Imaging With Magnification Test Procedure

by Michael J. Ferry, William Shensky III, and Andrew G. Mott

ARL-TN-0333 September 2008

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Army Research Laboratory

Adelphi, MD 20783-1197

ARL-TN-0333 September 2008

Imaging With Magnification Test Procedure

Michael J. Ferry, William Shensky III, Andrew G. Mott

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution is unlimited. 
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Imaging With Magnification Test Procedure

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Michael J. Ferry, William Shensky III, Andrew G. Mott

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U.S. Army Research Laboratory

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Adelphi, MD 20783-1197

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Approved for public release; distribution is unlimited.

13. SUPPLEMENTARY NOTES

14. ABSTRACT

Although modern cameras have advanced to the point where pixel size approaches 4 microns, we have yet to achieve the submicron size 

necessary to accurately resolve focused laser beams. Present methods such as pinhole scans and knife edge scans assume spatial wave forms 

to fit the transmission, and those assumptions may be erroneous. This technical note details a method of imaging with calibrated 

magnification and commercial laser beam profiling software to accurately measure beam sizes. Experimental layout is detailed, all reference 

material included, and the equations stated.

15. SUBJECT TERMS

Imaging, laser beam, magnification, USAF 1951

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Michael Ferry

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iii

Contents

Contents iii

List of Figures and Table iv

1. Introduction 1

2. Theory of Magnification 1

3. Test Procedure 2

4. Numerical Example 8

5. Summary 9

References 10

Distribution List 11
iv

List of Figures and Table

Figure 1. Layout of imaging with magnification experiment.........................................................3

Figure 2. Typical test patterns.........................................................................................................4

Figure 3. Relative dimensions of a test pattern element. ................................................................5

Figure 4. Example image of a magnified element. .........................................................................6

Table 1. Lookup Table for Spatial Frequencies of the USAF Test Pattern. ...................................5
1

1. Introduction

This technical note details a procedure for accurately measuring the laser beam diameter of a 

focused laser beam when the spot size is on the order of or less than the camera pixel dimension 

of a commercial beam profiling system. Imaging with magnification is used to expand the beam 

to a large enough size to have many pixels across the beam. Magnification is calibrated with test 

patterns, and the actual beam size is determined. This technical note is intended for the bench 

level scientist.

2. Theory of Magnification

The relevant equations (1) are

1/fim = 1/So + 1/Si (1)

M = - Si /So (2)

where,

fim = focal length of the imaging lens

So = object distance, the distance from lens to the Air Force test pattern

Si = image distance, the distance from lens to the camera

M = magnification

Solving equations 1 and 2 for Si gives

Si = fim × (1–M), (3)

which is the image distance as a function of the magnification and focal length. Solving

equation 1 for the corresponding object distance gives,

So = 1/(1/fim–1/Si). (4)

Negative magnification means an inverted image, which is what occurs with single lens imaging. 

Equation 3 makes apparent that image distance increases as a product of imaging lens focal 

length and magnification. When imaging with magnification, the negative lens with the shortest 

focal length available should be used to reduce the amount of bench space needed for 

measurement. For magnification, |Si| > |So|, the image distance is the limiting factor. 
2

A typical top-hat beam focuses to

dfocus = 2.44 × λ × fsys /dlens (top-hat beams) (5)

where dfocus = the diameter of the Airy disk in the focal plane; λ is the wavelength; fsys is the focal 

length of the lens of the system lens; and dlens is the near field diameter of the laser beam at that 

lens.

To determine how much magnification is required, assume a resolution requirement that the 

magnified beam is spread across at least 10 pixels on the CCD (2).

dmag > 10 × p (6)

where dmag is the beam diameter of the magnified beam on the camera, and p is the smallest pixel 

dimension. The magnified diameter of the laser beam on the camera is equal to the 

magnification times the real beam diameter at the focus of the system lens,

dmag = M × dfocus or M = dmag /dfocus. (7)

Solving equations 7, 6, and 5 for magnification gives

M > 10 × p/dfocus. (8)

Note: if plane waves are used instead of Gaussian beams, equation 5 becomes

dfocus = (4/π) × λ × fsys /dlens (Gaussian beams). (9)

Real beams are not diffraction limited, so the minimum required magnification predicted by 

equation 8 may be excessive.

