An ideal imaging system delivers a perfect, converging, spherical wavefront where the optical path length (OPL) along all rays from a given object point to the corresponding point on the image are identical. The Strehl ratio is a common and easily computed figure of merit for describing image quality in near diffraction-limited systems. Here we derive an approximate expression for the Strehl ratio and compare with exact results for a few cases.

#### The Strehl Ratio

The peak brightness in a perfect image is designated \(I_0\). When aberrations are present in an optical system the wavefront converging towards the image is no longer spherical but distorted. Aberrations distort the image and reduce the peak brightness, \(I\), compared to the ideal case. The ratio of achieved image brightness relative to \( I_0\) is known as the Strehl ratio, \[ SR \triangleq \frac{I}{I_0} . \] The Strehl ratio is easy to compute and therefore a convenient measure of image quality.

#### The Marechal Approximation

In "Principles of Optics," Born & Wolf^{1} derive a Taylor series approximation to the on-axis intensity in the image plane for the case of small wavefront aberrations. The first-order term in this expansion for the Strehl ratio yields,

^{2}. Here \( \sigma_\phi^2 =(2\pi\; \Delta OPL / \lambda)^2 \) is the optical path length variance; \(\sigma_\phi\) is measured in radians. The quantity, \[ \Delta OPL^2 = \langle OPL^2\rangle - \langle OPL \rangle^2 , \]

is the physical wavefront variance; \(OPL \) is measured in units of length.

#### Sums of Random Phasors and the Extended Mareschal Approximation

The approximation for the Strehl ratio can be understood and extended by considering image formation as the interference from \(N\) sub-regions into which the converging wavefront has been divided. The electric field amplitude, \(E\), at the image plane is the vector sum of the corresponding \( N\) equal amplitude phasors (see Figure 1),

\[ E= \sum_{j=1}^N \exp(\mathbb{ i}\phi_j) ,\]where \(\phi_j=(2\pi/\lambda ) OPL_j\), and \( \mathbb{i} = \sqrt{-1}\).

If all \(N\) optical path lengths are identical, i.e., the wavefront is spherical, all the phases are the same and the RMS phase error is zero. Consequently, all the phasors line up coherently to give intensity, \( I_0 \propto E_0E_0^* = N^2\) at the location of the geometric image.

If the optical paths vary and the associated phase errors are zero mean, uncorrelated, and normally distributed then we can compute the electric field in terms of the expectation value of the complex phasor. From the definition of the expectation value, \( \langle \cdot \rangle\), \[ E=N \langle \exp(\mathbb{i} \phi) \rangle, \] and using the result ^{3} for a normally distributed quantity with zero mean \[ \langle \exp(\mathbb{i} \phi ) \rangle = \exp(-\sigma_\phi^2 /2), \] we have \[ E= N \exp(-\sigma_\phi^2/2) ,\] and \[ SR = \frac{I}{I_0} = \frac{EE^*}{N^2} = \exp(-\sigma_\phi^2) . \] The Strehl ratio quantifies the peak intensity of an image formed by a distorted or aberrated wavefront relative to the peak intensity of an unaberrated wave; this last equation is known as the extended Marechal approximation. In the radio astronomy literature the Strehl ratio is analogous to antenna gain and this result is known as the Ruze formula ^{4}.

#### Some Examples

Figure 2 shows some numerical diffraction calculations for an unobscured circular pupil (left hand column). The far field diffraction pattern (center and right hand columns) is computed using the Fraunhofer approximation implemented using fast Fourier transforms. The figure shows three rows with increasingly large wavefront errors. These errors are normally distributed with a mean of zero and an RMS that is listed at the bottom of each surface plot in the left column. The central column shows a false-color image of the diffraction pattern or point spread function (PSF) displayed using a logarithmic scale. The right hand column shows a 1-d plot of a horizonal line cut through the center of the image.

The first row of Figure 2 shows the aberration free result, where the PSF is an Airy function. The first and second Airy rings (4.7% and 1.6% of the peak) are easily identifiable in the false color image. In the second row normally distributed wavefront errors are included with an RMS of 1/8 of a wave. In the corresponding image the central intensity has decreasd by about a factor of two relative to the Airy function (the numerically computed Strehl ratio is 0.54). Only the first Airy ring is clearly identifable and numerous speckles at 1% of peak brightness are scattered across the image. In the third row the RMS is increased to 1/4 of a wave. The core of the Airy function persists, but it is now only a tenth of its original brightness (\(SR\) = 0.1) and speckles are pervasive.

Figure 2 considers pure Gaussian random aberrations, with no spatial correlation. Comparison of the numerically computed Strehl ratios shows that the extended Marechal approximation is exact under these circumstances.

Because surface polishing defects leads to spatially correlated errors and and optical misalignment tends to yield smoothly varying wavefront shapes, polynomial desciptions are commonly adopted to describe actual wavefront shapes. Figure 3 shows an example of a low order aberration. In this case we have chosen Zernike spherical aberration described by \[ W(\rho ) = C_{40} ( 6 \rho^4 - 6\rho^2 +1 ), \] where \(C_{40}\) is the Zernike polynomial coefficient determining the strength of the aberration and \(\rho \) is the radial pupil coordinate. Zernike spherical aberration is a balanced aberration where spherical aberration, \(\rho^4\), is balanced with defocus, \(\rho^2\), to minimize the resultant RMS wavefront error. The peak-to-valley amplitude of this aberration is \( 3C_{40}/2\) and the corresponding RMS wavefront error for this aberration is \( C_{40}/ \sqrt{5} \simeq 0.447 C_{40}\).

The wavefront surface plots in Figure 3 are labeled by the value of \(C_{40}\) and the exact Strehl ratio is listed on the image of the PSF in the central column. The PSF images and line plots show the charactertics of spherical aberration which decreases the peak of the PSF fills in the dark zones between the Airy rings thereby reducing overall contrast.

Figure 4 compares the exact results for the Strehl ratio of Zernike spherical aberration with the extended Marechal approximation. Unlike the case of Gaussian errors, the analytic formula is not exact. However, for values of \(C_{40} < 0.4 \), the approximation for the Strehl ratio is better than 10%.

The well-known Rayleigh \(\lambda/4\) criterion for "diffraction limited" perfomance is based on the observation that a quarter wave of spherical aberration reduces the Strehl ratio to \(0.8\). One quarter of a wave corresponds to \(C_{40} = 1/6\), which consultation of Fig. 4 shows corresponds to this Strehl ratio.

**Footnotes:**

^{1. Born, M., & Wolf, E., 1980, "Principles of Optics", Sixth Ed., \(\S\)9.1.3, p. 464, Pergammon Press ↩}

^{2. Marechal (1947, Rev. d'Opt., 26, 257) showed that \( S\simeq (1 - \sigma_\phi^2/2)^2\). The expression cited in Born and Wolf is the result achieved when terms in \(\sigma^4\) and higher are neglected. ↩}

^{3. The expectation value of \(\exp({\mathbb{i}\phi})\) for zero mean, normally distributed errors, \(\phi\), is ↩} \[ \langle \exp({\mathbb{i}\phi})\rangle = \int_{-\infty}^{\infty} \frac{ \exp{(\mathbb{i}\phi}) }{\sqrt{2\pi}\sigma_\phi} \exp{\left( -\frac{\phi^2}{2\sigma_\phi^2} \right) }\; d\phi = \exp{(-\sigma_\phi^2/2)}. \]

^{4. Ruze, J. 1966 Proc. IEEE, 54, 633 ↩}

© *Eikonal Optics*