Properties of Closeup Lenses
This post is divided into 4 main sections:
A) Focal Lengths and Principal Planes of the Raynox250 and Raynox150
B) CloseupLens Magnification Formula with Infinite Focus of the Camera
 B1) The Formula
 B2) Actual vs. Nominal Focal Lengths of Telephoto Lenses
 B3) Magnification Test on the Raynox250
C) Depth of Field Using a Closeup Lens and Infinite Focus of the Camera
 C1) The Formula
 C2) DOF Test Shots
D) Appendix: Derivation of Magnification and DOF Formulas
A) Focal Lengths and Principal Planes of the Raynox250 and Raynox150
The nominal values of focal length of the Raynox250 and Raynox150 closeup lenses are:
f_Raynox250 = 1000 / 8 diopters = 125 mm (nominal value)
f_Raynox125 = 1000/ 4.8 diopters = 208.3 mm (nominal value)
Wanting to check the nominal values of the focal lengths, I performed a series of tests, using parallax to determine the locations of the images for various object distances. I mounted the closeup lens on a level tube on the floor next to a metrestick and located the image using a vertical pin and moving my head back and forth until there was no parallax with the image. At first, I tried using the thin lens formula, but could not get consistent values for the focal lengths. The closeup lenses are too thick to be treated as thin lenses. The PanoHelp site explains how to modify the thin lens formula for use with thick lenses:
http://www.panohelp.com/thinlensformula.html
The thin lens formula can still be used but object distance is measured from the object's principal plane and the image distance is measured to the image's principal plane. I could find nothing online that specifies the location of these principal planes.
Besides the series of tests using the parallax method, I measured the the object distance to the front of the camera barrel, with closeup lens attached and with my Canon SX30 set to maximum optical zoom (nominally 150.5 mm) and infinite focus. These distances were 138 mm for the Raynox250 and 223 mm for the Raynox125 and the image distance is infinity. For each closeup lens, I lumped this information together with my parallax measurements and fitted the three unknowns: focal length, principal plane of the object H_obj and principal plane of the image H_img. Here are the results:
Raynox250 Properties
f_Raynox250 = 125.0 mm +/ 0.5 mm
H_obj = 13 mm from back surface of universal adaptor
(5 mm from front surface of lens housing)
H_img = 9 mm from back surface of universal adaptor
(9 mm from front surface of lens housing)
So, the nominal focal length of 125 mm for the Raynox250 is accurate.
Raynox150 Properties
f_Raynox150 = 211.0 mm +/ 0.5 mm
H_obj = 12 mm from back surface of universal adaptor
(6 mm from front surface of lens housing)
H_img = 4 mm from back surface of universal adaptor
(14 mm from front surface of lens housing)
So, the focal length of 211.0 +/ 0.5 mm for the Raynox150 is a little larger than the nominal value of 208.3 mm.
B) CloseupLens Magnification Formula with Infinite Focus of the Camera
B1) The Formula
I often take closeup shots using the Raynox250 or Raynox150 or a homemade 600 mm closeup lens on my Canon SX30.
When focusstacking a series of shots, using macro rails, the camera focus is manually set to infinity. For that infinitefocus setting, the magnification of the closeup lens is:
m = f_b / f_c [1]
where
 f_b is the "base" focal length of the camera at infinite focus
 f_c is the focal length of the closeup lens
 m, the magnification, is the size of the image on the sensor divided by the size of the object being photographed. (In photography, it is called the subject but in optics, it is called the object.)
http://www.alanwood.net/photography/closeuplenses.html
(The formula describes the tabulated data on this website for infinite focus of the camera lens.)
I was not able to find a derivation of Equation 1 online, but it is derived below as Equation 11 in the Appendix.
If infinite focus is not used, then the nominal value of focal length is not achieved (as given in EXIF files). In that case you will not know the value of f_b, just that it is smaller than the value in the EXIF file and that the amount of magnification will be less than calculated from Equation 1 using the value of f_b in the EXIF file. (To observe this effect, Google "focus breathing" on YouTube.)
