What the f/#?

The Tamron 70-200 f/2.8 Di VC USD zoom lens has a focal ratio of f/2.8. This defines the largest aperture the lens is capable of having at all focal lengths throughout the zoom range. Operating at f/2.8, the focal length selected will be 2.8X the size of the aperture. While the focal length range is 70-200mm, the range of largest apertures is 25mm at a 70mm focal length to 71mm at 200mm focal length.

The Tamron 70-200 f/2.8 Di VC USD zoom lens has a constant focal ratio of f/2.8. This defines the largest aperture the lens will have throughout the zoom range. Operating at f/2.8, the focal length selected will be 2.8X the aperture. With a focal length range of 70-200mm, the widest aperture varies from 25mm at a focal length of 70mm to 71mm at a 200mm focal length. (Bill Ferris)

Let’s nerd out with some tech talk. Let’s chat about focal ratio.

Focal ratio is a rarely seen or heard phrase in online photography blogs and forums, which is surprising when you consider the important role focal ratio plays in photography. Focal ratio describes the size of a lens’s focal length relative to its aperture. It is typically expressed as an f-number, such as f/2.8. Ironically, when photographers start talking about lens aperture, it’s more than likely they’re actually discussing focal ratio. Let’s see if we can sort all this out.

We’ll begin at the beginning. Focal length is typically the first number mentioned when describing a lens. A 50mm lens has a focal length of, wait for it…50mm or roughly two inches. One may be inclined to think focal length is the distance from the front of the lens to the back, but it’s not. Focal length is the distance from the optical center of the lens to the image plane (film or sensor) where the image is formed. The optical center is usually inside the lens and is sometimes referred to as the point of convergence; the point where two light rays converge and cross.

The above diagram shows a cross section of the Nikkor 50mm f/1.4 lens. The focal length of the lens is 50mm, which is measured from the optical center of the lens to the image plane at the sensor.

The above diagram shows a cross section of the Nikkor 50mm f/1.4 lens. The focal length of the lens is 50mm, which is measured from the optical center of the lens to the image plane at the sensor.

Focal length determines how much the image is magnified. This is typically described as the angle of view produced by the lens. A 50mm lens produces a 47° (on a diagonal) angle of view at the image plane of a 35mm camera body. A 24mm lens delivers an 84° angle of view and a 200mm lens presents a 12° angle of view. Since the angle of view produced by a 50mm lens is similar to that of normal vision, it is known in 35mm photography as a normal lens. 24mm is a wide angle focal length and a 200mm is a telephoto lens.

Of course, 35mm is just one of many photographic formats. A photographic format is defined by the physical size of the medium used to record the image. In film photography, 35mm describes the length of the long side of a slide or film negative. Today’s digital cameras use light-sensitive CMOS sensors to record images. In full frame digital cameras, the sensor measures 36mm on the longest side. APS-C digital cameras have sensors that are about 23mm on the longest side. The camera in your smartphone or tablet is probably built around a sensor no larger than about 10 millimeters. What does sensor size have to do with this topic? A lot.

The above diagram illustrates the relative sizes of common digital camera sensor formats. The largest shown is a full frame (FX) sensor. The smallesst (lower left corner) is representaive of a typical smart phone (1/2.3") sensor.

The above diagram illustrates the relative sizes of common digital camera sensor formats. The largest shown is a full frame (35mm equivalent) sensor. The smallest (lower left corner) is representative of a smartphone (1/2.3″) sensor.

The smaller the sensor or film medium, the farther you need to be from your subject to match the field of view delivered by a given focal length lens. Imagine standing 10 feet from your subject with a full-frame DSLR camera and framing your subject head-to-toe using a normal 50mm lens. If you were to mount the same lens on an APS-C camera body, that camera’s smaller sensor would cut off or crop a portion of the image produced by the lens. You would need to step back to a distance of about 15 feet to reproduce the angle of view you had with the full frame camera body.

Another factor to consider when shooting with a “crop sensor” body is the effect of sensor size on depth of field. Depth of field (DOF) is the range of distances – nearest to farthest – in an image that appear acceptably sharp and in-focus. DOF is determined by magnification (lens focal length) and by the lens focal ratio or f-number. In a nutshell, bringing the subject closer decreases depth of field. Moving the subject farther away increases depth of field. As depth of field increases, a deeper portion of the image appears in focus. As depth of field decreases, only a narrow or shallow range looks sharp and in focus.

Both photographs were made using a Nikon D610 with Tamron 70-200 Di VC USD zoom lens at 125mm. The image on the left was shot at f/2.8 and has a much shallower depth of field. The image on the right was shot at f/32 and presents a much wider depth of field.

