"We faced so many deep coverages where teams were taking away all our deep throws," Mahomes said of the 2021 season during a Super Bowl LV media session. "I had to learn when to just take what's underneath. That's something where I've kind of grown and matured. We'll call deep plays still all the time, but if defenses are going to play that deep, even though I want to in my heart of hearts to throw it downfield and make those big plays, I'll take the underneath stuff and find ways to move the ball down the field that way."

With aerial yards down across the NFL over the past year-plus, we turned to NFL reporter Brooke Pryor for insight and context on the development. Listen here

Consider the phenomenon an offensive see-saw. While the volume of deep throws decreased across the league, the run game surged. In 2022, offenses averaged 4.5 yards per rush and 121.5 rushing yards per game -- both record-highs since at least 2006.

He beat Philadelphia Eagles cornerback James Bradberry, who started the play across from him at the line of scrimmage before dropping off into a zone defense. Nearly 30 yards down the field, Valdes-Scantling looked wide open for what could easily turn into the kind of momentum-swinging, home-run play that wins Super Bowls.

In practice, however, it is difficult to achieve numerical aperture values above 0.95 with dry objectives. Figure 2 illustrates a series of light cones derived from objectives of varying focal length and numerical aperture. As the light cones change, the angle µ increases from 7° in Figure 2(a) to 60° in Figure 2(c), with a resulting increase in the numerical aperture from 0.12 to 0.87, nearing the limit when air is the imaging medium.

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Bengals coach Zac Taylor did just that in his offensive adjustments from 2021 to 2022. A year after Joe Burrow and the Bengals stormed into the Super Bowl with 11 regular-season touchdowns that went at least 20 air yards, Taylor noticed defenses playing Burrow differently, taking away the deep balls that made the Bengals offense so lethal. After averaging 8.3 air yards per attempt in 2021, Burrow's average dropped to 7.2. And though he attempted nearly 100 more passes in 2022, Burrow attempted four fewer passes of more than 15 air yards. In his first outing of 2023, Burrow averaged 7.1 air yards per attempt -- but more alarmingly, 2.6 yards per passing attempt. It was the first time in his career that he averaged fewer than five yards per attempt.

"I felt like we saw more two-high defenses than any year I've been in the league," Taylor said of the 2022 season. "And, yeah, specifically to us, but also around the league more generally, you're seeing a lot of coordinators come in and play a lot more two-high looks.

By examining the numerical aperture equation, it is apparent that refractive index is the limiting factor in achieving numerical apertures greater than 1.0. Therefore, in order to obtain higher working numerical apertures, the refractive index of the medium between the front lens of the objective and the specimen must be increased. Microscope objectives are now available that allow imaging in alternative media such as water (refractive index = 1.33), glycerin (refractive index = 1.47), and immersion oil (refractive index = 1.51). Care should be used with these objectives to prevent unwanted artifacts that will arise when an objective is used with a different immersion medium than it was designed for. We suggest that microscopists never use objectives designed for oil immersion with either glycerin or water, although several newer objectives have recently been introduced that will work with multiple media. You should check with the manufacturer if there are any doubts.

The next season, Mahomes noticed defenses were starting to play him differently. If he was going to beat them, it wasn't going to be with go-balls to Tyreek Hill.

"The best [situation] is when you know the true wideouts are threatening people all the time on the outside, and when they do, that allows the rest of the game to become a lot easier," Shanahan said. "And then if they're tough with the ball in their hands, then they usually get to really change the game."

Shanahan's signature offense is predicated on those versatile, tough players, ones like Deebo Samuel and Christian McCaffrey. McCaffrey, who the 49ers acquired in a midseason blockbuster trade last year, racked up 695 yards after catch, while Samuel amassed 493 in an abbreviated season. The 49ers averaged a league-high 6.6 yards after the catch last season, marking the fifth consecutive season where Shanahan's offense averaged more YAC per reception than any other team in football. Predictably, the 49ers were 29th last season in pass attempts of more than 15 air yards.

