Now… isn’t that special?!?!?!   Leave a comment

We learned something unexpected recently.

Other than microscope objectives, I use three photographic objectives- a wide angle (15/2.8), a ‘normal’ (85/1.4), and a telephoto (400/2.8). For astrophotography, I have always used the 800/5.6 lens combination (400mm + 2x tele) because I wanted to image faint, distant, objects. The large entrance pupil (143mm diameter) of the lens acts as a large ‘light bucket’, and the long focal length provides high magnification:


As I’ve posted previously, using either the 400/2.8 or 800/5.6 configuration, I stack around 100 15-second exposures acquired at ISO2000- and those 100 images are the top 20% in quality of all acquired images. By contrast, using a shorter focal length lens would increase the acceptance rate because the residual alignment error is below the resolution limit of the lens. Using the 85mm lens, I can acquire 30s exposures at ISO100 and use 100% of the images.

Picture saved with settings applied.

One way to quantify the ‘efficiency’ of a lens is to compare the size of the entrance pupil to the size of an Airy disc- under ideal conditions, the amount of light entering the lens is concentrated into an Airy disc. Analysis shows that the efficiency scales as f^2/N^4, where f is the focal length and N the f-number.

Calculating the efficiency provides the following table:

focal length f/# entrance pupil diameter Airy disc radius irradiance concentration relative gain
400.0 2.8 142.9 1.7E-02 298714.3 1.0E+00
85.0 1.4 60.7 8.5E-03 107910.6 3.6E-01
800.0 5.6 142.9 3.4E-02 149357.2 5.0E-01
15.0 2.8 5.4 1.7E-02 420.1 1.4E-03

What is surprising is the effect of the N^4 dependence: the 85mm is nearly as efficient than the 400mm (72%) in spite of having an entrance pupil substantially smaller.

This chart shows that the 85mm lens is very well suited for astrophotography in terms of viewing faint objects. However, the lower angular magnification results in a loss of spatial detail, meaning the 85mm is not suitable for viewing nebulae, clusters, galaxies, planets, etc. Here are image pairs comparing the 85mm and 400mm lenses:

Picture saved with settings applied.




The chart also (apparently) shows that the wide-angle lens is very unsuited for astrophotography- as compared to the 400/2.8, the efficiency is 0.1%. So, I was very surprised when I recently used the 15mm lens to image the galactic plane around the constellation Cygnus, with individual exposures of 13s @ ISO 2000:


This was completely unexpected. Even though the 15/2.8 is only 0.1% as efficient as the 400/2.8, the image brightness is as high as what I can generate with the 400/2.8, using essentially identical exposure times. How can this be?

The reason has to do with the density of stars. If I am imaging in a direction pointing out of the galactic plane where the density of stars is low, I would indeed conclude that the 15mm lens is unsuitable. However, each airy disc at the image corresponds to an angular field of view (named the ‘instantaneous field of view’, named back when detectors were single pixels and a scanning system was used to create an image plane), and since the field of view of the wide-angle is considerably more than the telephoto, there is an amplification factor due to the presence of multiple stars ‘sharing’ a single airy disc.



That is to say, when the density of stars is high and the lens magnification low, many stars will be imaged to the same airy disc, resulting in brighter images. How do the lenses compare with this metric?

At this point I have to mention the standard disclaimer about Bayer filters and color cameras. That said, measuring the size of an airy disc from the acquired images results in this table:

f f/# field of view (deg) field per pixel (rad) back projected solid angle for Airy disc (sterad) density gain magnification
400 2.8 5.2 1.50E-05 8.15E-09 1.0
85 1.4 23.9 6.92E-05 7.65E-08 9.4
800 5.6 2.6 7.52E-06 4.09E-09 0.5
15 2.8 100.4 2.91E-04 3.04E-06 372.8

Taking both factors into account results in this metric:

focal length f/# Overall efficiency
400.0 2.8 1.00
85.0 1.4 3.39
800.0 5.6 0.25
15.0 2.8 0.52

When the density of stars is high, I only need to double the ISO when using the 15mm in order to match the performance of the 400mm. Even more surprising, the 85mm lens is more than 3 times as efficient- in practice, this means I can use a lower ISO setting (ISO 100 instead of ISO 400), reducing the gain noise.

The moral of the story is: try new things!


Posted September 16, 2013 by resnicklab in Physics, pic of the moment, Science

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