The above photographs were made using a Nikon D610 and Tamron 70-200mm f/2.8 Di VC USD zoom lens at 125mm. The image on the left was shot at f/2.8 and has a much shallower depth of field. The image on the right was shot at f/32 and shows much more of the field in focus.

As mentioned, focal ratio also has an effect on depth of field. For any given focal length, increasing focal ratio (making the f-number larger) increases depth of field while decreasing focal ratio (making the f-number smaller) reduces depth of field. We’ve already discussed the cropping effect of shooting with a smaller sensor. Stepping back to reproduce a desired angle of view increases depth of field. Zooming or changing lenses to shoot with a shorter focal length (to match the field of view provided by a full frame sensor body) increases depth of field.

One can compensate for the increased depth of field which results from the adjustments commonly made to expand the angle of view delivered by a crop sensor camera by shooting with smaller f-numbers. For example, shooting with a 35mm lens at f/1.4 will allow an APS-C sensor body to produce photographs having the same framing and depth of field as images made from the same position using a 50mm f/2.0 lens on a full frame body.

Let’s explore this in a bit more detail. Suppose you’re shooting with two cameras, one full frame and the other a crop sensor, and using the same 50mm lens with both. Its effective focal length (the focal length matching the angle of view delivered to the sensor) will be 50% longer or 75mm on the APS-C body. At f/4, the 50mm lens will have an aperture of 12.5mm. If we step back to compensate for the more narrow angle of view, the effective focal ratio (the focal ratio delivering an equivalent depth of field from the distance at which this lens matches the angle of view delivered to a full frame camera) will be f/6. Its effective 75mm focal length divided by the 12.5mm aperture equals six.

Do you see the relationship? We’re using an f/4 lens on an APS-C body. When the goal is to match the angle of view and depth of field produced by a full frame camera, we can determine the effective focal ratio at which a crop sensor camera needs to operate by dividing the focal ratio of the lens by the crop factor. The crop factor is 1.5 and the effective focal ratio (for depth of field) is f/6.

Here’s an illustration.

These images illustrate how to use a crop sensor camera to match both the angle of view and the depth of field delivered by a full frame body. I used a Nikon D610 and Nikon D90 to make photographs of the same toy caboose. Both cameras used a Tamron 70-200 f/2.8 Di VC USD lens. The lens was mounted on a tripod and the bodies switched out to ensure the lens would not move from its position during the test. The D610 uses a 36mm sensor and shot at 105mm, f/4 to make both images. The D90 uses an APS-C sensor with a 1.5X crop factor. I shot at 75mm, f/4 to make the first image. Comparing the first (top) images, we see that the D90 delivered a similar angle of view as the D610 but a comparison of the background shows the D610 to have a more shallow depth of field. The background in the D90 image is just skosh nearer to being in focus. For the second image, I applied the conversion factor and shot with the D90 at 70mm, f/2.8. A comparison of this image with the D610 image shows both to have delivered similar angles of view and similar depth of field. (Bill Ferris)

These images illustrate how to use a crop sensor camera to match both the angle of view and the depth of field delivered by a full frame body. I used a Nikon D610 and Nikon D90 to make photographs of the same toy caboose. Both cameras used the same Tamron 70-200 f/2.8 Di VC USD lens. The lens was mounted on a tripod and the bodies switched out to ensure the lens would not move from its position during the test. The D610 is built around a 36mm sensor and was used at 105mm, f/4 to make both images. The D90 has an APS-C sensor with a 1.5X crop factor. I shot at 70mm, f/4 to make the first image. Comparing the first (top) images, we see that the D90 delivered a similar angle of view as the D610 but a comparison of background detail reveals the D610 to have a more shallow depth of field. The background in the D90 image is just a skosh nearer to being in focus. For the second image, I applied the conversion factor and shot with the D90 at 70mm, f/2.8. A comparison of this image with the D610 image shows both to have delivered similar angles of view and similar depth of field. (Bill Ferris)

So, we’ve demonstrated that, in comparison with full frame cameras, crop sensor camera bodies produce images having narrower angles of view and, when adjustments are made to compensate for this, increased depth of field. We’ve also demonstrated that you can compensate for these performance factors. Either increase the distance between you and the subject or use a shorter focal length to increase the angle of view. Shoot at a smaller focal ratio (f-number) to make the depth of field more shallow. Next, we’ll explore the relationship between sensor size and length of exposure. Here’s a heads up, the outcome may not be what you expect.

