Examination, TEN1, in courses SK2500/SK2501, Physics of Biomedical Microscopy,

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1 KTH Applie Physics Examination, TEN1, in courses SK2500/SK2501, Physics of Biomeical Microscopy, , 8-13, FA32 Allowe ais: Compenium Imaging Physics (hane out) Compenium Light Microscopy (hane out) Extracts from (hane out) Pocket calculator Notes: Write your name on all papers. Write on one sie of the paper only. (Papers will be scanne) Only one problem solution on each paper. Give motivations to all your answers, an explain all symbols that are introuce. Data may be given that are not neee for solving the problems. You are encourage to use simple figures to clarify explanations. You may answer in English or Sweish. Each problem can give 10 points. (You nee 30p to pass.)

2 2 Problem 1. Lamp filament Luminous fiel iaphragm Aperture iaphragm Specimen Exit pupil y z Eyepiece Eye Collector lens Conenser Objective Fig. 1. Ray path in a microscope with Köhler illumination. You are using a transmitte light microscope with Köhler illumination accoring to Fig. 1. The objective is labelle 10/0.35 an the conenser is labelle NA = 0.7. First you ajust the illumination system for optimum performance accoring to the school book. Then you reuce the size of the aperture iaphragm so that it becomes very small. a) How will the image brightness, resolution an epth-of-fiel change when you make the aperture smaller? (3p) Using this very small size of the aperture iaphragm, you e-center the aperture more an more (for example by moving it in the y-irection in Fig. 1). b) What happens with the resolution in the microscope when the aperture is ecentere more an more? Explain why? (3p) c) When the aperture has move a sufficiently large istance from the center, something rastic happens to the appearance of the image. Explain what happens an why. (4p)

3 3 Problem 2. Toay many microscopes are equippe with etectors that are so sensitive that light emitte from a single fluorophore molecule can be etecte. Let s look at the following case: Each measurement is carrie out by counting the number of photons coming from a sample uring a preetermine length of time, for example 1 millisecon. We know that each sample can contain between one an ten fluorophore molecules. In the one-molecule case we enote the expecte number of etecte photons uring the measurement time by N (i.e. this is the average number recore if we repeat the measurement many times). Consequently, for the n-molecule case the expecte number is nn. We want to etermine the number of molecules in each sample with a reasonable amount of certainty. We efine reasonable amount of certainty in the following way: The 2 intervals of the one to ten molecule cases shoul not overlap, Fig. 2. Here stanar eviation for the n-molecule case. (There is a 95 % probability that the outcome of a measurement falls within a range set by the expecte value 2 ). What is the smallest number N that fulfills this conition for n-numbers from 1 up to 10? n is the Number of etecte photons Expecte values nn 2 n 2 n Must not overlap (n-1)n 2 n-1 n-1 molecules n molecules Fig. 2. Conition for etermining the number of molecules with a reasonable amount of certainty.

4 4 Problem 3. An epi-fluorescence microscope (wiefiel, not confocal) is to be use to stuy thin fluorescent specimens that contain high-contrast grilike structures with perio lengths own to 0.6 m. A requirement is that the grilike structures shoul be visible with goo contrast (something like 50% moulation), an at the same time the largest possible specimen area shoul be visible when looking through the eyepieces. The microscope is equippe with eyepieces labelle 10x/20. The following high-quality (= nearly iffraction-limite) objectives are available: 10/ / /1.3 60/1.4 The emission spectrum of the fluorophore is shown in Fig. 3, an 436 nm light from a mercury lamp is use for illumination. Intensity (nm) Fig. 3. Emission spectrum for the fluorophore. Which objective is best suite for this application, an how large is the specimen area that will be seen through the eyepieces?

5 5 Problem 4. a y a x Detector element (pixel) Fig. 4. Magnifie etail of a CCD sensor. The shae areas are sensitive to light (we assume that the sensitivity is uniform over each shae area). Figure 4 shows the pixel layout in a CCD sensor. The a ratio can vary a lot between ifferent sensors. a = 1 correspons to fill factor 100%, i.e. the entire sensor area is sensitive to light. We will then etect the maximum number of photons, an get the best possible signal-to-noise ratio. In reality the fill factor is always less than 100%; it can be as low as 20% in some cases. a) Can we expect that the a ratio influences the contrast of moiré (aliasing) patterns, if such patterns occur? If so, will we get the highest moiré contrast for high or low a ratios? Explain your answers! (3p) b) Let s assume that we have a sensor with an a ratio of 0.75, an that we project a line pattern with nearly 100% sinusoial intensity moulation onto the sensor. What are the maximum moulations (contrasts) we can get in moiré (aliasing) patterns for vertical/horizontal orientation (along x or y axes), an iagonal (45 egree) orientation of the projecte line pattern? (7p)