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"On a Texas high school football field, you're seeing a lot of spread," Smith said. "Guys are very talented, skilled position guys. You're not seeing some of the old school, intermediate drop back game."

"I don't know," Smith said. "That's a good question. It's like in basketball, right? You had the three-point revolution. People saw the Warriors win a certain way. But until you have another Steph Curry and Draymond Green and Kevin Durant and Klay Thompson playing together, you know, who can jack that many threes and be that successful? Is the basketball game ever going back to low post? It's a great debate."

"A lot of defensive schemes were around their cornerbacks," Peterson said. "...I won't say defenses have become more complicated, but defenses have said now we don't want to make the quarterback just find the easy throw right away. We want to make him think through the process of passing the ball, just not line up, 'OK, I got a press corner here, a single high, I'm going deep to one of these guys.'"

In a league seemingly obsessed with finding the biggest arm -- and the most explosive receivers to take advantage of it -- the moonshots might be a thing of the past.

The numerical aperture of an objective is also dependent, to a certain degree, upon the amount of correction for optical aberration. Highly corrected objectives tend to have much larger numerical apertures for the respective magnification as illustrated in Table 1 below. If we take a series of typical 10x objectives as an example, we see that for flat-field corrected plan objectives, numerical aperture increases correspond to correction for chromatic and spherical aberration: plan achromat, N.A. = 0.25; plan fluorite, N.A. = 0.30; and plan apochromat, N.A. = 0.45.

Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.

In 2022, Ekeler set a career high by leading the league with 843 yards after catch, exceeding his expected yards after catch of 762, per NFL Next Gen Stats. Ekeler was fourth in receiving yards on a Chargers team where no one had more than 900 receiving yards, despite having a cannon-armed quarterback in Herbert.

AT MIDFIELD LATE in the first quarter of a tight Super Bowl LVII, Kansas City Chiefs wide receiver Marquez Valdes-Scantling streaked down the field on a corner route.

Sitting at a round table during NFL owners meetings earlier this year, Arthur Smith's eyes lit up at a question about the deep ball's decline. He leaned back in his chair and pondered.

"I'm sure there's a myriad of reasons that outline (the deep ball decline), but one that jumps out to you is the uptick in rushing," Steelers coach Mike Tomlin said. "Passing down, rushing up, it's a reasonable kind of discussion."

Visitors are invited to explore changes in numerical aperture with changes in µ, using our interactive Java tutorial that investigates how numerical aperture and magnification are related to the angular aperture of an objective.

The NFL is cyclical. An offensive concept rises up through the levels of football, NFL defenses adapt to it and take it away, offenses find a new answer, on and on. It's a never ending carousel, and that's why after 25 seasons as a head coach between Philadelphia and Kansas City, nothing fazes Andy Reid. It's a never-ending chess match with a finite number of logical counter moves. It's just about having the right answer key to the right test at the right time.

"It's high risk, throwing the ball all the way down the field, maybe in double coverage or if it's 40 yards in the air," Steelers quarterback Mitch Trubisky said. "Rather, (you can) throw it five yards and let that guy run the rest of the way, so the run after catch has been a huge emphasis ...You see the really good teams who are scoring a lot of points. They're not necessarily always throwing it deep. They're throwing it underneath and letting the guy run the rest of the way."

Instead, defenses are using more two-high looks, putting two safeties back deep and making it more difficult for receivers to get open on the vertical routes. And it's not just the secondary adjustments that discourage quarterbacks from throwing deep. As the game evolved, defenses went from static zone coverage where defenders were only active if someone was in their assigned area to more dynamic, man-match coverages that blend zone concepts with varying degrees of man-to-man defense. Think Pete Carroll's Legion of Boom to Vic Fangio's counterpunch developed with the San Francisco 49ers.

