This lab will teach us basics of optics: using the optical bench, 3D optical alignment and dealing with rotation of elements, stray light control, and treating equipment well. We want to examine the blackbody spectrum of the lamp. We will use refraction in a collimating lens to collimate the light beam and refocus it on a small ccd. We'll examine the blackbody spectrum dependence on wavelength and angle.

Equipment

These are the steps for alignment of the optics we will use. For background and more details about the equipment, see the lab handout.

Part A

1. Attached rail to table aligned with translational stage.
2. Mounted photodiode assembly on translational stage and set micrometer to the midpoint of the focus travel
3. Attached camera lens *#32974 plano convex lens with 50mm diameter and 150mm focal length). Best optical performance when the convex side of the lens is facing the lamp and the flat side of the lens is facing the detector. Careful: both filters and lenses should only be touched on the edge, away from the path of the incoming beam.

Q. How far from the photodiode does this camera lens belong?
A. Approximately 150mm.

4. Attached empty filter holder
5. Attached iris and taped paper to block extra incoming light
6. Collimating lens (#31402 achromate with 63 mm diameter and 356mm focal length). The curvier surface faces the iris.
7. QTH lamp #63200. Mounted approximately 350mm from the collimating lens. We did not plug the power supply in and made sure the toggle on the back was turned to off. Then we took black/white wires from the lamp and connected the leads to the lugs on the power supply. Finally we plugged in the power supply and turned it on.
8. Collimating part i! We first took off the optical elements after the iris so we can check the size of beam on the wall relative to the size of the iris. The circle on the wall is 4.5cm at first. The one after the iris was about 4cm. We checked that the lens is exactly on axis. The y axis (of the beam on the wall) was aligned by first trying to maximize the size of the beam on the wall (rotated collimating lens leads to oblique circles) and also checking from the top. Then we aligned the source of the light, the lamp, with the optical array. After, we aligned the lens in the y and the z axis so the beam was centered on the iris.
9. Collimating part ii! We rechecked the beam size after the iris. The iris is 4.10 +/- 0.05 cm, and we got the beam on the wall, which was about a meter away, to the same size.
10. The filter was approximately aligned in the z/height direction, and the camera lens was put at the focal length away from the camera (checked). We also checked the angle of the planoconvex lens. We closed the iris a little so the filter holder would not get in the way.
11. The lamp was a bit low, so it means our z axis alignment is off. We realigned everything. We raised the lamp, centered it with the iris, knew our beam was collimated, and realigned the z axis of all the other pieces. Luckily, we were able to keep the beam collimated (we checked the beam diameter).
12. The camera lens ended up being 9 +- 0.5 cm from the photodiode.
13. Attached BNC coax cable from photodiode to the input terminal of the high dynamic range transimpedance amplifier, turned it on after plugging it all together.
14. Adjusted current range to the smallest non-(-1) reading for best precision. This was 200 microamps, so our unit is uA. This works, so we have a final iris diameter to 2.15 +/- 0.05 cm (measured twice).

Q. What is the reimaged size of the QTH filament on the diode?
A. We measured 1mm +- 0.5 diameter, so area < 1mm^2 which is the size of the photodetector.

15. Take measurements.

Part B

16. We removed the lens hlder and the camera lens. We removed the translational stage and replaced it with the rotational stage. We put the photodiode assembly back on the rotation axis, by using the post holder that can screw into the rotational stage, and being careful with the posts that are now a bit too small.
17. Alignment was easy this time. Do the same as yesterday, but because we do not need to move our setup much, only corrections were made.
18. Take measurements. See section 3B.

Measurements

We ran measurements over two days of lab.

Part A

Make two measurements (ie, multiple times of each):

translational stage micrometer focus reading
value of maximum current

We made the 400nm measurements and then stopped for the next day.

(Day 2)

We continued with the Part A Measurements.

1. Pick a filter (600nm). Record the current as a function of micrometer focus reading using stems of 0.5mm over a focus range of +/- 2.5mm from best focus.
2. We ran measurements to the low end, repeated the center one, and did two measurements of each.

Part B

1. We measured in steps of 5deg the current as a function of angle.

Q. You can determine how accurately you were aligned at the center position by comparing measurements at +- a few steps, why?
A. Because the maximum current implies the beam was orthoganal to the ccd.

Tables

The following tables are data we collected.

Table 1A: Column definitions and error

col	name	err	units	explanation
lambda	wavelength		nm
d1	distance	0.005	mm	translational stage micrometer focus reading
d2	distance	0.005	mm	translational stage micrometer focus reading
c1	current	0.0005	uA	value of maximum current
c2	current	0.0005	uA	value of maximum current
FWHM	FWHM	0.5	nm	filter FWHM
peak	peak	0.5	%	filter peak value
noise	noise	0.0005	uA	outside light
R	reponsivity	0.0025	A/W	diode efficiency, to help get QE
QE				unitless quantum efficiency QE = R* 1.24e-6 / lambda
d	distance	0.007	mm	mean(d1, d2)
c	current	0.0007	uA	mean(c1, c2)
I()	intensity	varies	kW m-2 sr-1 nm-1	Intensity of QTH lamp

