Subsequent apertures (i.e. ƒ/2 -> ƒ/2.8 or ƒ/2 -> ƒ/1.4) decrease or increase the diameter of the aperture's pupil by a factor of √2 to halve or double the total area of the aperture's pupil.

If your intuition is like mine, you'll probably do some naive math; like some very naive multiplication, and come up with the wrong answer. Sorry, nope! That's not how f-numbers work at all. If you got that right, then well, you probably already know everything, you may as well just be done here. Thanks for reading! Joking, you're not getting off that easily.

We highly suggest reading each of the following posts in order to learn how to better fractalize your images. Don't expect to shoot at your best without first understanding this information! Some of the facts you'll learn might surprise you (they sure surprised us 😲).

Learn the trending creative techniques that the world's most popular photographers are using to create inspirational images

This blog post is the first in a three part series explaining how photographers can use prisms like Fractals to capture creative images.

OK, this headline is purposely clickbaity. Of course ISO and shutter speed are important—otherwise they wouldn't have been introduced to begin with. Aperture though, is uniquely king over these two other settings—and its importance is generally overlooked by most photographers. Our reluctance to recognize the powerhouse that we can unleash by truly understanding our aperture is making our images worse!

The resolution achieved by a spectrometer depends on the angular dispersion provided by the grating, defined as the change in β due to a change in λ, or dβ/dλ, and how the varying angles image onto the detector array or DMD. Taking the derivative of (1) gives

Early lens manufacturers decided that sequential stops should double or halve the amount of light that enters the lens. As we discussed before, this is done by doubling or halving the area of the aperture. So how do we double or halve the area of the aperture? It's actually quite easy: it can be done by increasing or decreasing the aperture's diameter by a factor of square root of two!

Diffraction gratingexperiment

This was achieved through the use of a narrower aperture, which, again, reduced the amount of defocus that the prism's reflection could create. Had he shot with the lens wide open, the haze would produced by the prsim would be overwhelming. Kyun's choice was artistically appropriate given the context:

where Δλ is the minimum separation between two spectral lines that can be observed as distinct, individual lines at the output of the grating when operated around a central wavelength λctr. In diffraction theory, the output of the grating at a given wavelength produces a sinc2 function in space, which consists of an intense central lobe of some width and many less intense side lobes. The Rayleigh criterion dictates that two sinc2–shaped patterns can be distinguished from each other once the peak of the main lobe of one pattern moves to the zero in between the main lobe and the first side lobe of the second pattern. For spectrometers, this criterion and the geometry of the spectrometer sets the resolving power as

In the above example, when we decrease the depth of field by a factor of two, our circles of confusion become twice as large for rays that pass through the prism. The prism therefore becomes twice as defocused, and starts to blend in quite nicely with the surrounding scene:

Years ago, I ran into an issue during a wedding in which the electronic displays on my camera stopped working. Like, completely dark. Everything else functioned fine—the camera would still fire and capture shots perfectly, but, I had no idea what my shots looked like or if they were exposed correctly!

Does that mean you should always shoot at the fastest aperture you have? Is f-stop ƒ/1.4 always going to be king? Is more defocusing always going to be preferrable?

What isdiffraction gratingin Physics

where Wg is the illuminated width of the diffraction grating in mm. In practice, the minimum resolution obtained for a given spectrum depends more on the parameters of the detector array or DMD than the resolving power, as well as the width of the entrance slit.

The circles of confusion —i.e. amount of defocusing increases as your focal length increases. Why? Because your focal plane is typically further away from your lens at longer focal lengths, but your prism is still just a few cm from the front element of your lens! The increased delta between the prism and the plane of focus (where your subject lie) causes your prism to become more defocused as your camera's sensor sees it, which can make your subject just too blurry:

where Lpixel is the dimension of the detector or mirror along the horizontal direction, which in most cases is very close to the distance between detectors or mirrors. A better resolution (smaller Δλ) is obtained by either decreasing the pixel dimensions (smaller detectors or mirrors) or by increasing the linear dispersion of the grating – distributing the same number of wavelengths ∂λ over a much larger distance ∂x. In some cases, careful measurement can provide a sub-pixel/mirror resolution smaller than that in Equation (7), though in general the limit set in Equation (7) holds true. Combining the results of Equations (7) and (5) illuminates an important trade-off in the design and operation of spectrometers. To get better resolution requires a higher linear dispersion, while improving the range of the spectrometer requires a smaller linear dispersion. To balance the two conditions requires care in selecting the dimensions of the analyzing array for a given choice of groove density for the grating.

