f/# and Numerical Aperture. It can often be easier to talk about the overall light throughput as the cone angle, or the numerical aperture (NA), of a lens. The ...

According to the Petzval field curvature plot, the total field curvature from the center of the image (0 degrees) to the edge (9.4 degrees) is about 0.23 mm. We can measure the Petzval curvature on the corresponding through focus MTF chart by measuring the distance from the peak of the 0 degree line to the peak of the 9.4 degree line. These peaks as seen on the plot are about 0.15mm apart. The field curvature plot exaggerates the real field curvature by 150%! This is because Petzval field curvature plots rely on approximations.

MTFlens

The light from the lens comes to a focus and then expands again, as it should. The diameter of the illumination cone (called the spot size) is very small at the focus but it is larger at either side. The following figure, called a “through focus spot diagram”, shows its variation on either side of focus.

f/# (pronounced “EFF number”) is the focal length of a lens divided by the diameter of the beam of light that enters it. If you’d like to learn more about that concept, head to this page. For a simple lens, such as the one pictured above, the diameter of the beam is the diameter of the lens. More complicated lenses, like those on digital cameras, often have an entering beam diameter that differs from the lens diameter. The lens above has a focal length that’s twice the diameter of the entering beam, so the f/# is 2. This is written as f/2 (pronounced EFF two).

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Now that you understand the theoretical limit of through focus MTF, it’s time to get practical. A picture of the through-focus MTF of a real lens is displayed below.

Mtf curveangle

Researchers at Lehigh University in the United States developed a new thin-film solar cell absorber material that reportedly features an average photovoltaic absorption of 80% and an external quantum efficiency (EQE) of 190%.

MTFOptics

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These devices are believed to have the potential to exceed the Shockley-Queisser limit – the maximum theoretical efficiency that a solar cell with a single p-n junction can reach. It is calculated by examining the amount of electrical energy that is extracted per incident photon.

The CuxGeSe/SnS material features an intermediate energy bandgap ranging from 0.78 eV and 1.26 eV. With it, the group designed and modeled a thin-film solar cell with the proposed material as the active layer.

Mtf curvechart

Through-focus MTF can be used to determine the depth of focus for a lens. In the above plot, all three fields have an MTF above 0.2 from -0.14 units to +0.07 units. This means that the total depth of focus is 0.21 units. If the units are mm, this would be very easy for a mechanical designer to accommodate, but if the units are microns, it would be exceedingly difficult. That’s why it’s important for your optical engineer to communicate this information to you as you design your optical system.

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The new material consists of an heterostructure combining germanium, selenium, and tin sulfide, which also integrates atoms of zerovalent copper. It features an average photovoltaic absorption over 80% and could help photovoltaic cells break the Shockley-Queisser efficiency limit, according to its creators.

The simulation showed that the cell EQE may span from 110% to 190%. The researchers also found that by gauging the absorber's thickness the cell optical activity increases in wavelengths ranging from 600 to 1200 nm.

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As before, the blue curve represents data from the image’s center, red from the edge, and green from in between. To get these curves, the user specifies a certain spatial frequency that’s commonly half to three fourths of the maximum. Then the user has the lens design software plot the variation of the MTF at this spatial frequency versus focus. The zero point is typically the “best focus” for the lens. The plot shows how fast the MTF falls off as the image plane moves from its optimal location.

Looking forward, the research group said new research is required to identify a practical way to embed the novel material in real solar cells. However, they also noted that the experimental techniques used to create these materials are already “highly advanced.”

The first basic principle of through-focus MTF is the geometrical limitation. Any lens that focuses light produces an “illumination cone.” See below for an example:

The central spot is at the focus of the lens and the spots to the left and right correspond to the illumination cone before (to the left) and after (to the right) focus. As you can see, the spot size increases dramatically as you move away from focus. This forms an upper limit on the through focus MTF. The F# of the optical system will determine how fast this happens. Any photographers reading this may be familiar with this term, but let me explain it for the others.

The new material is a 2D two-dimensional Van der Waals (vdW) material, which means it features a crystalline planar configuration held together by ionic bonds. It consists of an heterostructure combining germanium (Ge), selenium (Se), and tin sulfide (Sns) with atoms of zerovalent copper (Cu) being inserted between the material's layers.

