2008 Semifinal Exam Part C 11 Optical Society of America Bonus Question Researchers have developed a lens made of liquid.The spherical lens consists of a droplet of transparent liquid resting on an electrically controllable surface.When the voltage of the surface is changed,the droplet itself changes shape;it either tries to "ball-up"more strongly or it becomes flatter.The figure below is a sketch of the liquid lens and several parameters that describe it,including the thickness of the lens (t),the radius of curvature of the top surface(R)and the contact angle(0),which represents the angle between the flat surface beneath the droplet and the tangent to the curved surface at the point of contact. R R a.When a certain voltage is applied,both the contact angle and lens thickness increase (and the lens becomes more curved).In this case,is the liquid attracted or repelled by the surface? b.Express the contact angle as a function of R and t. c.The total volume of the liquid lens is an important parameter because as the liquid lens changes shape, its volume is conserved.Calculate the volume of the lens as a function of R and t. d.Use your result to part(b)to eliminate the variable t from your expression for the volume and find V(R,). e.By changing the voltage on the control surface,the contact angle,6,can be changed,which in turn changes the focal length of the lens,f.The lensmaker's formula can be used to calculate the focal length and is given by 方-（忘） where niquid and nair are the refractive indices of the liquid in the lens and air around it,and R and R2 are the radii of curvature of the two surfaces of the lens.In figure 1.R is the curved face and R2 is the flat face.Use the lensmaker's formula to calculate the focal length of the lens in terms of the total volume of the liquid,the contact angle,and the relevant refractive indices. Sidenote:liquid lenses are interesting because they are electrically controllable,variable focus lenses that can be very compact.People are working on putting them into cell phone cameras for ultra- compact zoom lenses.For more information on this type of liquid lens,see T.Krupenkin,S.Yang, and P.Mach,"Tunable liquid microlens,"Appl.Phys.Lett.82,316-318(2003). Copyright C2008 American Association of Physics Teachers2008 Semifinal Exam Part C 11 Optical Society of America Bonus Question Researchers have developed a lens made of liquid. The spherical lens consists of a droplet of transparent liquid resting on an electrically controllable surface. When the voltage of the surface is changed, the droplet itself changes shape; it either tries to “ball-up” more strongly or it becomes flatter. The figure below is a sketch of the liquid lens and several parameters that describe it, including the thickness of the lens (t), the radius of curvature of the top surface (R) and the contact angle (θ), which represents the angle between the flat surface beneath the droplet and the tangent to the curved surface at the point of contact. R R t θ a. When a certain voltage is applied, both the contact angle and lens thickness increase (and the lens becomes more curved). In this case, is the liquid attracted or repelled by the surface? b. Express the contact angle as a function of R and t. c. The total volume of the liquid lens is an important parameter because as the liquid lens changes shape, its volume is conserved. Calculate the volume of the lens as a function of R and t. d. Use your result to part (b) to eliminate the variable t from your expression for the volume and find V (R, θ). e. By changing the voltage on the control surface, the contact angle, θ, can be changed, which in turn changes the focal length of the lens, f. The lensmaker’s formula can be used to calculate the focal length and is given by 1 f = (nliquid − nair)  1 R1 − 1 R2  , where nliquid and nair are the refractive indices of the liquid in the lens and air around it, and R1 and R2 are the radii of curvature of the two surfaces of the lens. In figure 1, R1 is the curved face and R2 is the flat face. Use the lensmaker’s formula to calculate the focal length of the lens in terms of the total volume of the liquid, the contact angle, and the relevant refractive indices. Sidenote: liquid lenses are interesting because they are electrically controllable, variable focus lenses that can be very compact. People are working on putting them into cell phone cameras for ultra￾compact zoom lenses. For more information on this type of liquid lens, see T. Krupenkin, S. Yang, and P. Mach, “Tunable liquid microlens,” Appl. Phys. Lett. 82, 316-318 (2003). Copyright c 2008 American Association of Physics Teachers