Table 1B: Current := f(lambda, distance)

lambda	d1	d2	c1	c2	FWHM	peak	noise	R	QE
400	11.22	10.245	0.105	0.108	70	65		0.087	0.2697
450	11.479	11.475	0.381	0.386	70	74	0.005	0.162	0.4464
500	11.462	11.369	0.671	0.669	69	73	0.003	0.232	0.57536
550	12.012	12.071	1.063	1.064	68	74	0.004	0.295	0.6650909091
600	12.662	12.155	1.706	1.703	64	74	0.006	0.354	0.7316
650	13.242	13.211	2.09	2.1	70	72	0	0.406	0.7745230769
700	13.355	13.726	3.19	3.2	77	67	0	0.451	0.7989142857

Table 1C: Intensity

lambda	d	c	err_d	err_c	I	err_I	err_I with SR
400	10.7325	0.1065	0.007092993139	0.0007073172262	1.292262008	0.1161014856	0.2620767432
450	11.477	0.3835	0.007071068134	0.0007071518569	2.528927713	0.2179880049	0.5088609278
500	11.4155	0.67	0.007071243806	0.000707109144	2.819060706	0.2426771338	0.5671034727
550	12.0415	1.0635	0.007071131471	0.0007071070156	3.339101487	0.2886122669	0.6722193223
600	12.4085	1.7045	0.007075496344	0.0007071076026	4.615058398	0.4104081881	0.9340911376
650	13.2265	2.095	0.007071082378	0.0007071128228	3.713162788	0.3157086576	0.7452916579
700	13.5405	3.195	0.007073058803	0.0007071093788	3.64042903	0.2953008377	0.7247821379

Table 2A: Column definitions and error

d	distance	0.005	mm	translational stage micrometer focus reading
c1	current	0.0005	uA	value of maximum current
c2	current	0.0005	uA	value of maximum current
c	current	0.0007	uA	mean(c1, c2)

Table 2B: Current := f(600nm, distance)

d	c1	c2	c	err_c
9.46	1.444	1.443	1.4435	0.0007071069084
9.96	1.539	1.593	1.566	0.0007074221644
10.46	1.616	1.618	1.617	0.0007071071868
10.96	1.69	1.673	1.6815	0.000707133885
11.46	1.72	1.714	1.717	0.0007071100192
11.96	1.803	1.79	1.7965	0.0007071206664
12.46	1.75	1.807	1.7785	0.0007073792147
12.96	1.74	1.798	1.769	0.0007073918985
13.46	1.724	1.765	1.7445	0.0007072532675
13.96	1.671	1.721	1.696	0.0007073372919
14.46	1.617	1.659	1.638	0.0007072811434

Table 3A: Column definitions and error

angle	angle	0.5	deg	rotational stage angle reading
c1	current	0.0005	uA	value of maximum current
c2	current	0.0005	uA	value of maximum current
c	current	0.0007	uA	mean(c1, c2)

Table 3B: Current := f(white light, angle)

angle	c1	c2	c	err_c
0	0.011	0.011	0.022	0.0007
5	0.011	0.011	0.027	0.0007
10	0.016	0.016	0.055	0.0007
15	0.039	0.039	0.141	0.0007
20	0.102	0.102	0.2855	0.0007
25	0.184	0.183	0.4055	0.0007
30	0.222	0.222	0.4655	0.0007
35	0.244	0.243	0.568	0.0007
40	0.326	0.323	0.801	0.0007
45	0.477	0.476	1.1455	0.0007
50	0.668	0.67	1.502	0.0007
55	0.834	0.832	1.761	0.0007
60	0.931	0.925	1.9615	0.0007
65	1.034	1.033	2.1695	0.0007
70	1.138	1.134	2.359	0.0007
75	1.224	1.222	2.5105	0.0007
80	1.289	1.286	2.6125	0.0007
85	1.327	1.323	2.662	0.0007
90	1.338	1.336	2.689	0.0007
95	1.353	1.351	2.6865	0.0007
100	1.335	1.334	2.642	0.0007
105	1.308	1.307	2.551	0.0007
110	1.245	1.242	2.41	0.0007
115	1.168	1.165	2.2775	0.0007
120	1.112	1.11	2.118	0.0007
125	1.009	1.005	1.899	0.0007
130	0.892	0.892	1.6605	0.0007
135	0.771	0.766	1.3445	0.0007
140	0.575	0.577	0.998	0.0007
145	0.423	0.421	0.75	0.0007
150	0.327	0.329	0.591	0.0007
155	0.262	0.264	0.4295	0.0007
160	0.165	0.168	0.231	0.0007
165	0.064	0.065	0.0875	0.0007
170	0.023	0.023	0.034	0.0007
175	0.011	0.011	0.019	0.0007
180	0.008	0.008	0.008	0.0007

Analysis

This section will have four parts; two are discussion on error, one on the equations used, and the last are plots and a short discription.

