# How do I calculate the integral for the point spread function of an optical system without the fresnel approximation?

For an object located at the $$(x_o, y_o)$$ plane, and a lens located at the $$(x, y)$$ plane, the image is produced at the $$(x_i, y_i)$$ plane. We can consider the image field $$U_i$$ to be the sum of the points of the object field $$U_o$$ with a weightage $$h$$. Written explicitly,

$$U_i(x_i, y_i) = \int^\infty_{-\infty}\int^\infty_{-\infty} h(x_i, y_i; x_o, y_o) U_o(x_o, y_o) \,dx_o\,dy_o\, .$$

We obtain an expression of $$h$$ by first considering a point object $$U_o(x_o, y_o) = \delta(x_o - x_p, y_o-y_p)$$. Meanwhile, on the RHS of the equation, we can use our knowledge of the propagation of a point source light field through a lens to get an expression for $$U_i$$. For a thorough derivation refer to Introduction To Fourier Optics by Goodman Chapt 5. The expression for $$h$$ is thus given by

$$h(x_i, y_i; x_o, y_o) \propto \int^\infty_{-\infty}\,dx\int^\infty_{-\infty}\,dy\, P(x, y) \, \\ \times \exp\left\{ -ik(n-1) \left[R_1 - R_2 + \sqrt{R_1 - x^2 + y^2} - \sqrt{R_2 - x^2 + y^2} \right] \right\}\\ \times \frac{exp\left\{ ik\sqrt{d_o^2 + (x-x_o)^2 + (y-y_o)^2} \right\}}{{d_o^2 + (x-x_o)^2 + (y-y_o)^2}}\\ \times \frac{exp\left\{ ik\sqrt{d_i^2 + (x_i-x)^2 + (y_i-y)^2} \right\}}{{d_i^2 + (x_i-x)^2 + (y_i-y)^2}}\,,$$

where $$n$$ is the refractive index of the lens, $$R_1$$ and $$R_2$$ are the radius of curvatures of each side of the lens. $$d_o/d_i$$ is the distance from the $$(x_o, y_o)/(x_i, y_i)$$ plane to the $$(x,y)$$ plane. $$k$$ is the wavenumber. All of which can just be treated as constants in the integral.

$$P(x,y)$$ is the pupil function, which basically adds limits to the infinite integrals. For a circular pupil of diameter $$l$$, we have $$P(x,y) = \frac{circ (\sqrt{x^2 + y^2})}{(l/2)}$$, the limits are changed from

$$\int_{-\infty}^\infty \, dx \int_{-\infty}^\infty\, dy \, P(x,y) \rightarrow \int_{-l/2}^{l/2}\, dx \int_{-\sqrt{(l/2)^2 - x^2}}^{\sqrt{(l/2)^2 - x^2}} \, dy\, .$$

There is a much simpler expression of $$h$$, in which you have to take certain approximations about the relative magnitude of the distance between the lens plane and the object/image plane, and the size of the object being imaged, known as the Fresnel and Fraunhofer approximations. These approximations lead to $$U_i$$ being a simple convolution of $$h$$ and $$U_o$$.

I am interested in studying the case where these approximations do not hold, and how that affects the image. I have tried to compute for $$U_i$$ numerically, however I found that to be too computationally draining. Thus I seek for an analytical solution for $$h$$.

How do I solve for the integral of $$h$$ for a given lens circular pupil of diameter $$l$$?

• try Huygens Principle--it does not use those approximations, although it has other problems. – user45664 Feb 24 at 17:13
• @user45664 , I believe I am using the Huygens principle in my workings. The wavelets in each propagation are the sum of spherical waves of the form $\exp (ikr)/r^2$. Could you enlighten me as to what problems the Huygens Principle imposes? Especially for a high NA microscope objective – Tian Feb 25 at 6:58
• Huygens Principle in its basic form has the problems of the wake and the backward wave. Also it used impulse type wavelets rather than sinusoidal ones. – user45664 Feb 25 at 16:39