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Strehlformula

Aug 4, 2014 — However, optical microscopes still have to overcome a critical limit in optical resolution caused by the diffraction of visible light wavefronts ...

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.

Strehlratio MTF

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Strehlratio Zemax

Strehlratio calculator

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

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If the upgrades to the lab are any indication, the future is bright for these two and the rest of the photonics students. That future might even find them working for Ciena. Case in point, Amanda Grassi, Prototype Line Supervisor/Technical Support and 2019 Algonquin College Alumni.

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Strehlratio for primary aberrations in terms of their aberration variance

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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 Bachelor of Information Technology – Optical Systems and Sensors (BIT-OSS) program (formerly Photonics & Laser Technology – BIT-PLT), equips the students with exposure to both hardware and software design skills and prepares them for many roles within the industry, including telecommunication system architects and operators. The specialized lab is accessible to all students and visitors and sees 20 dedicated students each year, alongside research and development work and corporate training sessions. “The Optophotonics Lab became a world-class research center to conduct several research projects, a place where SMEs test and integrate their products to the optical communications networks. The lab also provides corporate training”, said Dr. Almuhtadi.

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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),

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

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.

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.

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What is a goodStrehlratio

Eyepiece which contains a lens to magnify an image 10x that projected by the objective lens. Image: Ocular Lens. Body Tube.

​​​​​​​The roots of this partnership stem from when Dr. Almuhtadi and Rodney Wilson, Ciena’s Chief Technologist of External Research Networks, worked together at Nortel more than 20 years ago. Ciena went on to purchase Nortel’s Metro Ethernet Networks (MEN) business in 2009.

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.

Strehlratio formula

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. ↩

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,

Ciena contributed tremendously to the original lab upgrade in 2012, and eight years later, the partnership continues to grow. With Ciena’s continued support, Algonquin College students get a world-class lab on campus, complete with 600km of fiber cables to stimulate a long-distance, multi-city environment. In turn, Ciena, and other photonic industry employers get highly qualified graduates with both theoretical and applied knowledge and the latest and greatest skills while investing in our local economy and the industry. Ciena now boasts more than 100 Algonquin College alumni as part of their worldwide workforce.

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.

“My experience at Ciena has provided me with a lot of learning opportunities and the chance to work with the most cutting-edge technologies in telecoms right now,” Grassi said. “I love that there is always something new to pursue and I am encouraged to develop constantly. I enjoy working with various engineering teams to see the new products through from R&D conceptualization to deployment."

​​​​​​​For students like Sarah Monforton and Rohan Chopra, the lab was a big selling feature when first applying for the program. “The lab makes all the difference, it really elevates the program,” said Chopra. “We have a distinct advantage because of the equipment we’re using – it is commonly used throughout our industry. It puts us in another league." Monforton concurs: “Having the equipment ready to go in class is so beneficial, with all the hands-on training we have access to before we're even in the workforce.”

“With this upgrade, we can deliver an updated curriculum with the latest equipment so students can be ready for their future when they get employed in the industry,” added Dr. Almuhtadi. “Our graduates have an incredible employment rate as they usually get multiple offers from employers, even some (current students) before they become graduates."

With a booming industry, job prospects seem bright for graduates in this growing field. “It doesn’t hurt that the current outlook for the industry is quite positive,” added Monforton. “I feel like we’re on a great path to move with the future with all that we’re learning at Algonquin."

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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.

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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.

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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%.

They’re both third-year students in the joint program with Carleton University. This program is not for the faint of heart though. “Classes are tough, but you learn to like the challenges,” added Chopra. “With support from companies like Ciena, it helps us stay engaged and continue to grow as technology advances."

The recently upgraded Ciena Optophotonics Lab, can transmit data flows at 200 Gbit/s, compared to 2.5 Gbit/s previously. The lab features equipment used in major telecommunications facilities in Canada, allowing Algonquin College students to get hands-on experience from world-class and cutting-edge technology in this new teaching lab installation.

Strehlpronunciation

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We leverage photonics in numerous applications daily, in areas as diverse as communications, defense, manufacturing, entertainment, consumer equipment, aviation, agriculture, life sciences, and medicine. Algonquin College is home to the only photonics lab of its kind in the world – all thanks to a substantial upgrade and sponsorship donated by Ciena.

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