Math for macro photographers

I have collected some equations and examples that might be useful for macro shooters. The frequently asked questions like magnification ratios when using extension tubes or bellows, close-up lenses, reversed lenses and stacked lenses are covered here. I also have included the equation to calculate the light loss stemming from the use of extension tubes (or bellows) and in the final chapter are some fundamental geometric formulas used in the design of optical systems.

Those readers more familiar with this subject will notice, that in all but the last chapter I have used the "rounded" focal length of 50 mm when using the Nikkor  AF 50/1.8D in the calculations. The lens has actually a focal length of 52.5 mm when it is focused to infinity, but for the sake of clarity I decided to use the published values. This same applies to all lenses, the actual focal lengths and aperture values are rounded to the nearest convenient value. There propably is no lens in the world with a focal length of 50.0 mm and a maximum aperture of f/1.80.

In chapter 9 I present a actual test of all the formulas with accurate numbers, or as accurate as I could come up with.

Contents:

• extension
• reversed lens
• reversed lens with extension
• stacked lenses
• stacked lenses with extension
• close-up lenses
• light loss with extension
• basic geometric formulas
• testing the formulas in action

1. Extension

Extension rings are an inexpensive way to increase lens magnification. Extension rings do not have any glass elements or lenses, they are just empty tubes that add some distance between the actual lens and the camera. Current Nikon models do not provide autofocus or cpu contacts, but with all but the cheapest Nikon digital SLR cameras the exposure meter is still functional.

Nikon PK-13 (27.5 mm) extension ring

The formula is

where
M = magnification
m_L = the magnification of the lens alone (see lens manual)
F = the lens focal lenght in millimeters
E = added extension in millimeters

My Nikkor AIS 105/2.8 can achieve 1:2 (= 0.5) magnification by itself. We can calculate the magnification of the system, when using this lens and two extension tubes totalling 80 millimeters, the PN-11 (52.5 mm) and PK-13 (27.5 mm).

M = ( (0.5 x 105) + 80 ) / 105 = 1.26 » 1.3

Magnification of AIS Nikkor 105/2.8 with 80 mm extension is 1.3X.
 

2. Reversed lens

A lens can be mounted backwards, or in a reversed position, with a reversal ring. The effect is that the distance between the subject and the camera can be decreased and thus image magnification increased. With a reversal ring autofocus is not functional and exposure metering works only with selected cameras.

Nikon BR2A lens reversing ring

We cannot directly calculate the zero extension magnification (Z_m) of the lens we wish to mount in a reverse position. Instead we have to take a test photograph of a ruler to determine this value or find the information in the lens manual. In my case I reversed my Nikkor AF 50/1.8D using the Nikon BR2A reversal ring, focused the lens into infinity and took a photo of a ruler.

32.5 millimeters wide subject

The final image of the ruler scale showed 32.5 millimeters. The Nikon D200 camera sensor is 23.6 millimeters wide, thus the magnification is

Z_m = image width/subject width = 23.6/32.5 = 0.73

Magnification of a reversed Nikkor AF 50/1.8D is 0.73X.

3. Reversed lens with extension

The formula to calculate the magnification of a reversed lens with extension is simple, but one must first determine the zero extension magnification (Z_m) of the lens in question. That differs from lens to lens and cannot be directly calculated. See chapter 2 to determine the Z_m of your lens.

The formula is

where
M = magnification
Z_m = zero extension magnification of the reversed lens
E = added magnification in millimeters
F = the lens focal length in millimeters

Using Nikkor AF 50/1.8D reversed with a PK-13 extension tube (27.5 mm) we can calculate the final magnification to be

M = 0.73 + (27.5/50) = 1.28 » 1.3

Magnification of a reversed Nikkor AF 50/1.8D with 27.5 mm extension is 1.3X.

4. Stacked lenses

We can obtain high magnifications when we stack two lenses. We only need a cheap stacking ring with male threads on both sides to screw the lenses together.