The Eagles defense had two safeties covering over the top as C.J. Gardner-Johnson and Marcus Epps patrolled the turf that separated Valdes-Scantling from the end zone.

In 2006, about 43.5% of the passing yards generated by offenses were produced by receivers after catching the football. By 2022, that mark was north of 50%.

"It's the way the game's been played, and I think you're seeing a much more wide open game now coming from the youth level."

The evolution of Mahomes' aerial attack mirrors a league-wide trend that last season saw the fewest number of deep balls attempted -- defined as traveling at least 15 air yards -- since 2006.ESPN Daily podcastWith aerial yards down across the NFL over the past year-plus, we turned to NFL reporter Brooke Pryor for insight and context on the development. Listen here

"The really good, really disciplined quarterbacks know where the ball has to go," Orlovsky said. "They get the ball out quickly, they take your boring completions and they make sure the coach knows again, I need you to call that (deep pass) again at some point. I'll do the right thing with the football. Keep calling your shots and you've got to trust your quarterback to make the right decisions."

When light from the various points of a specimen passes through the objective and is reconstituted as an image, the various points of the specimen appear in the image as small patterns (not points) known as Airy patterns. This phenomenon is caused by diffraction or scattering of the light as it passes through the minute parts and spaces in the specimen and the circular back aperture of the objective. The central maximum of the Airy patterns is often referred to as an Airy disk, which is defined as the region enclosed by the first minimum of the Airy pattern and contains 84 percent of the luminous energy. These Airy disks consist of small concentric light and dark circles as illustrated in Figure 3. This figure shows Airy disks and their intensity distributions as a function of separation distance.

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But when quarterbacks did let it fly, they were pretty successful. Completion percentages on those attempts of at least 15 yards downfield was 45.1% -- the second-highest completion percentage in a single season since 2006.

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Where R is resolution (the smallest resolvable distance between two objects), NA equals numerical aperture, λ equals wavelength, NA(obj) equals the objective numerical aperture, and NA(Cond) is the condenser numerical aperture. Notice that equation (1) and (2) differ by the multiplication factor, which is 0.5 for equation (1) and 0.61 for equation (2). These equations are based upon a number of factors (including a variety of theoretical calculations made by optical physicists) to account for the behavior of objectives and condensers, and should not be considered an absolute value of any one general physical law. In some instances, such as confocal and fluorescence microscopy, the resolution may actually exceed the limits placed by any one of these three equations. Other factors, such as low specimen contrast and improper illumination may serve to lower resolution and, more often than not, the real-world maximum value of R (about 0.25 µm using a mid-spectrum wavelength of 550 nanometers) and a numerical aperture of 1.35 to 1.40 are not realized in practice. Table 2 provides a list resolution (R) and numerical aperture (NA) by objective magnification and correction.

Most objectives in the magnification range between 60x and 100x (and higher) are designed for use with immersion oil. By examining the numerical aperture equation above, we find that the highest theoretical numerical aperture obtainable with immersion oil is 1.51 (when sin (µ) = 1). In practice, however, most oil immersion objectives have a maximum numerical aperture of 1.4, with the most common numerical apertures ranging from 1.0 to 1.35.

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Patrick Peterson remembers coming into an NFL where truly elite defenses had lockdown corners, where they dared quarterbacks to try it: your best guy vs. our best guy.

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As Gardner-Johnson tried to bait Mahomes into throwing the deep ball, the quarterback instead found Travis Kelce underneath. He connected with the tight end on a pass that Next Gen Stats calculated as a 90% completion probability the second it left Mahomes' hand. The play went down as a 22-yard reception, but Mahomes' pass traveled just under 14 air yards, while Kelce picked up the remaining 8 yards after making the grab.

It's a debate that pervaded the 2023 NFL draft as teams evaluated first-round quarterbacks and carefully considered the offensive identity each pick could bring and what level of success it could have in an ever-evolving league.

