Finally, noting that , the power of the lens surface, we arrive at our goal: A formula that allows us to calculate the change in curvature from the image surface to the object surface as a function of the power of the lens surface. Since the image surface for lens 1 is the object surface for lens 2 (not shown in the figure), we can chain this calculation through several surfaces (just like the imaging equation). Applying this equation to the next surface, we get . Substituting for in the first equation, we get: , and for K surfaces: . If the initial object plane is flat (r0 = ∞), then we have the equation for the Petzval Curvature of the image plane: Petzval Curvature: This equation clearly shows that positive power surfaces cause the image plane to curve negatively (that is, with the center of curvature to the left of the surface), and negative power surfaces cause it to curve positively. Also, the only way that the image plane can be flat is if the surface powers (weighted by the indices on either side of the surface) sum to zero. It is equally clear (from the derivation) that this image plane curvature exists even for paraxial rays. We have chosen to ignore that in our development of ray ‘paraxial’ tracing, since our larger goal was to develop a simple mathematical model of a perfect imaging system. (It was in pursuit of this goal that we made the otherwise unjustified substitution of ray for ray -- so that the perfect model would also cover tangents angles finite height rays.) Implications of Petzval Curvature for lens design: In multiple lens systems that must have net positive power, the negative lenses can be situated so that they contribute relatively less (given their power) to the total power than the positive lenses. If we use the usual way of generating multiple-lens power formulas (tracing a ray parallel to the axis through the system), but leave the formula in terms of ray heights, instead of lens spacings, we get: By arranging the negative lenses where the parallel ray has a low height from the axis, we can minimize the negative contribution to the total power while balancing the Petzval sum. This is what is done in the Cooke Triplet style of camera lens: The image plane doesn’t have to be flat only flat enough that the exactly – Depth of Focus (DOF) of the optics includes the image plane:
Finally, noting that , the power of the lens surface, we arrive at our goal: A formula that allows us to calculate the change in curvature from the image surface to the object surface as a function of the power of the lens surface. Since the image surface for lens 1 is the object surface for lens 2 (not shown in the figure), we can chain this calculation through several surfaces (just like the imaging equation). Applying this equation to the next surface, we get . Substituting for in the first equation, we get: , and for K surfaces: . If the initial object plane is flat (r0 = ∞), then we have the equation for the Petzval Curvature of the image plane: Petzval Curvature: This equation clearly shows that positive power surfaces cause the image plane to curve negatively (that is, with the center of curvature to the left of the surface), and negative power surfaces cause it to curve positively. Also, the only way that the image plane can be flat is if the surface powers (weighted by the indices on either side of the surface) sum to zero. It is equally clear (from the derivation) that this image plane curvature exists even for paraxial rays. We have chosen to ignore that in our development of ‘paraxial’ ray tracing, since our larger goal was to develop a simple mathematical model of a perfect imaging system. (It was in pursuit of this goal that we made the otherwise unjustified substitution of ray tangents for ray angles -- so that the perfect model would also cover finite height rays.) Implications of Petzval Curvature for lens design: In multiple lens systems that must have net positive power, the negative lenses can be situated so that they contribute relatively less (given their power) to the total power than the positive lenses. If we use the usual way of generating multiple-lens power formulas (tracing a ray parallel to the axis through the system), but leave the formula in terms of ray heights, instead of lens spacings, we get: By arranging the negative lenses where the parallel ray has a low height from the axis, we can minimize the negative contribution to the total power while balancing the Petzval sum. This is what is done in the Cooke Triplet style of camera lens: The image plane doesn’t have to be exactly flat – only flat enough that the Depth of Focus (DOF) of the optics includes the image plane:
10)%20PetzvalTheorem%20(2-6-13).docx