A standard 52 to 52 mm stacking ring standing on top of a Nikon lens cap

Nikkor AIS 105/2.8 with a stacking ring and a Nikkor 50/1.8D reversed

The primary lens will be mounted normally to the camera and the stacked lens is screwed in reverse position to the end of the primary lens. The primary lens should have higher focal length than the stacked lens and the stacked lens cannot be a zoom. It is usually best to set the stacked lens aperture wide open. The reversed lens in the end of the primary lens acts as in a similar way as a close-up attachment lens, but it usually has a very high diopter value.

The formula to calculate the magnification of two stacked lenses is

where
M = magnification
F_P = focal length of the primary lens
F_R = focal length of the reversed stacked lens

Using my AIS Nikkor 105/2.8 as the primary lens and mounting the Nikkor AF 50/1.8D in a reverse position with a stacking ring we get

M = 105/50 = 2.1

The total magnification of a Nikkor AIS 105/2.8 with a stacked Nikkor AF 50/1.8D is 2.1X.

5. Stacked lenses with extension

To calculate the magnification when using stacked lenses with extension, it is easier if we first calculate the resultant focal length of the stacked lens system. The formula to calculate the focal length of a stacked lens system is

where
F_tot = total focal lenght of the stacked system
F_P = focal length of the primary lens
F_R = focal length of the reversed stacked lens

Using my AIS Nikkor 105/2.8 as the primary lens and mounting the Nikkor AF 50/1.8D in a reverse position with a stacking ring we get

F_tot = (105 x 50) / (105 + 50) = 33.8 » 34 mm

The total resultant focal length of a Nikkor AIS 105/2.8 with a stacked Nikkor AF 50/1.8D is 34 mm.

When we know the total focal length of a stacked lens system, we can add extension and calculate the final magnification with the normal extension formula presented above in chapter 1 and we find the necessary m_L with the formula in chapter 4. For example, adding a PK-13 extension tube (27.5 mm) to the stacked lenses presented here yields magnification

M = ( (2.1 x 34) + 27.5 ) / 34 = 2.91 » 2.9

Magnification of AIS Nikkor 105/2.8 with a reversed AF Nikkor 50/1.8D and 27.5 mm extension is 2.9X.
 

6. Close-up lenses

To calculate the magnification when using a close-up attachment lens follows the same rules as using a stacked lens. When the primary lens is focused to infinity, the subject-to-sensor distance is the same as the focal length of the close-up lens. The strength of close-up lenses is usually given in dioptres, but it is simple to calculate the focal length of a close-up lens.

Canon 500 D close-up lens (2 diopter)

The formula is

F = 1000/C

where
F = focal length of the close-up attachment lens
C = strength in dioptres

For example, the popular Canon 500D close-up lens has a strength of 2 dioptres, thus we find out that it's focal length is F = 1000/2 = 500 mm.

Case A. Infinity focus

If we attach this close up lens to the Nikkor AF-S VR 70-200/2.8G zoom lens zoomed to 200 mm and focused to infinity, we can calculate the magnification with the formula given in chapter 4.

M = 200/500 = 0.4

Case B. Minimum focus

The Nikkor lens has a magnification of 1:5.6 (0.18X) at the minimum focusing distance of 1.4 meters (1400 mm) and the focal length decreases to 180 mm at the longest zoom setting according to the formula (b) in chapter 8

F = 1400 / (5.6 + 1/5.6 + 2) = 180 mm

The lens-to-sensor distance T at minimum focus is according to formula (c) in chapter 8

T = (0.178 + 1) x 180 = 212 mm

(the lens has 12 mm of internal extension at minimum focusing distance)

We can calculate the magnification at minimum focusing distance of a Nikkor AF-S VR 70-200/2.8G lens with 500D using the formula (a) given in chapter 8 if we first determine the F_tot of the stacked system (chapter 5)

F_tot = (180 x 500) / (180 + 500) = 132 mm

M = (212/132) - 1 = 0.6

7. Light loss with extension

We can calculate the loss of light that results when using extension, even though modern cameras take this into account when using TTL (through-the-lens) exposure metering.

The formula for light loss (in stops) is

where
LL = light loss in stops
E = added extension
F = focal length of the lens
ln = the natural logarithm function

If we use 72.5 mm of extension (27.5 + 55 mm) with the Nikkor AIS 105/2.8 lens, the light loss caused by the extension is

LL = (2 x ln(1 + 72.5/105)) / ln 2 = 1.5 stops

This formula also explains the changes of effective aperture that modern cameras indicate when using cpu macro lenses with internal extension, for example the Nikkor AF-S 60/2.8G Micro.

