# Rapid Estimation of Lens Performance

This is a continuation from the previous tutorial - performance of representative lenses.

### Singlet

Figure 39 is a nomogram that allows quick estimation of the performance of a single refracting lens, with the stop at the lens, as a function of refractive index $$N$$, dispersion $$V$$, F-number, and field of view $$\theta$$.

• Chart A estimates the angular blur diameter $$\beta$$ resulting from a singlet with bending for minimum spherical aberration.
• The angular chromatic blur diameter is given by Chart B. The three rows of $$\text{FN}$$ values below the chart represent the angular blur diameter that contains the indicated percentage of the total energy.
• Chart C shows the blur diameter due to astigmatism.

Coma for a singlet bent for minimum spherical aberration with the stop at the lens is approximately

$\tag{50}\frac{\theta}{16\cdot(N+2)\cdot(\text{FN})^2}$

### Depth of Focus

The depth of focus of an optical system is expressed as the axial displacement that the image may experience before the resultant image blur becomes excessive.

Figure 40 shows the geometric relationship of the angular blur tolerance $$\Delta\theta$$ to the depth of focus $$\delta_{\pm}$$. If the entrance pupil diameter is $$D_\text{ep}$$ and the image distance is $$s_i$$, then the depth of focus is

$\tag{51}\delta_{\pm}=\frac{s_i^2\Delta\theta}{D_\text{ep}\pm{s_i\Delta\theta}}$

or when $$\delta\ll{s_i}$$, the depth of focus becomes

$\tag{52}\delta=\frac{s_i^2\Delta\theta}{D_\text{ep}}$

When $$s_i=f$$, then

$\tag{53}\delta=f\Delta\theta\text{FN}$

The depth of field is distance that the object may be moved without causing excessive image blur with a fixed image location. The distance at which a lens may be focused such that the depth of field extends to infinity is $$s_o=D_\text{ep}/\Delta\theta$$ and is called the hyperfocal distance.

If the lens system is diffraction-limited, then the depth of focus according to the Rayleigh criterion is given by

$\tag{54}\delta=\pm\frac{\lambda}{2n_i\sin^2u_i}$

### Diffraction-Limited Lenses

It is well known that the shape of the image irradiance of an incoherent, monochromatic point-source formed by an aberration-free, circularly-symmetric lens system is described by the Airy function

$\tag{55}E(r)=C_0\left[\frac{2J_1(kD_\text{ep}r/2)}{kD_\text{ep}r}\right]^2$

where $$J_1$$ is the first order Bessel function of the first kind, $$E_\text{ep}$$ is the diameter of the entrance pupil, $$k$$ is $$2\pi/\lambda$$, $$r$$ is the radial distance from the center of the image to the observation point, and $$C_0$$ is a scaling factor.

The angular radius $$\beta_\text{DL}$$ of the first dark ring of the image is $$1.22(\lambda/D_\text{ep})$$. A common measure for the resolution is Lord Rayleigh’s criterion that asserts that two point sources are just resolvable when the maximum of one Airy pattern coincides with the first dark ring of the second Airy pattern, i. e., an angular separation of $$\beta_\text{DL}$$.

Figure 41 presents a nomogram that can be used to make a rapid estimate of the diameter of angular or linear blur for a diffraction-limited system.

The modulation transfer function (MTF) at a specific wavelength $$\lambda$$ for a circular entrance pupil can be computed by

$\tag{56}\text{MTF}_{\lambda}(\Omega)=\frac{2}{\pi}[\arccos\Omega-\Omega\sqrt{1-\Omega^2}]\quad\text{for}\quad0\le\Omega\le1$

where $$\Omega$$ is the normalized spatial frequency ($$\nu/\nu_\text{co}$$) with the maximum or cut-off frequency $$\nu_\text{co}$$ being given by $$1/\lambda_o\text{FN}$$.

Should the source be polychromatic and the lens system be aberration-free, then the perfect-image irradiance distribution of a point source can be written as

$\tag{57}E(r)=C_1\int_0^{\infty}\tilde{\mathscr{R}}(\lambda)\left[\frac{2J_1(kD_\text{ep}r/2)}{kD_\text{ep}r}\right]^2\text{d}\lambda$

where $$\tilde{\mathscr{R}}(\lambda)$$ is the peak normalized spectral weighting factor and $$C_1$$ is a scaling factor.

A quick estimation of this ideal irradiance distribution can be made by invoking the central limit theorem to approximate this distribution by a Gaussian function, i. e.,

$\tag{58}E(r)\approx{C_2}e^{-(r^2/2\sigma^2)}$

where $$C_2$$ is a scaling constant and $$\sigma^2$$ is the estimated variance of the irradiance distribution.

When $$\tilde{\mathscr{R}}(\lambda)=1$$ in the spectral interval $$\lambda_S$$ to $$\lambda_L$$ and zero otherwise with $$\lambda_S\lt\lambda_L$$, an estimate of $$\sigma$$ can be written as

$\tag{59}\sigma=\frac{\mathscr{M}\lambda_L}{\pi{D_\text{ep}}}$

where $$\mathscr{M}=1.335-0.625b+0.25b^2-0.0465b^3$$ with $$b=(\lambda_L/\lambda_S)-1$$.

Should $$\tilde{\mathscr{R}}(\lambda)=\lambda/\lambda_L$$ in the spectral interval $$\lambda_S$$ to $$\lambda_L$$ and zero otherwise, which approximates the behavior of a quantum detector, $$\mathscr{M}=1.335-0.65b+0.385b^2-0.099b^3$$.

The Gaussian estimate residual error is less than a few percent for $$b=0.5$$ and remains useful even as $$b\rightarrow0$$.

Figure 42 contains plots of $$\mathscr{M}$$ for both cases of $$\tilde{\mathscr{R}}(\lambda)$$, where the abscissa is $$\lambda_L/\lambda_S$$.

A useful estimation of the modulation transfer function for this polychromatic lens system is given by

$\tag{60}\text{MTF}(\nu)\approx{e}^{-2(\pi\sigma\nu)^2}$

where $$\nu$$ is the spatial frequency.

This approximation overestimates the MTF somewhat at lower spatial frequencies, while being rather a close fit at medium and higher spatial frequencies.

The reason for this is that the central portion of the irradiance distribution is closely matched by the Gaussian approximation, while the irradiance estimation beyond several Airy radii begins to degrade, therefore impacting the lower spatial frequencies.

Nevertheless, this approximation can provide useful insight into expected performance limits.

The next tutorial introduces Gaussian analysis of afocal lenses