Astronomy 302

Gaussian Beam Optics Lab

Spring 2000

Introduction: In a typical optical system operating at near-infrared wavelengths or shorter, the physical size of optical elements like lenses, mirrors and apertures are all many orders of magnitude larger than the wavelength of light in interest. At radio and sub-millimeter wavelengths, however, the photons have large wavelengths, from a tenth of a millimeter to many meters. Since ``ray tracing'' optics assumes that beams of light propagate in straight lines with no diffraction effects, this approach is badly suited for radio-wavelength optics. A better approximation of how light actually behaves in the long-wavelength limit is called Gaussian beam optics. In this approximation, a beam of light in an optical system has a Gaussian intensity distribution:

$$I(r)=I_0 e^{r^2/\zeta^2}$$

$$\zeta(z)=\zeta_0 [1+(z/k \zeta_0^2)^2]^{1/2}$$

$$k=2 \pi/\lambda$$

$$\zeta_0=\omega_0 / \sqrt 2$$

This means that the light beam is brightest in the middle of the beam and slowly tapers off as the angle between the direction of propagation and the observer increases. In ray optics, we assume that all the light goes in one direction only, like a laser beam.

Figure 1: A hypothetical image seen by a single ``pixel'' in the ``ray-tracing'' and Gaussian limits, respectively

Also, in Gaussian optics, the focus of a beam has a finite size, and a parallel beam never stays parallel; it always wants to expand. Since we're only used to optical wavelengths of light, most of these effects can be pretty strange. In this lab, we will measure the sensitivity of three feed horns (``antennas'') as a function of angle and compare them to Gaussian profiles. As you learned in class, the Reciprocity theorem stipulates that the sensitivity pattern of an antenna is identical to it's intensity distribution if used as a transmitter, so we can think of the pattern we measure as a generic ``beam pattern'' no matter if we measure it as a receiving horn or a transmitting horn.

Experimental Apparatus: In this experiment, we will use a system developed to measure the beam patterns of 85 GHz feed horns and optical elements. It consists of the following elements:

1.

Anechoic Chamber: This is a wooden box lined with a material called it Eccosorb. This material absorbs radio waves to prevent reflections. The horn is placed inside this box on a computer controlled mount that can move the horn in $0.45^\circ$ increments in both altitude and azimuth. It can also be moved by hand.

2.

Crystal Detector: Attached to the back end of the feed horn being measured is a small box with an electrical connector on it. This device is a Crystal Detector, aka a Square Law detector. This device rectifies the electromagnetic wave at its input. The rectified signal is output as a DC voltage.

3.

Lock-In Amplifier with Chopper: This is a special type of amplifier designed to measure signals buried in large amounts of noise. It needs to be hooked up to the crystal detector and a reference clock from the chopper. The chopper looks much like a fan blade, with Eccosorb-covered blades. The Lock-In-Amp takes a reading when the blade does not obstruct the transmitting horn, and another measurement when the blade does obstruct the transmitter and subtracts them. What you see on the display of the amp (and what the computer reads) is an average of many cycles.

Figure 2: Image of the lock-in amplifier, with the receiver/feedhorn assembly inside the chamber, below

4.

Computer with motor driver box and Analog to Digital Converter: The computer is used to drive the motors on the mount and to read and log the data from the lock-in amp. The computer can be set to do cross-cut scans, where the mount moves one axis, leaving the other fixed, or to move both and map out the entire beam pattern. Given one of these tasks, along with a starting and ending angle and a step size, the computer moves the feed horn to a given angle, allows vibrations to settle and then uses the A/D converter to read the output of the lock-in amp. This repeats until all of the measurements have been taken. The data is written to a text file that can be put on a disk and plotted with your favorite plotting program or spreadsheet.

5.

Gunn Oscillator: The Gunn Oscillator is the solid state device that actually creates the signal in the transmitter chain. It is an electrical device that produces a signal whose frequency is dependent on the applied voltage. The diode itself is very small and is placed inside a waveguide equipped with a backshort tuner and E-plane tuner. This device can be very easily damaged, so be careful with it.

6.

Wave Meter and Attenuator: Between the Gunn Oscillator and the transmitter horn sit two devices which can modify the transmitted signal. The device closest to the Gunn Oscillator is the Wave Meter, which can measure the frequency of the signal going through it. It is a very narrow-band filter which BLOCKS signal at the frequency shown on the dial. To measure the frequency of the signal, slowly turn the dial until the power on the lock-in amp's display suddenly drops. The frequency shown on the meter is then the frequency in the waveguide. Remember to move the dial away from this frequency when you're done to make sure all the power is transmitted. The next device is an attenuator. This device lowers a small metal vane into the middle of the waveguide and lowers the amount of power that gets through. In this lab, the attenuator should probably be set to zero attenuation.