3. Test Procedure

The first part of the procedure is to block out the experiment for desired magnification and image 

the test pattern on the camera. To begin, obtain a rough estimate of object and image distances. 

Use the equations from the theory of magnification section to determine the object and image 

distances needed for the imaging lens. This will help block out the lens and camera positions on 

the optical bench. See figure 1. 
3

Figure 1. Layout of imaging with magnification experiment. 

NOTE: The system lensFsys focuses the laser beam near the location of the Air Force

 Test Pattern (AFTP). The imaging lens fim is located at the object distance So

 from the test pattern. The camera is located at the imaging distance Si from the 

 imaging lens.

Once the system is blocked out according to distances predicted by theory, it is then time to 

image the test pattern. To do this, begin by observing on the bench where the system lens 

focuses the laser beam. Mark the focal position and block or shutter the laser beam. Remove the 

system lens, fsys, and place an Air Force Test Pattern (AFTP) at the marked position. 

Illuminate the transparent AFTP with a fiber light. The fiber light is desirable since bright light 

is required to image the test pattern. Gray scale rather than color scale is used on the camera to 

get fine resolution of bar edges. It is helpful to have the test pattern on an X-Y stage to easily 

scan the pattern across the beam for the series/element that fills the camera array. Place the 

imaging and system lenses on a Z-axis translation stages. Adjust the object distance with the 

imaging lens micrometer until a sharp, focused image of a single 3 bar element fills the camera 

array. 

Sometimes it is difficult determining from the camera image which group and element is imaged 

on the camera. Translating the pattern with the X-Y micrometer to search for the group/element 

numbers, is one way to decide which element was imaged. Another quick way is by placing an 

8.5 by 11 in. white paper in front of the camera to display a large section of the magnified image 

of the test pattern. All the other series/elements that image outside of the active area of the 

camera can then be seen.

Spatial distances separating the bars on an AFTP are well known and can be acquired by 

referring to the appropriate table for an AFTP. Figure 2 shows what a typical AFTP looks

like (3).
4

Figure 2. Typical test patterns.

NOTE: Shown are commercially available examples of the standard USAF 1951 test pattern. Group number is the 

large number on top of the largest element of the group, and element number is either to the left or right of the 

pattern element.

The group number (large number above the first one in the series) and an element number 

(smaller number to the left of the series, ranging from 1 to 6) uniquely identify any 3 bar 

element. In general, the elements descend in group number in a clockwise inward spiral. The 

exception is the first element of the left most series appears in the lower right hand corner of the 

test pattern.

The relative dimensions on any given element (4) are found in figure 3.
5

Figure 3. Relative dimensions of a test pattern element. 

NOTE: The quantity x is a spatial frequency of line pairs per millimeter (units mm-1) and 0.5/x is

the width of a single bar (mm), with 2.5/x being the length of a bar.

Here x is the lines per millimeter, which may be obtained for any given group/element from a 

lookup table. The lookup table is found in table 1. Notice that standard 2 in. square commercial 

test patterns typically begin with group 0 and leave off groups –1 and –2.

Table 1. Lookup table for spatial frequencies of the USAF test pattern. 

Number of Line Pairs / mm in USAF Resolving Power Test Target 1951

Group Number For High Res only

Element -2 -1 0 1 2 3 4 5 6 7 8 9

1 0.250 0.500 1.00 2.00 4.00 8.00 16.00 32.0 64.0 128.0 256.0 512.0

2 0.280 0.561 1.12 2.24 4.49 8.98 17.95 36.0 71.8 144.0 287.0 575.0

3 0.315 0.630 1.26 2.52 5.04 10.10 20.16 40.3 80.6 161.0 323.0 645.0

4 0.353 0.707 1.41 2.83 5.66 11.30 22.62 45.3 90.5 181.0 362.0 -----

5 0.397 0.793 1.59 3.17 6.35 12.70 25.39 50.8 102.0 203.0 406.0 -----

6 0.445 0.891 1.78 3.56 7.13 14.30 28.50 57.0 114.0 228.0 456.0 -----

NOTE: Each number in this table is x/2, half a line pair per millimeter. Invert numbers from this table to get the width of a 

single bar.