I used my Nikon D610 (full frame) and Nikon D90 (APS-C) to take a series of exposures of a toy train engine. The toy steam engine was set up outside on a small tray table. The sky was overcast with nice, even lighting throughout the test. Both bodies used the same Tamron 70-200mm f/2.8 Di VC USD lens, which was set at 70mm. I selected ISO 200 on both cameras for all exposures. The zoom lens was set up on a tripod and the camera bodies were switched out without changing the position of the lens. I used each camera to make exposures at f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22 and f/32. I shot in aperture priority on both cameras and let their internal brains select the proper exposure.

Below, are pairs of images showing the photographs made at the same settings with the two bodies, side-by-side. All are unedited JPEGs. Keep in mind that the sensor in the D90 body cropped the image to match the angle of view produced by a 105mm lens.

In this comparison, photographs of the same subject made with a Nikon D610 (left) and a Nikon D90 (right) are shown, side-by-side. Both cameras shot at ISO 200. Both cameras used the same Tamron lens at 70mm. The lens was mounted on a tripod to ensure it would remain in the same position throughout the test. For each focal ratio, both cameras used the same exposure. (Bill Ferris)

In this comparison, photographs of the same subject made with Nikon D610 (left) and Nikon D90 (right) cameras are shown, side-by-side. Both cameras were set to ISO 200. Both cameras used the same Tamron lens at 70mm. The lens was mounted on a tripod to ensure it would remain in the same position throughout the test. At each focal ratio, both cameras metered the scene as having the same brightness and chose the same exposure. (Bill Ferris)

Let’s talk more about this f-number thing. You’ll recall that focal ratio describes the ratio of the focal length of the lens to the aperture of the lens. A 50mm lens at f/2.0 has a focal length that is 2-times its aperture. Therefore, the lens aperture at f/2.0 will be 25mm. At f/4.0, the aperture is 12.5mm; at f/8.0, 6.25mm and so on. The relationship between aperture and focal ratio is pretty straight forward: for any given focal length, decreasing aperture increases focal ratio and increasing aperture decreases focal ratio.

Rarely, however, do photographers talk about the f-number as a focal ratio. More commonly, they talk about it as a lens aperture. They talk about an f/2.0 lens having a larger aperture than an f/4.0 lens. It’s an accurate statement, if we’re talking about the same lens at different focal ratios. But this is just one of many scenarios where focal ratios are compared.

Let’s consider the scenario of discussing different lenses. Suppose we’re comparing a 50mm lens to a 100mm lens. Suppose the 50mm lens is being used at f/2 and the 100mm lens is set to f/4. One might think the 50mm lens, by virtue of having a smaller f-number, will have a larger aperture. In fact, both lenses have identical 25mm apertures. It simply isn’t the case that every f/1.4 lens has a larger aperture than every f/8 lens. In reality, it is quite common for a lens operating at a large f-number to have a larger aperture than a lens working at a small f-number. I would wager to guess that there isn’t a focal ratio at which a 600mm lens doesn’t have a larger aperture than the fastest focal ratio smartphone.

One quality that does translate across different lenses and cameras, is the speed of the imaging system. What does speed have to do with photography? To understand, it helps to think of a properly exposed photograph as one where a certain intensity of light needs to fall upon the sensor at the image plane. Think of light as water, the sensor as a container used to collect water (light) and the lens as the opening through which water is poured into the container.

That said – and this next point is critical – a properly exposed image is not determined by the total quantity of light delivered to the sensor. The length of a proper exposure is determined by the average brightness of the image falling on the sensor. To better understand this, we’re going to introduce a new concept: surface brightness.

The above illustrates the concept of Surface Brightness in photography. For a properly exposed image, the camera's optical system must collect and deliver light having a surface brightness (brightness or intensity per square millimeter) to the sensor. This is represented by the evenly deep layer of "blue" light collected bu the sensor. If you use the same lens on a crop sensor body, the same intensity of light (represented by the central red region) is delivered to the sensor. Being smaller, the crop sensor collects less total light. However, the surface brightness of the image (the brightness per square millimeter) is identical to that of the larger sensor. (Bill Ferris)

The above illustrates the concept of Surface Brightness in photography. For a properly exposed image, the camera’s optical system must collect and deliver light having a surface brightness (brightness per square millimeter) to the sensor. This is represented by the thick layer of “blue” light collected by the sensor. The thickness of the layer represents the intensity or average brightness of the image. If we use the same lens on a crop sensor body, the same intensity (thickness) of light is delivered to the sensor. This is represented by the central red zone on the sensor. Being smaller, the crop sensor collects less total light. However, the surface brightness of the image is identical to that of the larger sensor. (Bill Ferris)

Earlier, a correct exposure was described as one where a container (sensor) is filled to the correct depth (intensity) with water (light). It doesn’t matter if the container is large enough to hold one gallon or 100 gallons. As long as it’s filled to the proper depth, the exposure will be good. In this example, the depth of the water represents the average brightness of the image at the image plane. Another way to describe the average brightness or intensity of light, is to talk about image surface brightness.

