can agree more with the form F=kx as long as incearsing the distance to the source (high field intensity) will increase the force, but if we increase the distance far from the source then the force will decrease... But then the potential is not U=0.5kx^2 anymore but U=-0.5kx^2 which will have like a peak and not a well.

I can agree more with the form F=kx as long as incearsing the distance to the source (high field intensity) will increase the force, but if we increase the distance far from the source then the force will decrease... But then the potential is not U=0.5kx^2 anymore but U=-0.5kx^2 which will have like a peak and not a well.

If I understand your point, then my response is that modeling the trap as a harmonic potential (or a spring) is only intended for regions near the center of the trap- in other words, the restoring force term is a decent model for particles already held in the trap, not for particles 'falling into' the trap from a distance.

Plus, if the particle has a refractive index lower than the surrounding medium, the trap does indeed act as a 'repulsive spring'- Bessel beam traps can work this way, keeping a particle held in the low-intensity center.

Does that help?

• Sciencestd
Sciencestd
If I understand your point, then my response is that modeling the trap as a harmonic potential (or a spring) is only intended for regions near the center of the trap- in other words, the restoring force term is a decent model for particles already held in the trap, not for particles 'falling into' the trap from a distance.

Plus, if the particle has a refractive index lower than the surrounding medium, the trap does indeed act as a 'repulsive spring'- Bessel beam traps can work this way, keeping a particle held in the low-intensity center.

Does that help?

Your answer has a lot of sense... actually I can agree with it..
I explained it quite different:
positive sign because the force is increasing in the same direction of increasing the displacement toward the source. The linearity is valid for short distances. For long distance displacement if we equate equation F=kx with equation of gradient force we find that "x" takes the spatial form of the squared electric field and in any case "k" equals to the constants of the equation.
in case that the particle is already trapped in the potential well of the electromagnetic field, and if we consider that other opposite forces act on the particle such as dragging force by the solution flow due to evaporation or the scattering force, then we add minus to the equation and it turns to be F=-kx, the particle then will be like in harmonic oscillator...

I don't know how much it is correct or if it's correct at all...! Maybe you can share me your opinion please.

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