Alternatively, the spatial frequency for a group/element may be calculated by

x = 2(g + 1 + (e–1)/6) (10)

where g = group number, e = element number, and x is the line pairs per mm.

After the group and series numbers of the imaged element are known, the next step is to measure 

the size of the image on the camera. Typical commercial laser beam profilers permit the user to 
6

manually slew a cursor across the image, and record the cursor position. To avoid noise issues 

with imaging, measure 2 line pairs to average out flaws in image position. 

For example, assume the 3 vertical bar element is imaged on the camera (see figure 4). Most 

beam profilers have a cross section feature. In this case, position the cursor and observe the 

horizontal cross section where it is a simple matter to go between identical points on the square 

wave form (shown in red along bottom of figure 4). To measure the image size in x, move the 

cursor to the left edge of the first vertical bar element, record the horizontal location, then move 

the cursor horizontally to the left edge of the 3rd element and record the location. The difference 

in these two positions divided by 2 is the magnified width of a line pair: 1/x. Looking up x from 

table 1, the magnification, M, may be then calculated from equation 7 by letting the magnified 

width of a line pair equal dmag and the actual width of the line pair from the lookup table be dfocus.

Figure 4. Example image of a magnified element. 

NOTE: Shown is group 2, element 5 as captured by a COHU 6612 digital camera. The cursor presently is at position (4861, 

2416 μm) at the left edge of bar 3, and the left edge of bar 1 was at (1020, 2416 μm). Two periods of this element are 

315 μm, and the magnified distance for two periods was 4861–1020 = 3841 μm, implying a 12.2 × magnification.

The next part of the procedure involves putting the system lens focus at the AFTP position.

Return the system lens, fsys, and remove the test pattern. During this part, do not move the 
7

imaging lens or camera. Moving either the imaging lens or camera changes the magnification

and forces recalibration of magnification. Put filters in front of the system lens to reduce the 

laser intensity to a very low level that will not damage the camera or unshutter the laser. Make 

slight adjustments to the system lens position with the translation stage to put the focus where the 

AFTP used to be. The system lens is in the correct position when the beam size on camera is at a 

minimum.

The final part of the procedure is to measure the magnified image size on the camera. With the

system lens position optimized, use the beam profiling software of the camera to determine the 

magnified image diameter on the camera, and then divide that by M to get the actual image size.

Beam profiles may be properly scaled by dividing by the magnification.

The procedure outlined above calibrates magnification first, and then measures the spot size. 

Sometimes it is advantageous to do the reverse, where the spot size is measured and then the 

experiment disassembled to measure magnification. In this “calibrate last” variant, it is 

important to place the AFTP exactly at the focus of the system lens, and rather than translate the 

system lens in z, the AFTP is translated in z to bring it into focus on the camera. Just as in the 

“calibrate first” method, the calibrate last procedure requires the image distance to remain 

constant after the magnified laser beam has been measured, and the imaging lens and camera 

should not be moved to avoid having to re-measure the magnified beam diameter. 

The steps for the calibrate last procedure are (1) with the laser on, adjust the imaging lens 

position to minimize spot size on the camera and record the magnified diameter, (2) block the 

laser and pull the system lens, (3) insert the AFTP on a 3 axis stage, adjusting x - y to get the 

appropriate element on the array then z to focus that element on the array, and (4) determine the 

magnification by measuring 2 periods of the 3 bar pattern in a manner similar to that described 

for the calibrate first procedure. The calibrate last procedure is most useful when it is desirable 

to hold the system lens position fixed. Calibrating first permits multiple system lenses to be used 

and resulting focal diameters measured with a single calibration.