Total area decreases by half every time we increase our aperture by one stop. That's why we can deterministically state that exposure will decrease by a factor of .5 everytime we increase our aperture, and increase by a factor of two everytime we decrease our aperture.

Diffraction gratingPDF

No! Prism photography is slightly more nuanced than that, and it is possible to create so much defocusing that it crowds out the subjects in our images. This becomes increasingly true as our focal lengths increase.

I scrambled my way home, plugged the images into the computer, aaaaaand *drumroll*... most of the images were... salvageable. The modern era has stressed the importance of shooting RAW over JPEGs, a law I luckily abided by, so I had 3 stops in either direction to pull my exposure up and down. Yes, 3 stops, this was in the early 2010's; none of that crazy, 20 stop exposure compensation stuff that's available today.

Equation (4), along with information about the detector array or DMD, provides the information necessary to determine the wavelength range and the minimum possible wavelength resolution of the spectrometer.

The fundamental operation of the diffraction grating is depicted in Figure G-1. The period of the grating is given by d, though most gratings are specified by the groove density D = 1/d quoted in typical units of grooves/mm. Incoming light strikes the grating at an incident angle α and leaves the grating at a diffracted angle β. Unlike a flat mirror, β does not equal α. In fact, the grating diffracts light into several different angles, called diffraction orders, each with a different diffraction efficiency. The condition given by α = β from Snell’s Law of reflection corresponds to the m = 0 diffraction order. The angles βm corresponding to the m = 1, 2, and higher order diffractions are found using the following fundamental grating equation:

We have about half the depth of field at ƒ/1.4 .645 ft than we do at ƒ/2.8. This is a big deal. Without getting too esoteric, we must understand that the circle of confusion for light rays passing through the prism continue to grow larger, the more out of focus the prism becomes.

where f is the effective exit focal length of the optics in mm and ∂x is the distance across the target surface in mm. Equation (3) applies exactly only to wavelengths incident perpendicular or very nearly perpendicular to the target surface. For other wavelengths, Equation (3) must be modified to account for the tilt angle γ between the wavelength’s ray and the surface, as shown in Figure G-2. The modified equation becomes

You can however, look at a photo, and say that it could be a stop brighter, or a stop darker. This brings us to an important golden rule...

You can think of circles of confusion as the amount of 'defocusing', or blurring, that occurs. Larger circles of confusion allow light to defocus more dramatically as it hits the sensor; which give you your bokeh—like effects. Larger circles of confusion = larger bokeh. Except, this is not really bokeh because it's happening in front of the lens. It's more like front-bokeh. Fokeh, anyone?

How does a diffraction grating workin physics

Okay great, but why does any of this matter? Especially for prisming? Well, it's a little known fact that your depth of field amount also sequentially increases and decreases by a factor of √2 for each stop changed. (!!!)

Diffraction gratingpattern

Most of us probably feel as though we know enough about aperture to get by. Under critique, though, how well do these beliefs hold up? Well, let's start off with a little quiz. Here's a seemingly straightforward question that many will find to be remarkably counterintuitive:

Transmissiondiffraction grating

The subject might strike you as too complex, esoteric, granular.. Name your favorite adjective, but I will argue that understanding these minutae actually will help you create better images. You will forever understand your equipment better, be able to make better photographic small talk, and wrapping your head around this subject will only take you a minute. Sounds like a pretty sweet deal, right? Knowledge is power; let's get into it.

After my fear began to settle, I started to think. Before the era of digital photography, the phenomenon I had just weathered through would have been completely normal to the average photographer. Before digital metering, there was no automatic way to know how to expose your images!