Surface brightness is defined as a brightness per unit area. In photography, we can define surface brightness as the brightness of light per square millimeter falling on the film or sensor. It is not a total volume or quantity of light. Rather, it is an average intensity of light. Surface brightness is strictly determined by the focal ratio of the optical system. The lens f-number determines the length of the exposure needed to deliver light of a certain intensity to the sensor. A full frame camera, crop sensor camera and smartphone camera focused on the same subject – and all operating at f/2.0 – will deliver the same light intensity per square millimeter (the same surface brightness) to their respective sensors during the same length exposure.

The relative sizes of full frame (pink) and APS-C (blue) sensors is illustrated above. The effects of a crop frame sensor include an increase in effective focal length and an increase in effective depth of field.

The relative sizes of full frame (pink) and APS-C (blue) sensors is illustrated above. The effects of a crop frame sensor include an increase in effective focal length and effective depth of field.

Despite the fact that a crop sensor doesn’t collect as much total light during an exposure as a full frame sensor, the intensity or surface brightness of the images formed on both sensors will be the same. We saw this at work in the above illustrations comparing exposures made with the D610 and D90. Despite the fact that, during each set of exposures, the D90’s smaller sensor collected less total light than the full frame sensor of the D610, the image made by the D90 was still properly exposed. This is because the exposures made by both cameras produced images having identical surface brightness at the image plane.

This set of images compares performance between crop sensor and full frame DSLR bodies. The images in the left column were made with a Nikon D90. Images in the right three columns were made with a Nikon D610. Both cameras used the same Tamron 70-200mm f/2.8 Di VC USD zoom lens, which was set up on a tripod to ensure it would not change position during the test. Both cameras used ISO 200, center point average metering and were operated in Aperture Priority. The subject in these photos is a scale model of the Lunar Excursion Module (LEM) from the Apollo program.

This set of images compares performance between crop sensor and full frame DSLR bodies. The images in the left column were made with a Nikon D90. Images in the right three columns were made with a Nikon D610. Both cameras used the same Tamron 70-200mm f/2.8 Di VC USD zoom lens, which was set up on a tripod to ensure it would not change position during the test. Both cameras used ISO 200, center point average metering and were operated in Aperture Priority. The subject in these photos is a scale model of the Lunar Excursion Module (LEM) from the Apollo program.

The above illustration allows us to compare the performance of crop sensor and full frame cameras. The first column of D610 exposures matches the settings of the D90 images in the left-most column. Focal length and focal ratio are the same. In most cases, both cameras’ metering systems selected the same exposure. The most obvious difference between the D90 and first set of D610 images is the wider angle of view delivered by the full frame sensor. For the second set of D610 images, I zoomed in to match the effective focal length of the D90. The angles of view of these images closely match the corresponding D90 exposures. The second set of D610 images were shot at f/2.8 and clearly display a more shallow depth of field. For the third set of D610 photographs, I changed the focal ratio to match the depth of field presented in the D90 images. Notice that the exposures for these images are all 1/800-second. They’re longer to compensate for the larger focal ratio.

Focal Ratio is the key to understanding how different cameras, lenses and sensors are able to make good photographs using the same or similar length exposures. Focal ratio determines the length of time needed to collect enough light to make an image having the required surface brightness. For any two cameras operating at the same ISO and delivering the same angle of view, the exposure times will typically be the same.

So, the next time you read or hear a photographer talking about an f/1.4 lens having a larger aperture than an f/2.0 lens, stop and give that statement some thought. If the lenses being compared are a 20mm f/1.4 and a 50mm f/2.0, the 50mm lens will be operating with a larger aperture. The 50mm lens will have a 25mm aperture at f/2.0 and the 20mm, f/1.4 lens aperture will be just over 14mm. However, due to its faster focal ratio, the 20mm lens will deliver more light per square millimeter to the sensor, faster. Because the f/1.4 lens produces a brighter image – an image having a higher surface brightness – the length of the exposure will be shorter.

In photography, the objective is not to deliver the largest volume of light to the sensor. The objective is to deliver the needed intensity (surface brightness) of light to the sensor. Speed is everything and focal ratio is the key.

Now, get out there and shoot!

Bill Ferris | August 2015

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