<blockquote>
<p>...if we illuminate a 1-d aperture of, say, size 1, the resultant power pattern is a sinc function in the sine of the angle ...</p>
</blockquote>
<p>While you've not yet provided a link to the discussion you're reading, nor a block quote of it, nor at least some page with a similar derivation, I'll take a stab at an answer guessing that you're referencing <a href="https://en.wikipedia.org/wiki/Fraunhofer_diffraction#Single-slit_diffraction_using_Huygens%27_principle" rel="nofollow noreferrer">single slit diffraction</a>.</p>
<p>In this case the &quot;forward&quot; problem is a plane wave incident on a 1D slit, and the radiated power in the far field on the other side, as a function of angle.</p>
<p>The &quot;reciprocal&quot; problem would be a plane wave incident on the slit at various angles resulting in a plane wave on the other side <em>normal to the slit</em>.</p>
<p>It's just the same exact diagram except that the arrows point in the opposite direction.</p>
<p>In my experience, we choose the direction that's easiest or most convenient to calculate, and then if that's not the direction we are ultimately interested in, we invoke reciprocity.</p>

CC BY-SA 4.0
2024-12-25T02:17:50.030Z
2024-12-25T02:17:50.030Z
uhoh
<p>I am studying the radioastronomy notes contained in the NRAO website (Essential Radio Astronomy).</p>
<p><a href="https://www.cv.nrao.edu/%7Esransom/web/Ch3.html#S1.SS5" rel="nofollow noreferrer">https://www.cv.nrao.edu/~sransom/web/Ch3.html#S1.SS5</a></p>
<p>I have a terrible confusion in mind. The fact is that in these notes everything is computed in transmission and then applied to the receiving situation thanks to the reciprocity theorem (which states the proportionality between the power gain and the effective collecting area). But I do not understand clearly how to use the theorem.</p>
<p>For example, in transmission, we find that if we illuminate a 1-d aperture of, say, size 1, the resultant power pattern is a sinc function in the sine of the angle <span class="math-container">$\theta$</span> between the axis perpendicular to the aperture and the direction of interest (so, we transmit most power inside the main lobe -essentially on axis-, and less in the sidelobes).</p>
<p>Illuminating an aperture uniformly means that the feed generates on the aperture a uniform electric field. Now, how should I translate this situation into receiving terms (for example for the observation of a point source) by using the reciprocity theorem (especially the concept of uniform illumination)?</p>
<p>What does the sinc function represent in the receiving situation? Does it tell me what is the output power of the antenna depending on the source position wrt the axis?</p>
<p>Please, if you can, explain this step by step and as clear as possible because I am extremely confused.</p>

CC BY-SA 4.0
2024-12-25T23:04:28.370Z
I added the link!
Al01
<telescope><radio-astronomy><radio-telescope><radio><education>
Power pattern and effective area single dish radio telescope- reciprocity theorem
Al01
# Power pattern and effective area single dish radio telescope- reciprocity theorem

(Asked by: username_0 on 2024-12-23)

I am studying the radioastronomy notes contained in the NRAO website (Essential Radio Astronomy).

[https://www.cv.nrao.edu/~sransom/web/Ch3.html#S1.SS5](https://www.cv.nrao.edu/%7Esransom/web/Ch3.html#S1.SS5)

I have a terrible confusion in mind. The fact is that in these notes everything is computed in transmission and then applied to the receiving situation thanks to the reciprocity theorem (which states the proportionality between the power gain and the effective collecting area). But I do not understand clearly how to use the theorem.

For example, in transmission, we find that if we illuminate a 1-d aperture of, say, size 1, the resultant power pattern is a sinc function in the sine of the angle $\theta$ between the axis perpendicular to the aperture and the direction of interest (so, we transmit most power inside the main lobe -essentially on axis-, and less in the sidelobes).

Illuminating an aperture uniformly means that the feed generates on the aperture a uniform electric field. Now, how should I translate this situation into receiving terms (for example for the observation of a point source) by using the reciprocity theorem (especially the concept of uniform illumination)?

What does the sinc function represent in the receiving situation? Does it tell me what is the output power of the antenna depending on the source position wrt the axis?

Please, if you can, explain this step by step and as clear as possible because I am extremely confused.

**Tags:** telescope, radio-astronomy, radio-telescope, radio, education
> **Comments:**
> **username_0**: I added the link!  

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## 1 answers

### Answer by username_1 on 2024-12-25. Score: 0

> ...if we illuminate a 1-d aperture of, say, size 1, the resultant power pattern is a sinc function in the sine of the angle ...

While you've not yet provided a link to the discussion you're reading, nor a block quote of it, nor at least some page with a similar derivation, I'll take a stab at an answer guessing that you're referencing [single slit diffraction](https://en.wikipedia.org/wiki/Fraunhofer_diffraction#Single-slit_diffraction_using_Huygens%27_principle).

In this case the "forward" problem is a plane wave incident on a 1D slit, and the radiated power in the far field on the other side, as a function of angle.

The "reciprocal" problem would be a plane wave incident on the slit at various angles resulting in a plane wave on the other side *normal to the slit*.

It's just the same exact diagram except that the arrows point in the opposite direction.

In my experience, we choose the direction that's easiest or most convenient to calculate, and then if that's not the direction we are ultimately interested in, we invoke reciprocity.


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