That's exactly the goal, Peterson said. Blanketing over the top, forcing the quarterback to hold on to the ball longer and progress through his reads gives the pass rush the extra tenth of a second needed to get to the quarterback.

Correct alignment of the microscope optical system is also of paramount importance to ensure maximum resolution. The substage condenser must be matched to the objective with respect to numerical aperture and adjustment of the aperture iris diaphragm for accurate light cone formation. The wavelength spectrum of light used to image a specimen is also a determining factor in resolution. Shorter wavelengths are capable of resolving details to a greater degree than are the longer wavelengths. There are several equations that have been derived to express the relationship between numerical aperture, wavelength, and resolution:

With the rise in popularity of skill players like McCaffrey, Samuel and Hill, it makes more sense for quarterbacks to take the safer throw knowing their receiver is plenty capable of picking up the yards on the back end of the play.

"Do you want Anthony Richardson, the big play superstar, massive arm? Or do you want C.J. Stroud who's the 'death by a thousand paper cuts' guy? People were torn," Orlovsky said. "Now obviously the ideal world, you want both. But it's interesting to see what style of quarterback you prefer with what kind of style of offense your team may want to run."

"It's not a Tampa-2 oriented defense, but it's very soft zones and the deep ball isn't there," 49ers coach Kyle Shanahan said. "If you make a living of trying things that a defense is really trying to take away, it's kind of an uphill battle, and you're wasting a lot. So sometimes you take what they give you."

Herbert averaged 6.7 air yards per attempt in 2022, nearly a full yard less than each of his first two seasons in the NFL. His percentage of passes attempted of more than 20 air yards was also its lowest last season, dropping from 10.3% in 2020 to 8.2% in 2022.

"A lot of times that's good quarterback play, not forcing the ball into those looks. It's not because those calls aren't coming in, they're just not forcing the ball down the field."

With the Titans, Smith leaned heavily on All-Pro running back Derrick Henry, and in Atlanta, he's continued to make his offenses more horizontal. Not only did the Falcons use a run-heavy offense in 2022 behind a strong offensive line and running back Tyler Allgeier, they also utilized the run-pass option with positionless offensive skill players like Cordarrelle Patterson and Kyle Pitts. The Falcons ranked 25th last season in passes attempted of more than 15 air yards, and inversely, ranked third in rushing yards.

There were 3,416 attempts at least 15 yards downfield during the 2022 regular season, the fewest in any season since 2006.

Figure 3(a) illustrates a hypothetical Airy disk that essentially consists of a diffraction pattern containing a central maximum (typically termed a zeroth order maximum) surrounded by concentric 1st, 2nd, 3rd, etc., order maxima of sequentially decreasing brightness that make up the intensity distribution. Two Airy disks and their intensity distributions at the limit of optical resolution are illustrated in Figure 3(b). In this part of the figure, the separation between the two disks exceeds their radii, and they are resolvable. The limit at which two Airy disks can be resolved into separate entities is often called the Rayleigh criterion. Figure 3(c) shows two Airy disks and their intensity distributions in a situation where the center-to-center distance between the zeroth order maxima is less than the width of these maxima, and the two disks are not individually resolvable by the Rayleigh criterion.

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And once those skilled position high school players make their way to the NFL, they become positionless players. The new wave in offensive threats, positionless players are the perfect vehicles to carry a scheme predicated on the run-pass option. The concept leans heavily on the run game to set up the deep ball, and it's a key part of Smith's offenses.

Hill's prowess as a speedy, vertical threat forces defenses to back up, but his quick-footed shiftiness makes him -- and players like him -- uniquely capable of taking advantage of the space created underneath with the defenses protecting the deep ball.

The angle µ is one-half the angular aperture (A) and is related to the numerical aperture through the following equation:

So what gives? There's not a simple answer. Around the league, head coaches, coordinators and players all point to different reasons.