"It is a little known fact that telephoto lenses can have a radically different actual focal length depending upon focus distance (Google 'focus breathing' for more info). For example, a Nikon AFS 28300mm f/3.55.6G ED VR lens set to 300mm will have the expected actual focal length near to 300mm when focused far away  but has an actual focal length less than half that when focused at 18 inches!" http://www.panohelp.com/lensfov.html
B2) Actual vs. Nominal Focal Lengths of Telephoto Lenses
A problem that I encountered when using the formula is that the Canon SX30's focal lengths appear to be smaller than nominal values, at least at the telephoto end. I reached this conclusion after several tests with different closeup lenses and several with the camera alone, all at the cameras maximum optical zoom (nominally 150.5 mm) and infinite focus. In all cases the image was too large to fit on the camera's sensor. The 1/2.3" sensor of the Canon SX30 has dimensions 6.17 mm x 4.55 mm, but the largest 4:3 frame that could fit in it would be 6.067 mm x 4.55 mm. My practice now is to use a telephoto focal length that is 95% of the nominal value.
Here is some confirmation that this may be a legitimate assumption:
"Lenses are rarely exactly as marked when it comes to focal length (or aperture, though that's another issue...), especially telephoto lenses." [table] "As you can see, it doesn't much matter who made the lens or what the maximum focal length was, in every case the measured maximum focal length is less than the marked maximum focal length, and the average difference is about 5%. The difference is always on the short side. I didn't find any examples where the actual focal length of a telephoto was longer than marked!"
 Bob Atkins, discussing the focal lengths of SLR lenses http://www.bobatkins.com/photography/technical/focus_breathing_focal_length_changes.html
If SLR telephoto lenses have actual focal lengths that are on average 5% smaller than specified values, it seems likely that this practice may also apply to fixedlens cameras. I have therefore assumed that when my Canon SX30 is set to its maximum optical zoom of nominally 150.5 mm (840 mm effective) at infinite focus, its actual focal length is:
f_b = 0.95 x 150.5 mm = 143.0 mm
B3) Magnification Test on the Raynox250
Here is a test shot of a reticle with 0.1 mm gradations taken with the Raynox250 closeup lens on my SX30 with focal length set to nominally 150.5 mm and infinite focus:
From Equation 1, using an actual focal length of 143 mm, the magnification is:
m = 143 mm / 125.0 mm = 1.144
The image size in this case is the width of the frame on the sensor, which is m times the object size (5.32 mm in the photo):
Frame width on sensor = m x 5.32 mm = 6.09 mm
This is about 0.4% larger than the width of 6.067 mm, which is the largest frame width that will fit on the sensor. The reason for this small remaining discrepancy is that use of 95% of the nominal focal length was an average of several tests, both with the camera alone and with closeup lenses mounted. For the Raynox250 and Raynox150, the percentage of nominal focal length was 94.6%; for a 600 mm closeup lens, it was 95.7%; for the camera alone at maximum zoom, it was 95.4%. These variations may be my experimental error and I have chosen 95% +/ 1%. As in the test jpeg shot above, the image width was always close to the full frame width, so the calculated magnifications represent an average over almost the whole frame width. It is possible that there is some distortion and that magnification may be larger at the centre of the frame than at the edges. The incamera RAWtojpeg conversion removes the distortion and may crop the image but I would expect both the distortion and the cropping to be very slight at the telephoto end. For SLR lenses, focal lengths that are 95% of nominal are a real property of the lenses and not an effect of RAWtojpeg conversion. My present thinking is that the conversion plays only a very minor part in the 95% factor. If there is cropping, the frame width on the sensor becomes less than 6.067 mm, resulting in a factor of less than 95%  say 94%.
The magnifications for the SX 30 at full optical zoom (0.95 x 150.5 mm = 143.0 mm) and infinite focus are given by:
m_Raynox250 = 143.0 mm / 125.0 mm = 1.144
m_Raynox150 = 143.0 mm / 211.0 mm = 0.678
C) Depth of Field Using a Closeup Lens for Infinite Focus of the Camera
C1) The Formula
Many websites deal with depthoffield but usually say that their formulas are valid "except for closeup and macro photography"
http://encyclopine.org/en/Depth_of_field
I used the methods described in the PanoHelp reference, above, to model the closeup lens and the camera lens as two thick lenses with a separation d between the closeup lens' imageprincipalplane and the camera lens' objectprincipalplane.
As shown in the Appendix (Equation 14), below, the depthoffield (DOF) is given by:
DOF = 2NC / m^2 [2]
with the camera set to infinite focus,
where
 N is the F number
 C is the circle of confusion = 0.005 mm
 m is the magnification as calculated above from Equation 1
For the SX30 set to full optical zoom (nominally f = 150.5 mm but actually 143 mm and infinite focus, with the Raynox 250:
m = 143 / 125 = 1.144
DOF = 2 x 5.8 x 0.005 / 1.144^2 = 0.044 mm @ f/5.8
= 2 x 8.0 x 0.005 / 1.144^2 = 0.061 mm @ f/8.0
This result surprised me, as I had thought a separation of 0.1 mm was adequate for focusstacking under these conditions (full zoom + Raynox 250). Fortunately, I do not take many focusstacked shots under these extreme conditions, but I decided to check.
