# Field Curvature : A Guide for Photographers

2023-01-09

This series of articles conceptually explains aberration, which is a performance index of a lens.

Of all the aberrations, this section introduces "curvature of field" in a simple and easy-to-understand manner using figures and simulations.

Related terms such as "astigmatic difference", "sagittal direction" and "tangential direction" are also explained.

Introduction to Lens Design

## Aberration and Curvature of Field

First, aberration is an index that shows the deviation from the ideal imaging relationship.

The camera lens is designed to focus the light emitted from the "subject" precisely on the imaging surface (film or CMOS).

However, it is difficult to focus light accurately, and the small amount of deviation is called "aberration".

These aberrations are classified and named according to their characteristics.

In the first article, we introduced spherical aberration, but this time we will explain "curvature of field".

The spherical aberration and axial chromatic aberration explained in the past are aberrations that show the performance of the center of a photograph, but the field curvature is an aberration of the peripheral part of a photograph.

In more detail, curvature of field is expressed by dividing into "sagittal direction" and "tangential direction", and the difference between the two is called "astigmatic difference".

In this section, I would like to explain the "astigmatic difference", "sagittal direction", and "tangential direction", which are deeply related to "field curvature", so that we can understand them as simple concepts as possible.

## Let's imagine

Let's say there are some stars in the pitch-dark sky and you want to take a picture.

There were three stars, in the center, periphery, and in the corners of the screen. For the sake of illustration, let's name the stars.

Now, let's take a picture of three stars. First, let's focus on the star in the center.

If you look at the result of the shooting, stars are separated by hundreds of millions of light-years, so they should be shot as extremely small points. However, the "center star" is shot as a small point, but the "periphery stars" and "corner stars" are slightly blurred.

Next, I focused on the "nearby stars" and took a picture.

This time, the "surrounding stars" appear as small dots, but the "center stars" and "corner stars" are blurred.

The presumed phenomenon from this result is that the focal point is out of focus at the center / periphery / corner.

In other words, this phenomenon is named "field curvature" because the plane on which light is focused (the image plane) is not flat (curved).

I will explain how the curvature of field appears in the optical path diagram that appears every time in the lens analysis article on this blog using a simulation.

As an optical path diagram, the above example of "photographing a far-off star" is shown below.

The image above shows a 50 mm F1.4 lens consisting of only one spherical lens.

Imagine using an inexpensive magnifying glass, such as those sold in 100-yen shops, as a lens for your camera.

The true distance to the star is hundreds of millions of light-years away, but since it is impossible to draw on the diagram, please note that it is quite close.

First, let's observe how the light passes through the lens.

The left side of the figure is the star side. Light emitted from the star passes through the lens and is focused on the image sensor (CMOS or film) on the right side.

Since it is only one lens, various aberrations are large and it is difficult to see, but there is a large gap between the image formation position of the "central star" and the image formation position of the "surrounding stars".

## Conceptual diagram of Field Curvature

Since it is somewhat difficult to see in the overall drawing, we expanded the range from the lens to the image pickup device and drew an auxiliary line in the optical path diagram to create a conceptual image of field curvature.

Since it is only one lens, the spherical aberration is large and it is difficult to see, but the image formation point (focus position) is connected with a vermillion line.

The difference between the focus point in the center and the focus point around it is connected to make a graph.

This curve shape corresponds to the aberration diagram of field curvature.

## Represented by an aberration diagram

Now, let's try to express it with the aberration diagram introduced in the usual analysis article.

I annotated the usual graph in Japanese.

The graph in the center of the longitudinal aberration diagram shows the field curvature.

In addition, the horizontal axis scale of the aberration diagram in the usual lens analysis article is ± 0.5 mm, but the aberration amount of this single lens is too large, so the horizontal axis scale is ± 10 mm. Please note that it is 20 times the normal value.

The graph also shows two line types, the dashed line is tangential and the solid line is sagittal.

