# Does using a focal reducer shrink depth of field?

When I use a 50mm f/2 lens on an APS-C camera with a focal reducer (speed booster), the angle of view is similar to a full-frame camera. Will I also get shallower Depth of Field (more background blur), or does it remain the same?

The combined lens (50mm f/2 and a 0.71x speedbooster/telecompressor) will act like a 35mm/f1.4 lens in all regards (Same applies to teleconverters, in the other direction. lens designs exist that consist of a teleconverter or telecompressor behind a main lens, eg many 100mm macro lenses are 50mm macro lenses with a teleconverter in the back...).

DOF will be in the end dependent on aperture and magnification (subject to sensor). That truth is often arrogantly misinterpreted to say focal length doesn't matter re: background blur. It does, read on.

At THE SAME FRAMING, the actual DOF (in metres of depth) of the in-focus subject will, counterintuitively, be a bit shallower since you gained an f-stop of aperture.

A smaller acceptable circle of confusion for a smaller sensor will counteract this lightly, but in practice not by a whole stop.

The beginning and end of the DOF area around the focal plane might shift due to a change of camera to subject distance, which can also introduce apparent changes of DOF depending on what areas behind or before the focal plane matter in the picture.

Apparent background blur will still be less, since the change in distance needed for the same framing will decrease the magnification of the background.

Focal reducers convert both the focal length and aperture of the lens. You can think of it as multiplying by 1 = 0.7/0.7. Your 50/2 lens will behave like a 35/1.4 (= 50×0.7 / 2×0.7) lens when used with the focal reducer.

Although DOF and background blur tend to be inversely related (minimizing DOF will maximize background blur), they are different concepts. The formulas used to calculate them are different.

• Depth of field is based on focal length, aperture, distance, and a predefined acceptable sharpness level. It is concerned with what parts of the image are expected to be sharp, not what parts of the image will be blurry, or how blurry unsharp portions of the image will be.

What happens to DOF depends not only on focal length and aperture, but also what else you do to accommodate the lens change.

DOF = 2 u2 N C / f2  (from Wikipedia)

N = aperture F-number
C = circle of confusion
u = distance to subject
f = focal length

• Background blur – How blurry is the background? How big are bokeh balls a given distance from the lens? Different lenses with the same focal lengths, apertures, and distances can create different, though similar, amounts of blur because of different amounts of distortion, aberration, and field curvature. (There's also foreground blur, but people tend to be less interested.)

b = f ms xd / (N (s + xd))

b = blur
f = focal length
N = aperture F-number
ms = subject magnification
xd = distance between subject and background
s = subject distance

As the subject-background distance increases, the formula simplifies to: b = f2 / N s.  If the changes in focal length (f) and subject distance (s) are proportional, to maintain subject size within the frame, background blur is proportional to f/N (the physical aperture size).

Other related concepts are Bokeh and Subject-background isolation.

• Bokeh is a qualitative description of the blur that is produced. Are the bokeh balls round? Do they vary in shape throughout the frame? Are they smooth? Do they have edge highlights? Are they smeared? Some people refer to how lenses "render" images.

• Subject-background isolation refers to (subjectively) how well the subject stands out from the background. This can be achieved with depth of field and background blur, as well as appropriate lighting (such as rim lighting and creative use of "glow"). The common formula is to try to use narrow depth of field with high background blur. However, some types of bokeh can achieve good subject isolation with high depth of field and low background blur. For instance, Sonnar lenses create bokeh balls with an edge highlight toward the center of the frame, but a smear toward the edge of the frame. This tends to emphasize the sharpness of the subject toward the center, while also emphasizing the blurriness of the background toward the periphery.

We measure the potential “speed” of a camera lens using the f-number system. The f-number is derived by dividing the focal length of the lens by the working diameter of the lens aperture (iris).

As an example, a 50mm focal length lens with its aperture set to 25mm is operating at f/2.

If we wish this lens was faster, we can mount a “speed buster”. What happens is, this supplemental lens shortens the focal length of the primary lens. Most “speed buster” lenses reduce the focal length about 70%. Thus, a 50mm camera lens is converted 50mm X .7 = 35mm. Now on a full frame “FF” 50mm on a full frame is considered “normal”. By “normal”, we mean it will deliver an image with a perspective that closely replicates what we see with our unaided eyes.