C2) DOF Test Shots
The image is of a single grain of table salt on 1 mm graph paper.
The first image is the result of focusstacking 37 shots, using a 0.05 mm separation.
The shots were focusstacked using Zerene Stacker, then cropped.
The second image stacks every other shot to obtain a 0.1 mm separation:
The third image stacks every fourth shot to obtain a 0.2 mm separation:
My conclusion is that the formula works pretty well. If you want the best quality, use 0.05 stacking separation, which is approximately what the formula indicates. At 0.1 mm, the results are still acceptable. The deterioration is slight and can only be noticed by looking at the shots alternately. I would say that 0.2 mm separation is not acceptable.
D) Appendix: Derivation of Magnification and DOF Formulas
The lens formula for the closeup lens is:
1/ Oc + 1 / Ic = 1 / fc [3]
where
 Oc is the object distance for the closeup lens (measured from the object principal plane)
 Ic is the image distance for the closeup lens (measured from the image principal plane)
 fc is the focal length of the closeup lens
When used with the camera lens set to infinite focus, the object distance Oc for the Raynox250 has to be within about 0.1 mm from the focal point.
Rewriting Equation 3,
1 / (fc + delta_Oc) + 1 / Ic = 1/ fc [4]
where delta_Oc is the object distance measured to the focus.
Solving for the image distance:
Ic ~ fc^2 / delta_Oc [5]
 For delta_Oc >0, the object is slightly farther from the lens than the focus and a real inverted image is produced far behind the camera. For example, the Raynox250 with delta_Oc = 0.1 mm, Ic = 125^2 / 0.1 = 156,250 mm.
 For delta_Oc < 0, the object is slightly closer to the lens than the focus and a virtual erect image is produced far in front of the camera the camera.
The closeup lens, considered alone, produces a very large magnification:
mc = Ic / Oc = Ic / (fc + delta_Oc) ~ Ic / fc [6]
The image produced by the closeup lens becomes the object for the "base" camera lens, where the object distance is given by:
Ob = d  Ic [7]
where d is the separation between the closeup lens' image principal plane and the camera lens' object principal plane.
The camera lens with infinite focus produces its image close to its focus according to the formula:
1 / Ob + 1 / (fb + delta_Ib) = 1/ fb [8]
where
 Ob is the object distance for the "base" camera lens (measured from the object principal plane)
 delta_Ib is the image distance for the camera lens (measured from the focus)
 fb is the focal length of the camera lens
Solving for delta_Ib:
delta_Ib = fb^2 / Ob ~ fb^2 / (d  Ic)
~ fb^2 / Ic [9]
The magnification of the camera lens alone is very small because the object distance is so large. The magnification is given by:
mb = Ib / Ob = (fb + delta_Ib) / Ob
= (fb + delta_Ib) / (d  Ic) ~  fb / Ic [10]
The total magnification from the closeup lens and camera lens is:
m = mc mb = fb / fc [11]
The lens equation, from Equations 5, 7 and 9, for both lenses together is given by:
delta_Oc = fc^2 / Ic = fc^2 / (d  Ob)
= fc^2 / [ d ( fb^2 / delta_Ib )] [12]
Because the image produced by the closeup lens is so far away, the value of d has negligible importance. I verified this in spreadsheet calculations, where it made a negligible difference whether d was set to 0 or 1000 mm. I also verified it experimentally with both the Raynox250 and Raynox150 by increasing the gap between the closeup lens an the camera up to 1 m, with negligible change of image size.
Setting d to zero in [12] and substituting [11] yields:
delta_Oc =  delta_Ib / m^2 [13]
with the camera set to infinite focus.
The near and far boundaries of the DOF are defined by focus distances delta Ib of +NC (behind the sensor) and NC (in front of the sensor),
where
 N is the F number
 C is the circle of confusion = 0.005 mm. Jeffrey's EXIF viewer gives 0.005 mm shots taken with my SX30 http://regex.info/exif.cgi This agrees with the formula C = sensor diagonal / 1500 http://en.wikipedia.org/wiki/Circle_of_confusion
The total DOF is given by:
DOF = 2NC / m^2 [14]
with the camera set to infinite focus.
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Aug 10, 2014  4 