"This thing is all a big circle, so I'm sure it'll come back somewhere," Reid said. "It's like a dog chasing its tail. You just gotta stay ahead of it. It'll be back around somewhere."

The numerical aperture of a microscope objective is a measure of its ability to gather light and resolve fine specimen detail at a fixed object distance. Image-forming light waves pass through the specimen and enter the objective in an inverted cone as illustrated in Figure 1. A longitudinal slice of this cone of light shows the angular aperture, a value that is determined by the focal length of the objective.

In day-to-day routine observations, most microscopists do not attempt to achieve the highest resolution image possible with their equipment. It is only under specialized circumstances, such as high-magnification brightfield, fluorescence, DIC, and confocal microscopy that we strive to reach the limits of the microscope. In most uses of the microscope, it is not necessary to use objectives of high numerical aperture because the specimen is readily resolved with use of lower numerical aperture objectives. This is particularly important because high numerical aperture and high magnification are accompanied by the disadvantages of very shallow depth of field (this refers to good focus in the area just below or just above the area being examined) and short working distance. Thus, in specimens where resolution is less critical and magnifications can be lower, it is better to use lower magnification objectives of modest numerical aperture in order to yield images with more working distance and more depth of field.

Even though Herbert's overall volume of deep passes decreased, the Chargers were still in the top half of the league with 61 attempts of passes of at least 20 air yards, though they were 20th with a 36.1 completion percentage. In theory, the more success a team has with picking up yards after catch on the short passes, the more likely an opportunity opens up for an occasional deep ball.

The resolution of a microscope objective is defined as the smallest distance between two points on a specimen that can still be distinguished as two separate entities. Resolution is a somewhat subjective value in microscopy because at high magnification, an image may appear unsharp but still be resolved to the maximum ability of the objective. Numerical aperture determines the resolving power of an objective, but the total resolution of a microscope system is also dependent upon the numerical aperture of the substage condenser. The higher the numerical aperture of the total system, the better the resolution.

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Sometimes that means throwing the ball to the running back and letting him do the dirty work. While Trubisky had running back Tarik Cohen as his YAC playmaker in Chicago, Chargers quarterback Justin Herbert has one of his own in Austin Ekeler.

That was Mahomes' longest completion of Super Bowl LVII. Despite possessing an arm capable of launching a football 70 yards, Mahomes attempted just four passes of at least 15 air yards -- and completed three of them -- in his second Super Bowl victory.

Careful positioning of the substage condenser aperture diaphragm is also critical to the control of numerical aperture and indiscriminate use of this diaphragm can lead to image degradation (as discussed in the section on substage condensers). Other factors, such as contrast and the efficiency of illumination, are also key elements that affect image resolution.

While Smith's units in Tennessee and Atlanta averaged some of the deepest throws in the league, they simply didn't throw the ball all that often because of the strength of the run game and the types of players on the roster.

"It's directly attached to Patrick Mahomes," ESPN analyst and former NFL quarterback Dan Orlovsky said, explaining the phenomenon. "Teams have sat there and said, 'We are going to take these home-run hitting quarterbacks and make them hit singles.'"

The smaller the Airy disks projected by an objective in forming the image, the more detail of the specimen that becomes discernible. Objectives of higher correction (fluorites and apochromats) produce smaller Airy disks than do objectives of lower correction. In a similar manner, objectives that have a higher numerical aperture are also capable of producing smaller Airy disks. This is the primary reason that objectives of high numerical aperture and total correction for optical aberration can distinguish finer detail in the specimen.

This feature of increasing numerical aperture across an increasing optical correction factor in a series of objectives of similar magnification holds true throughout the range of magnifications as shown in Table 1. Most manufacturers strive to ensure that their objectives have the highest correction and numerical aperture that is possible for each class of objective.