I'll explain what the numbers actually mean, and I'll also explain why they matter for your photography. By the time you're finished, you'll be a lens number ...

“The material’s efficiency leap is attributable largely to its distinctive ‘intermediate band states,' specific energy levels that are positioned within the material's electronic structure in a way that makes them ideal for solar energy conversion,” the scientists explained. “These states have energy levels within the optimal subband gaps—energy ranges where the material can efficiently absorb sunlight and produce charge carriers.”

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Stand in front of a window with a view. Hold the lens in between the window and a piece if paper. Change the distance between the lens and the ...

The spot size at a given distance from focus can never be smaller than the diameter calculated from the f/#. Take, for example, the f/2 lens shown above. The spot size 1 mm away from the focus can never be smaller than 1/2 mm. If the lens were f/2.5, the spot could be no smaller than 1/2.5 or 0.4 mm. Applying what we learned about MTF, the MTF at 2 cy/mm would be essentially zero for an f/2 lens 1 mm from focus. An f/2.5 lens would have a nonzero 2 cy/mm MTF at 1 mm from focus, but it would drop to zero at 1.25 mm from focus.

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“The rapid response and enhanced efficiency in Cu-intercalated samples, strongly indicate the potential of Cu-intercalated GeSe/SnS as a quantum material for use in advanced photovoltaic applications, offering an avenues for efficiency improvements in solar energy conversion,” they concluded.

MTFformula

Mtf curveexplained

The Petzval curvature plot doesn’t always differ significantly from the through focus MTF plot, but it’s always a good idea to specify the maximum allowable focal shift in MTF when designing a custom lens. This is because just specifying the Petzval curvature approximation may differ significantly from the real field curvature.

A concave lens is the opposite, being thicker at the edge than the center and spreading out a beam of light. Microscopes use convex lenses in order to focus ...

In the paper “Chemically tuned intermediate band states in atomically thin CuxGeSe/SnS quantum material for photovoltaic applications,” published in ScienceAdvance, the academics explained that the new quantum material may be an ideal match for intermediate band solar cells (IBSCs).

We can measure Petzval field curvature several different ways, but we prefer to rely on the through focus MTF plot. It generally gives the most accurate representation of the real field curvature. See the two plots below as an example. One is a plot of the through focus MTF and the other is a simple field curvature plot of the same lens. Both of these plots are for monochromatic light at 550nm.

Unfortunately, the best focus for the light at the edge of the image (red) and even part way out (green) does not lie at the same place as it does for the center of the image. This condition, known as “field curvature”, is caused by one of two aberrations: Petzval curvature or astigmatism. See our Aberrations page to learn about these. Also, the sagittal and tangential curves do not peak at the same focal position. This is typical of astigmatism. However, the difference between the sagittal and tangential curves is much less than the difference between the center and edge of the image. This means your optical engineer would tell you that the Petzval curvature is more of a problem than the astigmatism. If you can’t get the entire image in focus at the same time, this is likely the problem. If you focus on the edge of the image, the center looks fuzzy (and vice versa) as seen here.

Mtf curveexample

The device was assumed to be based on an indium tin oxide (ITO) substrate, an electron transport layer (ETL) based on zinc oxide (ZnO), CuxGeSe/SnS absorber, and a gold (Au) metal contact. “In our design, atomically thin GeSe and SnS are vertically stacked, facilitating the easy integration of the hybrid structures through van der Waals interactions,” the research team specified.

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Let’s start by observing the blue line. On it we can see that the MTF peaks at around 0.06 units to the right of the “best focus” and drops off on either side. The dropoff is not symmetric, which is common for systems with spherical aberration. For more information on aberrations, click here. Assuming that an MTF of 0.2 is acceptable, the image plane can be moved about 0.4 units to the left of best focus or 0.16 units to the right before the resolution at the center of the image is no longer acceptable.

Now that you understand MTF, the next important concept is how refocusing a lens affects it. This is called the through focus MTF. It is important that you get this information from your optical engineer, because without it you could end up with an optical system that does not meet your needs.

Th EQE is the ratio of the number of electrons collected by the solar cell to the number of photons that hit it. It defines how well a solar cell converts photons into electrical current. “In traditional solar cells, the maximum EQE is 100%, representing the generation and collection of one electron for each photon absorbed from sunlight,” the research's lead author Chinedu Ekuma said in a statement.