Finally, it is worth noting the methods developed for imaging with magnification (i.e., expanding

very small laser beams onto cameras), may also be used for the inverse case of imaging with 

demagnification, where the size of a very large laser beam needs reducing to fit on a small 

camera array. The main differences in imaging with demagnification are (1) the object distance 

is now much smaller than the image distance, and (2) the large beam may be rough measured and 

compared to the known array size to estimate the required demagnification.
8

4. Numerical Example

A two-times diffraction-limited 532 nm top-hat laser beam, with a dlens = 60 mm, is focused by 

an fsys=100 mm lens. From equation 5, the predicted diffraction limited diameter is

dfocus= 2.44 × (0.532 μm) × (100 mm)/(60 mm) ~2.2 μm. 

Since the beam is two-times diffraction limited, the real diameter should be approximately 4 μm 

at focus. Assume the camera available to measure this beam has a p = 5μm pixel dimension.

From equation 8, the desired magnification is:

M > 10 × (5 μm)/(4 μm) = 12.5.

With an fim = 25.4 mm imaging lens, equation 3 can be used to determine the image distance,

Si = (25.4 mm) × (1–12.5) = 292 mm ~11.5 in.

where distances have been converted to inches because lab benches are ruled in inches and the 

point of this calculation is to estimate layout spacing. From equation 4, the corresponding object 

distance is:

So = 1/(1/(25.4 mm) –1/(292 mm)) = 23.4 mm ~1 in.

If the available bench space is 30 in., and the camera takes up 4 in of that, 26 in. is left for the 

object and image distances, which is more than the required ~12.5 in. The camera is placed 

So = 24 in. from the imaging lens and the AFTP is mounted on a translation stage just under 1 in.

from the imaging lens.

Assume the camera has a 4 by 4 mm active area. To accurately measure the laser beam, the

largest magnified beam used on this camera should be ~3 mm, or 75% of the active area. In 

principle, it could go as high as M = 3 mm/0.004 mm = 750, but bench space limits the 

magnification to a much lower number ~24/1 or M = 24. The beam diameter should be about 

dmag = 4 × 24 = 96 μm on the camera, and have ~96/5 = 19 pixels across the beam.

To get a single 3 bar element to fill up 75% of the active area, the bar length, when magnified,

should be 3 mm. For a magnification of 24, the actual bar length should be 3 mm/24 = 125 μm. 

From figure 3, the bar length is related to the spatial frequency by

2.5/x = 125 μm = > x = 2.5/0.125 mm = 20 line pairs per mm. 

The chart in table 1 shows that group 3, element 3 has x/2 = 10.1 = > x ~20 line pairs per mm. 

Therefore, group 3, element 3 should be roughly the correct element that fills up the 4 by 4 mm 

array.
9

Typical beam profiling software yields the most accurate results when data > 2.5 times the beam 

diameter is excluded from the calculation. Therefore, a 96 μm diameter beam should have a

calculation exclusion diameter of 240 μm. On some profiling systems it is possible to define an 

exclusion diameter, while on others exclusion is achieved by digitally zooming in the profiling 

software to push extraneous data outside the calculation region. In the latter method to achieve

an exclusion diameter filling 75% of the screen, then the 3 mm/0.24 mm = 12.5 × camera zoom

would be required.

5. Summary

A procedure has been detailed for the quantitative measurement of small laser beams by the 

method of imaging with magnification. Equations were supplied to estimate approximate 

distances for the layout of the actual experiment. Usage of USAF 1951 to quantify 

magnification was detailed. A numerical example was provided to underscore the method.
10

References

1. Hecht, E.; Zajac, A. Optics; 4th ed.; Addison-Wesley Publishing Company, Inc. Reading, 

MA, 2002; p 108, p 112.

2. Macgregor, A. Beam Profiling: Know Your Beam, Laser Focus World.

http://www.laserfocusworld.com/articles/255504 (accessed Mon May 01 00:00:00 CDT 

2006).

3. Earl F. Glynn’s Image Processing Tech Note on USAF 1951 and Microcopy Resolution 

Test Charts and Pixel Profiles.

http://www.efg2.com/Lab/ImageProcessing/TestTargets/#USAF1951 (accessed 20 June 

2002).

4. Edmund Optics Online Catalog. Specification Table and Technical Images of 1951 USAF 

Glass Slide Resolution Targets. 

http://www.edmundoptics.com/onlinecatalog/displayproduct.cfm?productID=1790&search=

1#SpecificationTable (accessed 20 June 2008). 
11

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