Figure 3: The wave meter and attentuator readouts have several numerical dials, but the only one of concern is the one marked between two white sleeves. For example, the wave meter is tuned to 85.0 GHz and the attenuator is removing 3 dB of signal (i.e. half the power).

7.

Feedhorns: We will measure three types of feed horns in this lab: The standard gain horn, the Potter horn and the corrugated horn (each group will measure only one, though). The standard gain horn is a simple, slab sided pyramidal affair with very well defined properties. It has HUGE sidelobes, as you will see. The Potter horn is a long cylindrical element that is designed to cancel out sidelobes at one particular frequency. One group may measure it at the operating frequency, and another would measure it at another frequency to see the difference. The third is the corrugated horn, which is cone shaped and has a bunch of stepped-grooves inside (hence the name). This is the type of horn we use in our receivers for the South Pole and the SMT.

Figure 4: Images of three feedhorns: pyramidal, Potter, and corrugated.

8.

Lens and lens mount: We will also measure the effects of putting a lens in front of the feedhorns. We have a Low Density Polyethelyne lens (the first order of a Fresnel lens) designed for 85 GHz that can be placed on the mount in front of the feed horn.

Tasks: Each group will be assigned a particular feed horn to measure. You should do cross cut scans for $\theta$ with $\phi=0$ and for $\phi$ with $\theta=0$. Full beam pattern maps take far too long, so we'll have to limit you to cross cuts. I will instruct you on how to run the computer software when you make the measurements. You will want these scans for the horn by itself and with the lens in place. Also, you should make any measurements of the apparatus you'll need to answer the questions in the lab. Also, you should look carefully at how the apparatus is put together, and draw a block diagram of the whole system. Make sure you understand how each of the pieces of the setup work (this is why I exist; be sure to ask me all the questions you need). I will web-publish the data taken by both Tuesday and Wednesday groups for easy access.

What to Turn In: The write-up for this lab will be combined with next week's exercise, ``Receiver Characteristics'', and is due February 7th. While you should do the lab in groups and you can answer the questions in the lab as a group, each person should hand in their own typed lab report. Plots of all scans should be included and labeled; you can make them with a plotting program like SuperMongo, GNUplot or PGplot, a programming language like IDL or Matlab, or a spreadsheet like Excel or Quattro Pro. The lab should have an introduction where you explain the purpose of the lab and what you hope to discover. The procedure section should contain a detailed description of the experimental apparatus, with a block diagram (can be hand drawn) of the setup. You should also include a detailed description of how you collected the data, and where you think errors could creep in. Estimate the magnitude of the errors where you can. The analysis section should include plots of all your scans, and any other measurements you took while taking the data, like the frequency/wavelength you used, and any relevant apparatus dimensions you used in answering any of the questions.

Questions:

1.

Calculate the wavelength for the frequency you used. Comment on why diffraction should be important given this wavelength and the size of the optical elements you tested.

2.

Why do you think the Eccosorb is necessary in the setup? How does the corrugated absorber at the back of the chamber prevent standing waves? (remember, standing waves form in a cavity of length l when $l=\frac{n \lambda}{2}$). Also, use this same idea to explain how a backshort tuner works in the Gunn Diode, or any other waveguide based transmitter or detector.

3.

How does the measurement strategy of a lock-in amplifier reduce noise? What frequencies of noise can it cancel out?

4.

Measure the half power beamwidth from your plots for both the azimuthal and altitude scans. Use this measurement to calculate the horn beamwaist using the formulae included with the lab. Also, comment on how symmetrical your beam pattern is. Also, measure the angular position and power in the sidelobes on your scan. Why would sidelobes be bad in a receiver used for astronomy?

5.

Calculate the ratio of power between the bare horn and the horn with lens (the voltage read on the lock-in is proportional to the power). Remember to factor in any attenuation you have to add when the lens is in place. Also, from what you know about the propagation of radiation, calculate this ratio theoretically. Your answer should depend on the area of the horn, the area of the lens, the distance between the horn and transmitter and the distance between the lens and horn. Also, calculate the beam growth angle of a beam emerging from the lens you measured. Compare this to the growth angle you measured. Remember, if there was no diffraction, the beam coming out of the lens would be a cylinder with a radius the same as the lens.