If we mount a 35mm focal length lens on a full frame the resulting view is now classified as wide-angle. Please note: a 35mm mounted on a full frame delivers a moderate wide-angle. The shorter the focal length the wider the view.

The point is, the “speed buster” makes an adjustment, it shortens the focal length, and this widens the angle of view. Because the working focal length has been shortened, we must recalculate the f-number. Should the working diameter of iris remain unchanged, the math is 35 ÷ 25 = 1.4. The key point is the original and unmodified 50mm lens with a 25mm working iris diameter is operating at f/2. With the “speed booster” in place it becomes a 35mm operating at f/1.4. This is a gain of 1 f-stop in speed. A 1-f-stop gain doubles the amount of light that traverses the lens. This the “speed boost”.

The angle of view of a camera lens is a function of focal length and format size. If a 50mm is mounted on a FF (full frame 24mm by 36mm format), the angles of view will be 27° height, 40° length, 47° diagonal (the diagonal is the value most often published.

Mount a 35mm on a FF and the angles of view are 38° height, 55° length, 63° diagonal.

Mount a 50mm on a APS 16mm height by 24mm length the angles are 18° height, 38° length, 32° diagonal

Mount a 35mm on a APS angles are 26° height, 38 length, 45° diagonal As to depth-of-field – As a general rule-of-thumb, shorter focal length lenses ill exhibit a greater span of depth-of-field. Thus the “speed booster” causes the lens to gain speed while at the same time reducing the focal length which grants increased depth-of-field.

• There is no change in DOF (lens gives same amount of background bloor) when using speed booster, only angle of view change? Jan 11 at 18:46

## I. Short and step by step

1. When you put a 0.71 focal reducer (speed booster) on a a 50mm f/2 lens, the combo (lens + reducer) act as a 35mm f1.4 lens (1).

2. When you use a combo (lens + focal recuder) on an APS-C camera:

• The angle of view is similar the one when the 50mm is used on a full-frame camera.

• A gain of 1 stop in exposure is made: if the lens as an aperture ring: f2 on ring is becomes 1.4 on the sensor(2)(3).

This point is often overlook but has a very practical consequence: you gain 1/stop for ISO so at the end regarding the noise, this one will be more or lens similar than the on on the full frame.

• The depth of field is similar the one of the 50mm on a full-frame camera(4)(5)

## II. On the depth of field

The depth of field on a digital camera is a delicate subject because there are two different approaches.

The first, which has no equivalent in the field of analog photography, corresponds to the best possible image taking into account the characteristics of the sensor, i.e. the optical image on a sensor of a point in the scene must be smaller than the size of a pixel on this sensor.

The second, which is the one that will be discussed next, s related to the perception of a person looking at the image, the observer, looking at the image on a print, or possibly a screen assuming that the definition of this screen is such that the observer does not perceive the pixels of that screen.

It is the first definition that is used here.

## III. Formulas

Assuming that the distance to subject does not change

Values with a k index correspond to the case of a crop factor k;

With:

• Focal (mm): f
• Aperture: N
• Entrance pupil diameter (mm): d
• Circle of confusion (mm): c
• Distance to the subject (mm): D

### Aperture

From `N = f / d`, it follows that `Nk = fk / dk = (f / k) / d = (f / d) / k = N / k`.

### Depth of field

The size of the spot of the real point is a function of d (see Depth of Field Outside the Box by Richard F. Lyon), to have the same blur effect it is thus necessary that `dk = d`.

To be convinced of this, one can make the calculation in the formulas for the limit sharp plans in standard cases, i.e. outside of macro photography:

• Front Sharp Plan (mm) : `FSP = D H / (H + D)`
• Back Sharp Plan (mm) : `BSP = D H / (H - D)` and
• Hyperfocal (mm) : `H = f²/(c N)`

In order to maintain the size of the object point during visualization (on a print), the relationship between the confusion circles `c` and `ck` must be such that `ck = c / k`

It follows that:

``````Hk = fk² / (ck Nk) = (f / k)² / ((c / k) (N / k)) = f² / (c N) = H
``````

QED

## Notes

(1) Obviously the size of the image is also reduce, but this point is not relevant when a smaller sensor is used.

(2) I do not know if specific reducers modify the data so that the aperture displayed by the camera is or not modified.

(3) With a small approximation as the simple presence of new glass elements will reduce very marginally the light

(4) With a small approximation because 0.7 is not exactly equal to 1/1.5.