After light passes through the slit, a set of lenses or mirrors collimates the light (makes all light rays parallel to the optical axis) and delivers the light to the diffraction grating. A diffraction grating consists of a material containing a periodic variation in one of its optical properties. The diffraction grating separates the wavelength components of the light by directing each wavelength into a unique output angle. The change in output angle as a function of wavelength, called the angular dispersion, plays an important role in determining the wavelength resolution of the spectrometer. The period of the variation in the optical property of the grating strongly determines the available angular dispersion. The efficiency of the diffraction grating determines the amount of optical power available to the DMD (in the Nano) or detector array as a function of wavelength. The geometry of the periodic variation strongly influences the available efficiency. Both the period and geometry of the grating must be carefully selected in order to meet the needs of your spectroscopic application.

The depth of field with a subject 10ft from the camera, at ƒ/2.8, is 1.29 ft. What will happen to our depth of field if we decrease it by two stops, from ƒ/2.8 to -> ƒ/1.4? Let's take a look:

To determine the wavelength range, the linear dispersion is multiplied by the length LS of the active surface, effectively shown in Fig G-3(a). For a linear detector array, LS is the number of detectors times the average width of the detectors. For a DMD, the number of mirrors in one row of the DMD times the horizontal dimension of the mirrors gives LS. Using Equation (3) for simplicity, the wavelength range of the spectrometer is given by

where the wavelength λ is in nanometers and the groove density D  is in grooves/mm. As the wavelength varies, so the does the output angle, and thus a range of wavelengths are deflected into a range of different angles and spatially separated, allowing the spectrometer to act on small ranges of wavelengths individually.

What isgrating

So, the rules that we discussed earlier will apply to your effetive focal length. At an effective focal length of 50mm, try starting out at around ƒ/1.4. For an effective focal length of 85mm, try starting around ƒ/4. Again, your mileage may vary, and you'll want to play with your aperture a lot as you shoot to produce a result that's desirable for your scene/subject combo.

Lux is an absolute measure of brightness. Imagine that the measurable amount of brightness on a sunny day measures in at 100,000 lux, and we have our camera set as to allow all 100,000 lux to pass through to the camera's sensor at an aperture of ƒ/1.

There's actually nothing spooky about it, and it's more intuitive than we may have guessed. The first important thing to understand is that apertures are just fractional numbers. Specifically speaking, they are a fraction of the focal length of your lens. The math is easy:

Since the accuracy and speed of a spectroscopic measurement depend on the amount of optical power available to the detector, the grating directs must be able to efficiently direct optical power into the chosen diffraction order sent to the detector array or DMD. The power delivered to a given diffraction order depends on the geometry of the periodic variation of the optical properties of the grating. For gratings with symmetrical geometries – a symmetric triangle pattern for ruled gratings, or a sinusoidal pattern for holographic gratings – the m = 0 order contains, in general (though not for all grating cases), a larger percentage of the optical power, and the m = 1 and higher orders used by the spectrometer contain only limited power. Changing the geometry of the variation, called “blazing,” can optimize the power coupled into the diffraction order of choice. The geometry of a blazed ruled grating is shown in Figure G-4, where θ is defined as the blaze angle. Since the output angle β depends on the incident wavelength, the blaze angle optimizes the diffraction efficiency only in a range of wavelengths around the blaze wavelength λB. Figure G-5 shows the center of the diffraction enveloped can be moved to coincide with one of the diffraction orders. As a general rule, the efficiency at λB can be greater than 85%, and the diffraction efficiency falls to 50% of the maximum efficiency at 0.6λB and 1.8λB. The decrease in efficiency imposes an additional constraint on the overall operating range of the spectrometer. It is important to select the correct blaze wavelength for a given application. Based on the typical 50% efficiency range, the selected blaze wavelength should lie within the lower half of the desired operating wavelength range of the application.

This blog post is an overview for our four part series explaining how photographers can use prisms like Fractals to capture creative images.

The only way I can recount the experience was that it was like reading while blind... or, like, driving while asleep... while texting.. you get the idea. There would be an important event, say, the bride performing the bouquet toss. I'd snap images furiously, as if I had a clue what I was doing, and then a wedding guest would walk up to me:

Diffraction gratingformula

Manual shooting mode will be OK too, but might require extra work. In either case, the goal is absolute control and comfortability when setting your camera's aperture. Don't be afraid to experiment and shift your aperture up and down a stop, even in a live shoot.