The line color indicates the wavelength of light (color of light) as in the case of axial chromatic aberration in the spherical aberration diagram. For a basic explanation of the wavelength, see the article on axial chromatic aberration.

What is axial chromatic aberration?

## What is astigmatism?

In the explanation of the field curvature graph, I explained "sagittal direction" and "tangential direction" as a matter of course, but this will be supplemented.

"Sagittal direction" or "tangential direction" refers to the direction in which a ray is viewed: vertically, horizontally, or diagonally.

First of all, please look at the usual optical path diagram.

This path diagram shows only the "surrounding stars" mentioned in the previous "Shooting Stars" story. The usual path diagram shows the lens viewed from the side.

Since a real lens is a three dimensional object, there is also a world in the depth direction as shown in this figure.

## 3D optical path diagram

Let's display the light path as a three dimensional model that is easy to understand.

Since this is a single lens, the aberration is large and it is difficult to see, but the optical path of the "nearby stars" mentioned earlier is drawn in three dimensions.

The front side is the subject side and corresponds to the surface side of the lens, and the back side is the image pickup device side.

The 3-D drawing looks interesting, but it's hard to see the light rays, so I usually only show the cross-section in the horizontal direction in the analysis article.

## Sagittal and tangential directions

In this case, only the rays in the radial direction are seen in the tangential direction, and the rotation direction is in the sagittal direction.

This is what it looks like: When you look at the lens from the front, the rays in the radial direction of the screen are tangential.

The tangential direction is the direction connecting the central axis of the lens and the image height where the light hits.

The direction of the of the arrow shown above for the position of the "surrounding stars".

A ray that is perpendicular to a ray that is tangential is sagittal.

Now that we know the sagittal and tangential directions, let's take a look at the actual path diagram.

## Tangential direction as seen in a 3D optical path diagram

In the 3D view, if you draw only the rays in the tangential direction, it looks like this:

The above diagram shows the light path only in tangential direction.

If you are drawing a ray of light to the image height straight up, it will look like the usual path diagram when viewed from the side.

## Sagittal direction as seen in a 3D optical path diagram

Then, in the 3D view, draw only the sagittal ray as shown below.

If you look at this ray from above, it looks like the following figure.

Do you understand what a sagittal ray and a tangential ray look like?

If you look at the light rays in more detail, the focal point of the light rays differs between the tangential direction and the sagittal direction.

The difference between the focal point in the tangential direction and the focal point in the sagittal direction is called "astigmatic difference".

It is sometimes written as astigmatism.

If the astigmatism is large, the focal point in the tangential direction does not match the focal point in the sagittal direction.

Then, what originally appears at the point is shaped like an ellipse.

## Correct the Field Curvature

Up until a while ago, I explained with 1 lens, but in reality, such a bright Fno lens is composed of many lenses.

Let's look at the characteristics of the lens corrected for spherical aberration.

Earlier, I introduced the aberration of 50 mm F1.4 lens composed of only one lens.

The figure below shows the spherical aberration of a double Gaussian type 7-piece lens on the same spherical aberration graph scale.

### Longitudinal aberration

Graphs of spherical aberration, field curvature, and distortion

Aberration of the first single-component lens The graph is made at 20 times the normal size according to the horizontal axis scale of the figure.

Looking at the spherical aberration diagram, there is some doubt as to whether it was necessary to display it as an approximate straight line, but this lens is in the lowest price class. However, it utilizes the characteristics of the double Gaussian type lens, which is the crystal of human wisdom, and is able to correct aberrations with a very small number of lenses, so you can understand how amazing it is.

Check out the original article if you want to see it in its usual graph scale.

Analysis: NIKON NIKKOR 50 mm f1.4D

## Conclusion

To put it simply, curvature of field is the defocusing of the peripheral part of the screen.

I think that you can enjoy our blog 20 times by understanding the field curvature.