Edmund Optics USAF Calculator: Resolution & Chart

The Edmund Optics USAF calculator works out the limiting resolution of an optical system from the USAF 1951 test target line pair spacing, and it converts between line pairs per millimeter (lp/mm), line widths, and the equivalent pixel or sensor size needed to resolve them. You enter the target group and element numbers (or the spatial frequency directly), and the calculator returns the line pair spacing in micrometers and the resolution in lp/mm. This page explains the math behind the USAF 1951 target and how to use the calculator correctly for lens testing and optical system evaluation.

Formula

The USAF 1951 target consists of groups and elements. Each group has six elements, numbered 1 to 6. The spatial frequency (resolution) in line pairs per millimeter is given by:

Resolution (lp/mm) = 2^{(group + (element – 1)/6)}

Where:

• group = the target group number (can be negative for large patterns, e.g., group -2)
• element = the element number within the group (1 to 6)
• Resolution = number of line pairs per millimeter (lp/mm)

The line pair width (the distance from the start of one dark line to the start of the next dark line) is the reciprocal of the resolution in mm, converted to micrometers:

Line pair width (µm) = 1000 / resolution (lp/mm)

Each individual line (dark or light) is half the line pair width, so the line width is 500 / resolution.

Worked Example

Using the default values: group = 2, element = 4.

1. Compute the exponent: group + (element – 1)/6 = 2 + (4 – 1)/6 = 2 + 0.5 = 2.5
2. Resolution = 2^2.5 = 5.65685 lp/mm (rounded to 5.66 lp/mm)
3. Line pair width = 1000 / 5.65685 = 176.78 µm (rounded to 176.78 µm)
4. Line width = 500 / 5.65685 = 88.39 µm

The calculator shows the same: 5.66 lp/mm, 176.78 µm line pair width, and 88.39 µm line width.

How the USAF 1951 Target Works

The USAF 1951 resolution test chart is a standard pattern used to measure the resolving power of optical systems. It consists of groups of three-bar patterns, with each group containing six elements. The bar sizes decrease logarithmically as the group number increases. The spatial frequency doubles every group, and within a group each element is spaced by a factor of 2^1/6 (about 1.1225).

To use the target, you photograph or view it through the optical system and find the smallest group and element where the three bars are still distinguishable. That group and element number give you the resolution limit of the system.

How to Use the Calculator

To get the resolution from a USAF target reading:

• Enter the group number (can be negative, e.g., -2 for very large patterns).
• Enter the element number (1 to 6).
• The calculator outputs the resolution in lp/mm, line pair width in µm, and line width in µm.

If you have a measured resolution in lp/mm and want to find the corresponding group/element, you can work backwards: find the group by taking the base-2 logarithm of the resolution, then determine the element from the fractional part.

Common Mistakes and Misconceptions

A frequent mistake is confusing line pairs per millimeter with line widths per millimeter. A line pair consists of one dark line and one light line, so there are twice as many individual lines as line pairs in a given distance. For example, 5 lp/mm means 10 lines per mm (5 dark and 5 light).

Another misconception is that the USAF target resolution directly equals the lens’s diffraction limit. In practice, the resolution you measure depends on the sensor pixel size, the lighting, the contrast of the target, and the quality of the optical system. The USAF target gives an overall system resolution, not just the lens resolution.

Also, be careful with group numbers: group 0 has a specific base size, and group -1, -2, etc. are larger patterns used for low-magnification systems. The formula works for all integer groups, but the physical target may not include all groups.

Practical Applications

The USAF 1951 target is used in many fields:

• Lens testing – to compare the resolving power of different lenses.
• Microscopy – to calibrate magnification and resolution.
• Machine vision – to verify that a camera system can resolve the required detail.
• Astronomy – to test telescope optics.

For example, if a machine vision camera has a sensor with 5 µm pixels, the Nyquist limit is about 100 lp/mm (since one line pair needs at least two pixels). Using the target, you can check if the lens can actually deliver that resolution.

Limitations of the USAF Target

The USAF 1951 target has some limitations:

• It is a binary (high-contrast) target, so it doesn’t measure contrast at different spatial frequencies like a sine-wave target would.
• The subjective judgment of when bars are resolved can vary between observers.
• It gives a single resolution number, not a full modulation transfer function (MTF) curve.
• The target is typically printed on paper or film, and the print quality can limit the accuracy.

For more detailed analysis, optical engineers often use MTF measurements with sinusoidal patterns or edge targets.

Frequently Asked Questions

What does lp/mm mean?

lp/mm stands for line pairs per millimeter. It is a measure of spatial resolution: how many pairs of one dark and one light line fit in one millimeter. Higher lp/mm means finer detail can be resolved.

How do I convert lp/mm to line width?

Line width (the width of a single dark or light line) is half the line pair width. So line width (µm) = 500 / lp/mm. For example, 100 lp/mm corresponds to a line width of 5 µm.

What is the difference between group and element?

The group number sets the base spatial frequency, doubling each group. The element number (1-6) subdivides each group into six steps, each increasing the frequency by a factor of 2^(1/6). So group 2 element 1 is lower resolution than group 2 element 6.

Can I use this calculator for a specific Edmund Optics target?

Yes, the formula works for any USAF 1951 target, regardless of the manufacturer. The group and element designations are standardized. Just enter the group and element numbers you read from the target.

How do I determine the group and element from a resolution value?

Take the base-2 logarithm of the resolution in lp/mm. The integer part is the group number. Multiply the fractional part by 6 and round to get the element number (1-6). For example, 5.66 lp/mm gives log2(5.66)=2.5, so group 2, element 4.

What is the smallest feature the USAF target can measure?

It depends on the target version. Standard targets go down to group 7 element 6, which is about 228 lp/mm (line width ~2.2 µm). Some high-resolution targets go to group 9 or higher. The calculator works for any group number you enter.

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in g | Group number | - | 2 | 0
in e | Element number | - | 4 | 0
out lpmm | Resolution | lp/mm | 2^(g + (e-1)/6) | 2
out um | Line pair width | µm | 1000 / (2^(g + (e-1)/6)) | 2
out linew | Line width | µm | 500 / (2^(g + (e-1)/6)) | 2