An important concept to understand in image formation is the nature of diffracted light rays intercepted by the objective. Only in cases where the higher (1st, 2nd, 3rd, etc.) orders of diffracted rays are captured, can interference work to recreate the image in the intermediate image plane of the objective. When only the zeroth order rays are captured, it is virtually impossible to reconstitute a recognizable image of the specimen. When 1st order light rays are added to the zeroth order rays, the image becomes more coherent, but it is still lacking in sufficient detail. It is only when higher order rays are recombined, that the image will represent the true architecture of the specimen. This is the basis for the necessity of large numerical apertures (and subsequent smaller Airy disks) to achieve high-resolution images with an optical microscope.

The trend continued through the first week of the 2023 regular season, too. Both air yards per attempt and number of passes attempted of more than 20 air yards were lower than the 2022 averages -- and the 2022 Week 1 numbers. Just 9.2% of all passes attempted were deep balls, down from 10.4% in Week 1 of 2022, while air yards per attempt decreased from 7.3 to 7.1. Passers produced the fewest yards (6,225) and TD passes (37) in a Week 1 since 2006. They averaged 6.25 yards per attempt, the lowest Week 1 output since 1996.

Pee Wee football concepts in the NFL? Yes, and it's not as crazy as it sounds, Smith said. The NFL isn't just a copycat league, football is a copycat sport. Concepts start at the grassroots level, and as a generation of players advances through the ranks, so, too, do the schemes and styles.

where n is the refractive index of the imaging medium between the front lens of the objective and the specimen cover glass, a value that ranges from 1.00 for air to 1.51 for specialized immersion oils. Many authors substitute the variable α for µ in the numerical aperture equation. From this equation it is obvious that when the imaging medium is air (with a refractive index, n = 1.0), then the numerical aperture is dependent only upon the angle µ whose maximum value is 90°. The sin of the angle µ, therefore, has a maximum value of 1.0 (sin(90°) = 1), which is the theoretical maximum numerical aperture of a lens operating with air as the imaging medium (using "dry" microscope objectives).

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Figure 4 illustrates the effect of numerical aperture on the size of Airy disks imaged with a series of hypothetical objectives of the same focal length, but differing numerical apertures. With small numerical apertures, the Airy disk size is large, as shown in Figure 4(a). As the numerical aperture and light cone angle of an objective increases however, the size of the Airy disk decreases as illustrated in Figure 4(b) and Figure 4(c). The resulting image at the eyepiece diaphragm level is actually a mosaic of Airy disks which we perceive as light and dark. Where two disks are too close together so that their central spots overlap considerably, the two details represented by these overlapping disks are not resolved or separated and thus appear as one, as illustrated above in Figure 3.

When the microscope is in perfect alignment and has the objectives appropriately matched with the substage condenser, then we can substitute the numerical aperture of the objective into equations (1) and (2), with the added result that equation (3) reduces to equation (2). An important fact to note is that magnification does not appear as a factor in any of these equations, because only numerical aperture and wavelength of the illuminating light determine specimen resolution. As we have mentioned (and can be seen in the equations) the wavelength of light is an important factor in the resolution of a microscope. Shorter wavelengths yield higher resolution (lower values for R) and visa versa. The greatest resolving power in optical microscopy is realized with near-ultraviolet light, the shortest effective imaging wavelength. Near-ultraviolet light is followed by blue, then green, and finally red light in the ability to resolve specimen detail. Under most circumstances, microscopists use white light generated by a tungsten-halogen bulb to illuminate the specimen. The visible light spectrum is centered at about 550 nanometers, the dominant wavelength for green light (our eyes are most sensitive to green light). It is this wavelength that was used to calculate resolution values in Table 2. The numerical aperture value is also important in these equations and higher numerical apertures will also produce higher resolution, as is evident in Table 2. The effect of the wavelength of light on resolution, at a fixed numerical aperture (0.95), is listed in Table 3.

"The institution of the run-pass option has just provoked quicker throws," Orlovsky said. "And more space created laterally and horizontally, just spreading offenses outward, promotes easier underneath completions."