To contrast this, below we demonstrate an image shot at a high aperture which has created too strong a contrast between the 'prismed' part of the image and the non prismed part. We quickly notice there's an object in front of the lens, which removes some artistic integrity. The abstractity and nuance is lost. Aaand it wasn't a good image to begin with, but we're just trying to illustrate a point. You can clearly make out which part of the image is encompassed by the prism:

Ok, but how exactly does changing the aperture modify the intensity of light? Well, it's actually as intuitive as you'd guess it would be. An increase of one stop (i.e. ƒ/2.8 -> ƒ/4) halves the area (as in, the geometrical formula for obtaining the area of a circle, area = πr²) of the aperture. Inversely, a decrease of one stop (i.e. ƒ/2.8 -> ƒ/2) doubles the area of the aperture. Doubling the area, quite literally doubles the amount of light hitting the sensor, making the light twice as intense. Just like eating a cookie twice as big as another cookie, will make you twice as fat 😨.

We can observe that this is true with the simple proof. If we're using a 100mm lens at ƒ/2, the diameter of our aperture is 50mm, as per the equation:

Both ruled and holographic gratings work on the same fundamental diffraction principles and are generally governed by the same sets of equations. Therefore, the discussions of angular dispersion, resolution, and efficiency apply to both types of gratings.

Here we present a shot by Kyun Se Yawn, which demonstrates proper use of a narrower aperture at an extended focal length. Notice that the defocus from the prism around the couple in the image is relatively in focus; allowing the perfect amount of haze to be produced by the prism.

This is why equivalent exposures are so easy to calculate. Equivalent exposures are two different exposure settings that allow the exact same amount of light to reach the sensor, resulting in the exact same (equivalent) exposure. For example, the setting f/2.8 @ 1/500 a second would expose a photo exactly the same as ƒ/2 @ 1/1000 a second. Why? ƒ/2.8 -> f/2 is a doubling of the area (πr²) of your aperture, letting in twice as much light, so you need to double your shutter speed, from 1/500 to 1/1000 to maintain the same exposure.

A question arose from reader John Harvey Perez—for non-full frame (crop) sensors, what focal length and aperture combination is best? Many of us use cameras that are not full frame, and therefore use specialty lenses which are designed for non-full frame cameras, or use full-frame lenses and fail to appreciate the full viewing angle of the elements in the lens.

We observe that the area 981mm² is roughly half the area 1,963mm², which allows half the amount of light to enter the lens, exactly as we were expecting.

The wavelength resolution of the spectrometer has a theoretical minimum and a practical minimum value. The resolving power R of the grating itself defines the theoretical minimum resolution. The resolving power is defined as

Aperture confuses people, especially the f-stop scale—it doesn't seem to make any sense. Why is ƒ/2.8 a stop, and why does a stop of ƒ/2.8 signify a bigger diameter than an f-stop of ƒ/4?

Observe how the shallow aperture in this image lends to an overall dreamy effect; this is a direct result of the low aperture creating larger circles of confusion, a high amount of defocusing, and large fokeh (front bokeh).To contrast this, below we demonstrate an image shot at a high aperture which has created too strong a contrast between the 'prismed' part of the image and the non prismed part. We quickly notice there's an object in front of the lens, which removes some artistic integrity. The abstractity and nuance is lost. Aaand it wasn't a good image to begin with, but we're just trying to illustrate a point. You can clearly make out which part of the image is encompassed by the prism:

By this measure, if we were to increase our aperture by one stop, from ƒ/1 -> ƒ/1.4, we'd decrease the amount of brightness that reaches our camera's sensor by half: 50,000 = 100,000 / 2

And I started really really obsessing over this concept. How did they do it? What did they know? Having joined the photography movement during the digital era, there was so much about the internal mechanics of exposure that I was not aware of.

A longer detecting surface or a smaller linear dispersion (spreading a range of wavelengths ∂λ over a smaller distance ∂x) allows the spectrometer to analyze more wavelengths in a single measurement.

Being on the scene is more important than worrying about technical details. This may be true in many a case, but when we're dealing with art as delicate and as intricate as prisming, we have to be a bit more mindful.

Ok, but numbers like f/2.8 or f/5.6 are still weird. Why not just f/1, f/2, f/3...? Does math exist purely to make life confusing?

This continues in a linear fashion as we continue increasing our aperture, leading to only 3,125 lux reaching our camera's sensor at ƒ/5.6, a dramatic decrease!

