What is a goodStrehlratio

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.

1) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley and Sons, 1977), pp. 718-21, 727, 736-7, 761.

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Strehlratio Zemax

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.

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.

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Strehlratio calculator

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Strehlratio MTF

Dec 31, 2014 — Suppose you want to frame an object of size a at distance l . Then, roughly, a/l = s/f where s is the sensor size and f is the focal length.

The force on a charged particle moving through a magnetic field is F = qv × B. If a test charge, q, moves with a velocity v through a region where a magnetic field is present, and it experiences a force F, we can define the magnetic field, B, as the vector that satisfies this equation. From this equation, we can see that the units of B are newton/(coulomb(meter/second)), or newton/(ampere·meter). This combination of units is called the tesla. The magnitude of the force is F = qvB sin θ. In the photograph above, the electrons are, of course, coming towards the front of the table. The magnetic dipole, μ, associated with an electric current flowing in a coil is μ = NiA, where N is the number of turns in the coil, i is the current (in amperes) and A is the cross-sectional area of the coil. For points along the axis of the coil, at a distance much greater than the radius of the coil, the magnitude of the magnetic field is B = (μ0/2π)(μ/x3), where μ0, the permeability constant, equals 4π × 10-7 tesla·meter/ampere. Its direction is given by the right-hand rule. If the current flow in the coil in the photograph is upwards in front and downwards in back, or clockwise if we face the right side of the coil, then the north pole is to the left, and the south pole is to the right, and the field lines at the CRT point to the left. (See the explanation for 68.13 -- Right-hand rule model.) With the magnetic field oriented this way, since q is negative, F points upward, which, as we can see from the spot on the CRT screen, is the direction in which the beam is being deflected.

Strehlformula

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.

You can also hold the coil edge-on, that is, with the plane of the coil intersecting the axis of the CRT. This is equivalent to showing the deflection caused by the electric field around a bundle of parallel wires. With the coil held horizontally, the field lines at the CRT are vertical, and the beam is deflected horizontally. Varying either the distance between the coil and the CRT, or the current flowing through the coil (by adjusting the voltage on the power supply), changes the magnitude of the beam deflection. Flipping the orientation of the coil 180° or reversing the current, of course, reverses the direction of deflection.

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.

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Not only was the cathode ray tube a useful research tool by itself, but in suitably modified and refined form, it was the heart of the oscilloscope (first known as the cathode ray oscilloscope), the television set and displays for computers and video equipment. While many such devices still contain cathode ray tubes, most now use liquid crystal or LED flat-panel displays instead.

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.

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.

In the apparatus used in this demonstration, the voltages on the cathode and the anode are, respectively, approximately -110 and +350 volts, so the electrons acquire a total kinetic energy of about 460 electron-volts by the time they exit the electron gun of the CRT. In joules, this energy is (460 eV)(1.602 × 10-19 J/eV) = 7.37 × 10-17 joules. Kinetic energy, K, equals (1/2)mv2, so v = √(2K/m), and v = √(2(7.37 × 10-17 J)/9.11 × 10-31 kg), or 1.27 ×107 m/s.

Strehlpronunciation

by HH Nasse · Cited by 22 — Modulation transfer function of a 50mm lens of 35mm format in the image center, measured at f/ 2 and f/ 5.6, for purposes of comparison the diffraction-limited.

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 \( C_{40}/ \sqrt{5} \simeq 0.447 C_{40}\).

Measurement with a Hall probe shows that with the coil oriented as in the photograph and current adjusted so that the beam is deflected to the top of the screen, the magnetic field at the center of the face of the CRT is about 2.0 gauss, or 2.0 × 10-4 tesla. So the force that an electron experiences on its way to the screen of the CRT is on the order of (1.602 × 10-19 C)(1.27 × 107 m/s)(2.0 × 10-4 T) = 4.1 × 10-16 N.

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. ↩

Strehlratio for primary aberrations in terms of their aberration variance

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

Both this demonstration and the previous one in the catalogue (68.33 -- Cathode ray tube, magnet) show the deflection of an electron beam by a magnetic field. In demonstration 68.33, a bar magnet provides the magnetic field. In this demonstration, a current flowing through a coil generates the magnetic field. In both of these demonstrations, there is no potential applied to either pair of deflection plates in the CRT, so without an external magnetic field, the electron beam is undeflected and hits the center of the screen.

In "Principles of Optics," Born & Wolf1 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,

Flip the switch on the CRT power supply from “Standby” to “On.” A green spot should appear in the center of the screen. (If it does not appear, touch the pins on the double banana plug; this will discharge the deflection plates, and the spot will appear.) Now turn on the power to the coil. If the coil is oriented and connected exactly as shown, the spot now moves upwards, the distance depending on how much current is flowing through the coil, and on the distance of the coil from the CRT. You can move the coil around to change the direction in which the spot moves, and vary the current through the coil, and its distance from the CRT, to change the distance over which the spot moves.

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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.

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Strehlratio formula

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.

The earliest cathode ray tube, or CRT, enabled J.J. Thomson to establish the identity of the electron as a fundamental particle, and to determine its charge-to-mass ratio, e/m. (In a footnote to the section about the discovery of the electron, in Physics, Part Two, Robert Resnick and David Halliday write that evidence exists that a German physicist named Weichert discovered the electron several months before Thomson.) A cathode ray tube is an evacuated glass envelope having a cathode at the back, and an anode some distance in front of it. A filament near the cathode heats it, “boiling” electrons off it. A potential placed between the cathode and the anode accelerates these electrons through the tube, at the front face of which they strike a phosphor screen. Where the electrons hit the screen, they excite the phosphor, causing it to fluoresce and produce a (bright) spot on the screen. Pairs of plates, placed at right angles to each other between the anode and the screen, allow one to deflect the electron beam, or “cathode ray,” up, down, left or right as desired. By using a pair of deflection plates, measuring the beam deflection for a particular deflection potential, and then applying a uniform magnetic field to cancel the deflection, Thomson was able to calculate e/m for the electron. (It is -1.75882001076 × 1011 C/kg. Its charge is -1.602176634 × 10-19 C, and its mass is 9.1093837015 × 10-31 kg.*)

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),

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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%.