8. Basic geometric formulas

The basic formula is that of the reproduction ratio or magnification, which states that

Magnification = size_on_sensor / actual_size

That is to say, if we shoot a 10 cent coin with actual diameter of 15 millimeters and it is projected to the sensor (or film) with a diameter of 7.5 millimeters, we have a reproduction ratio or magnification of 7.5:15 = 1:2 = 0.5.

According to the laws of geometrical optics we find that

where
M = magnification
T = distance between lens and sensor (from the lens' optical center)
F_tot = the total focal lenght of the lens system (camera objective)

According to this formula (a) we find that magnification increases if we increase the distance (extension) between film and lens and/or we decrease the focal length. The formula indicates also that when a lens is focused to infinity, T = F. With a subject in infinity, it will be produced to sensor (or film) with nil size, because T/F = 1 and thus M = 0.

Many lenses have the property that focal length shortens when the lens is focused closer than infinity. We can use this basic relation

where
D = focusing distance (distance between subject and sensor)
F = focal length
M = magnification

If we solve this equation in respect to F, we arrive at

We can calculate the focal length of the Nikkor AF-S 70-200/2.8G at the closest focusing distance, where M=0.18 (1:5.6) and D=1400 mm (1.4 meters)

F = ( 1400 / ( 5.6 + 0.18 + 2 ) ) = 180 mm

If we solve the equation (a) in respect to T, we arrive at

With all lenses focused to infinity T = F but at closer distances the T increases due to the internal extension built into lenses. For example the Nikkor AF-S VR  70-200/2.8G has

T = (0.18 + 1) x 180 = 212 mm

because M = 0.18 and F = 180 mm at minimum focus

Thus we can see that the lens has 12 mm of internal extension at the minimum focusing distance.

9. Testing the formulas in action

To verify the accuracy of the formulas, I made some testing. Here is one such example, which uses almost all the formulas presented here. This time no approximations or rounding is done, so I used Nikkor lens database to find exact values and used several decimal places in the calculations.

Sample image showing 3.25 millimeters high subject

I decided to use Nikkor AIS 105/2.8 as the primary lens and a Nikkor AF 50/1.8D stacked to it in reverse position, and to top it all I added 107.5 mm of extension (PN-11 + 2 x PK13) between the camera and the primary lens.

We will use the formula above (first presented in chapter 1), but first we need to know the focal lenght of  the full stacked lens system (F_tot) and the magnification of the stacked lenses (m_L). To calculate the F_tot we need first to know the focal length of the AIS 105/2.8 (let's call it F₁₀₅) when it is focused to the minimum focusing distance of 410 mm (and at magnification of 0.5).

Formula (b) in chapter 8 gives focal length in minimum focusing distance as:

F₁₀₅ = (410/(0.5 + 2 + 2)) = 91.1 mm

The exact focal length of the Nikkor AF 50/1.8D when focused to infinity is 52.5 mm (Nikkor lens database). According to the formula in chapter 5 we get the total system focal length of these 2 lenses:

F_tot = (91.1 x 52.5) / (91.1 + 52.5) = 33.31 mm

The m_L magnification of this lens system can be calculated with the formula in chapter 4:

m_L = 91.1/52.5 = 1.74

Now we have all the necessary values that the formula requires:

M = magnification
m_L = 1.74
F = 33.31
E = 107.5

Solution:
M = ( (1.74 x 33.31) + 107.5 ) / 33.31 = 4.97

Conclusion

The sample image in the beginning of this chapter is shows 3.25 mm high subject. It is actually the same ruler that I used earlier in chapter 2. The sensor in D200 is 15.8 mm high, so the actual magnification is

M = 15.8/3.25 = 4.86

The actual magnification appears to be 4.86 and the calculation arrived at 4.97  which isn't all that bad, since the error is only 2.2%. This error most likely stems from small inaccuracies in the published values of the AIS 105/2.8, as well as some rounding in the calculations.

Last edit: March 22, 2008 3:24 PM