An estimate of the practical resolution of the spectrometer combines the linear dispersion with the physical dimensions of the detector array or the DMD, effectively shown in Figure G-3(b). In either case, a group of wavelengths incident on a single detector or on a single DMD mirror are measured simultaneously as a single unit with total power attributed to the nominal central wavelength of the group. Therefore, an estimate of the wavelength resolution of the spectrometer, considering only the grating and the analyzing array parameters, can be approximated by

Two different types of gratings can be used for spectroscopic applications. A ruled grating consists of a material into which a large number of parallel grooves are etched and then coated with a highly reflective material such as gold. A holographic grating is created by interference of two laser beams within a responsive material which results in a periodic variation in the refractive index of the material. Many processes can produce holographic gratings in a variety of materials, though the most common type found in spectrometers consists of a glass substrate exposed to interfering ultraviolet beams. Holographic grating can provide much higher angular dispersion due to the ability to write variations with very small periods, perform better in the ultraviolet end of the spectrum, and can be written onto curved surfaces to provide focusing capabilities in addition to the angular dispersion. However, holographic gratings are less efficient and more expensive than ruled gratings and absorption becomes an issue in the infrared end of the spectrum. Therefore, spectrometers operating in the infrared typically employ ruled gratings.

The answer is that effective focal length should be the factor which you pay attention to. That is to say, if you have a 25mm lens, and a crop factor of two, the effective focal length of your lens is:

Worried that will break your bank? Check out our writeup on why the nifty-fifty is the best and most inexpensive way to prism. The nifty fifty will allow you to shoot at ƒ/1.4 for a mere few hundred bucks—it's a prismer's dream!

Ok, before we get into the deep end.. let's back up a bit. What is a stop? Well, it's essentially a relative measure of light. By relative, I mean that a stop can only be judged in relation to an absolute exposure. For example, you can't look at a photo and describe it as having been shot at 12 stops.

Lens choice: the fixed-range 50mm lens (aka the "nifty fifty") might be the only lens you need for prisming. Here's why.

An interesting tidbit, the fastest lens in film history was the ASA/Zeiss 50mm f/0.7. We then know that wide open, the diameter of the aperture was:

We've demonstrated that you'll create larger circles of confusion and more defocusing (fokeh 🤭), when your aperture is shallower; in layman's terms:

So what aperture should you shoot? There is no strict answer to this question; but generally speaking, to emulate the prism effect that you get at f-stop ƒ/1.4 at 50mm, you may want to try an f-stop of around ƒ/4 if you're shooting at 85mm or f-stop ƒ/5.6 if you're shooting at 135mm. Your mileage may vary, and you want to be sure that you're moving your aperture up and down a lot to achieve an effect that's desirable for your scene, as these things tend to be highly context depending. The correct setting is in there—you just have to find it!

Note that a larger groove density D increase the angular dispersion, as does the use of higher diffraction orders. After diffraction by the grating, focusing optics direct the now dispersed light onto the linear detector array or DMD. The focusing optics may be the curved surface of the substrate of the grating or a combination of external lenses or mirrors. The optics convert the angular dispersion into a linear dispersion at the target surface, defined as the change in position on the surface as a function of the wavelength, as shown in Figure G-2. In general, the linear dispersion is given by

Bam! This fact matters a lot. Why? Well, most people think ƒ/2.8 is fast enough. But... if we're being prudent, we can dispel this with some math.

It's therefore important to note that longer focal lenghts require narrower apertures when prisming. In these scenarios, increasing your aperture will allow you to create smaller circles of confusion and create more attractive front bokeh.

Why halves, or doubles? Well, that's purely a matter of convention. Early lens designers decided that doubling or halving the current aperture, stop by stop, would be an easy way to move between different apertures. There's no law of nature that requires stops to be defined in this way—but it was defined that way, and it's a convention that we've all agreed on, it's here to stay.

We must strictly control our aperture. I've worked with prism photographers for many years now, and many of them pay little attention to their aperture, or fail to shoot aperture priority entirely. In prism photography, this won't work! We must be mindful of our aperture, and we'll often need to dial it up or down a stop to get the results that we're looking for. But we absolutely must make sure that we shoot as wide as possible, and wide focal lengths often prefer apertures like ƒ/1.4 or shallower.