Mathematical Error

First, I will explain all of our error estimates. We used propagation of errors, ie, for any variable $X$ (like Intensity $I$ or mean distance $d$ ),

(err_{X}/X)^{2}=\sum (err_{i}/q_{i})^{2}

where $q_{i}$ is the ith components of $err_{X}$ . The error is either recorded as 0.5 the smallest seeable tick mark for collected data, or error is found using propagation of variables.

Human Error

Hopefully, systematic error is noted and corrected for (like the variable $noise$ which is our measurement of how the room light affected current readings), or accrued into the error bars (measurement error and propagated error). However, due to imperfect alignment, and other things that may not be idea (for instance, our camera lens was not the given focal length away from the photodiode, and the light beam hitting the face-on photodiode was slightly angled) we have human-made error inherent in the data.

Data analysis

lambda is the wavelength in nm
d1, d2, and d are all measurements on the translational stage micrometer focus (mm)
c1, c2, and c are peak current measurements (uA)
FWHM is the FWHM of the transmittance from the filter
peak is the peak % of the filter
noise is a measurement of how outside light changes our measurements
R is the responsivity from the lab handout
QE is calculated quantum efficiency
SR is steradians of light (sr)
I is intensity (kW m-2 sr-1 nm-1)
angle is the angle of rotation (deg)

recall

r_{beam}=1.1+/-0.05cm

R_{guess}=110cm

Intensity equation

I=(c/R)*(peak/FWHM^{2})/(SR*lambda)

To get the m^-2 dependence in the intensity, divide the current by an approximation to the transmittance there, which is peak/FWHM^2. noise is consistantly <1% of the current measurements and pretty stable; we will fold it into our equation. SR is more complicated. I didn't measure the distance from the light source to the photodiode, so SR is a guess. Because of this, I did not fold in the error from this, as it would be on order of 20cm/20% of the value.

SR=A/R^{2}=pi*(r_{beam}/R)^{2}=3.801*R_{guess}^{-2}=pi*10^{-4}

Note, error of the intensity is calculated by propagation of errors of four elements (current, peak, R, FWHM, but not SR) and still ends up at almost 10%. With the SR error included, this rises to 25%-93% of each intensity point (see err_I with SR column).

Plots

The first plot is distance as a function of wavelength.

Distance for optimal current as a function of wavelength.

The next two plots are looking at current as function of distance and angle.

Current as a function of mm distance from the source using 600nm light. Errors are small.

Normalized current of white light at different angles from orthogonal. Errors are small in both x,y directions.

From the normalized current as a function of angle, we can find an offset of at most +2.5 degrees; if you look at the table, you'll see that the maximum value should be between 90 and 95 degrees, but close to 90.

The next two plots are calculated intensity of the lamp compared to ideal blackbody curves.

Plank function and our lamp blackbody, 1e14 decreased in orders of magnitude. Errors are large.

The actual data I got, which is giving intensities of about 1e14 kW m-2 nm-1 sr-1. Compare that we're now using T=1e15K.

Discussion

The discussion covers what the results mean, mistakes, and conclusions.

Part A

The first plot shows how as wavelength increases, the distance/focus measurement that leads to the highest current increases as well. The variation in optimal focus with wavelength is maybe caused by the changing focal point of the camera lens due to different refractions in the glass as a function of wavelength. Also, the handout mentions that the detector can be underfilled, which might cause huge variation (typically it is 1-2% variation in filling efficiency).

The next two plots show there is an optical advantage to a certain point, whether the detector is angled or closer or farther from the source. Note that they show we found the maximum current/center/peak pretty well.

The next two plots show intensity. The one on top gives my intensity * 1e-14, which matches well with 4000K. The second plot shows the actual numbers I get for intensity, which does not. The intensity I got is 14 orders of magnitude different from the blackbody ideal values I calculated. This seems, then, like I just can't calculate the intensity or BB right, because I was pretty confused about them already. Instead of letting the computer do them (because it easily ran into infinities as we were taking an exponent) I wrote all constants in exponential form and added/subtracted the exponents on paper and added them in; I did the same thing when calculating intensity. I would like to spend more time thinking about this, and checking where I lost or gained powers of ten.

The sensitivity function of the instrument has accounted for the transmittance of the filter and the variation in responsivity of the photodiode. It hasn't accounted for variation in the lamp (stability of light output should increase with time) and any carelessness we were about protecting the photodiode from changes in ambient light. I think these should not be that significant, because we measure the baseline light from the environs and it seems to hover around 1% or less.

QTH lamp temperature seems to agree with 4000K (ignoring orders of magnitude), which is hotter than the 3300 posited based on my values.

Part B

The handout on Power and Energy Measurement says that responsivity of the detector falls off as $R_{\theta }=R\cos(\theta )$ . By eye, the results agree very well with this model. At large angles it is assumed to be harder to completely fill the photodetector and for reflection off the pane of glass of the detector (see the explanatory supplement).

Conclusion

We learned how to use the optical table and assorted optical elements. We aligned things in three dimensions, took multiple measurements of data, quantified errors, and tried to explain the data after plotting it and analyzing it. The blackbody spectrum I got was extremely off (in order of magnitude), but alright in terms of numbers; I am not sure where these extremely different orders of magnitude come from yet.