6 6 Problem 5. Quantum ots have properties similar to fluorophore molecules, but they are more resistant to photobleaching an have narrower emission spectra. A specimen labelle with two types of quantum ots, green an re, is to be scanne with a confocal laser scanning microscope. The excitation an emission spectra for the two types of quantum ots are shown in Fig. 5. The confocal microscope is equippe with lasers, filters an beam splitters accoring to Fig. 6. Green emission Re emission Re excitation Green excitation nm Fig. 5. Excitation an emission spectra for the two types of quantum ots use in the experiment. Beam splitters: R R R R Dichroic mirrors: R 458 R 488 R 458/514 R 458/543 R 488/633 Specimen Barrier filter: LP475 LP505 LP530 LP560 LP630 Barrier filter: BP BP BP BP BP Laser wavelength: 458 nm 488 nm 514 nm 543 nm 633 nm Detect. Ch. 1 Detect. Ch. 2 Fig. 6. Laser an filter arrangements in the confocal microscope use. The ichroic mirrors reflect the wavelengths inicate an transmit all other wavelengths. The beam splitters reflect the wavelength ranges given, an transmit longer wavelengths. Barrier filters BP transmit the wavelength ranges given. Barrier filters LP transmit wavelengths longer than the numbers given. Note: Both ichroic mirrors an beam splitters are non-ieal in the sense that a small proportion of the light that shoul be reflecte is transmitte an vice versa. The barrier filters, on the other han, have negligible transmission outsie the inicate wavelength ranges.

7 7 When recoring the specimen, you want the two etection channels (Ch. 1 & 2 in Fig. 6) to etect only one type of quantum ot each (negligible crosstalk). You also want to etect as many emitte photons as possible to get a goo signal-to-noise ratio. Suggest a goo combination of laser wavelength(s), ichroic mirror, beam splitter, an barrier filters for ch.1 an ch. 2. Explain your choice! Problem 6. The optical section thickness is of great importance in confocal microscopy. An ambitious researcher wante to test a Fluar 60/1.1 water immersion objective (without coverglass correction), to see if it ha satisfactory performance. To o this, a very thin homogeneous fluorescent layer was evaporate onto a specimen glass. The fluorophore use was very resistant to photobleaching. A rop of water was place above the fluorescent layer (the fluorophore was not water-soluble), an the focus was ajuste so that the objective mae contact with the water. The image intensity as a function of focus position was then recore, Fig. 7. A laser wavelength of 488 nm was use, an a nm banpass filter was place in front of the etector. The etector aperture size was 0.1 Airy units. Does the result in Fig. 7 inicate that the objective performs well concerning optical sectioning (i.e. is the result reasonably close to what is preicte for a iffraction-limite objective)? The refractive inex of water is Normalize intensity Focus position ( m) Fig. 7. Intensity as a function of focus position for the 60/1.1 water immersion objective. Goo Luck! from Anna, Ilaria & Kjell

8 8 Solutions to examination in course SK2500/SK2501, (Also other reasonable solutions may be acceptable) (Sometimes extra comments are given in the solutions that are not necessary to get maximum score at the examination.) Problem 1. a) Reucing the aperture size means that less light gets through, an therefore the image brightness will be reuce (but the size of the illuminate area will be the same). The resolution will be reuce, because the maximum illumination angle φ in Fig. 10 in the microscopy compenium will be reuce. The epth-of-fiel will increase, because objects out of focus will be image with a smaller blur circle. b) Decentering the small aperture means that the angle between the illuminating light rays an the optical axis will increase (angle φ in Fig. 10 in the microscopy compenium). Accoring to eq. 8 in the compenium, this means that the smallest perio length that can be resolve will become smaller. Therefore the resolution will increase graually as the aperture is more an more ecentere (up to the point where φ equals the maximum collection angle set by the N.A. of the objective). c) When the small aperture is ecentere so far that the angle φ mentione above is larger than sin 1 (0.35) no illumination light can enter irectly into the objective. Only light that is scattere, iffracte or refracte by the a specimen can enter the objective. The image backgroun will therefore be ark an light-eflecting specimen parts will appear bright. Thus we have obtaine ark-fiel imaging (chap in the compenium). For this to be possible, the N.A. of the conenser must be higher than the N.A. of the objective (which is the case here). Problem 2. The stanar eviation σ is equal to the square root of the expecte number of etecte photons (Imaging Physics compenium, chap. 4). Using this, we can re-write the conition (n 1)N + 2σ n 1 nn 2σ n as N 2σ n + 2σ n 1 = 2 nn + 2 (n 1)N = 2 N( n + n 1) Solving for N, we get N 4( n + n 1) 2 The require number of etecte photons, N, is highest for the maximum n-number that we nee to hanle. With n = 10 we get N 4( ) 2 = 152 So to be on the safe sie, we coul aim for something like 160 etecte photons for the one-molecule case. Problem 3. Looking at the MTF curve in Fig. 8 in the microscopy compenium, we see that the MTF value is > 50% for frequencies lower than N.A.. The highest freq. in the specimen is 1 λ m-1. We then get the conition N.A. 1 λ The peak of the fluorescence emission curve is at 530 nm, so we can use this as the wavelength. We then get N. A. 9 6 = 1.1. This is satisfie only

9 9 for objectives 40/1.3 an 60/1.4. Since we want the largest area possible to be viewe, we shoul minimize the magnification. Therefore we shoul use the 40/1.3 objective. This will isplay a specimen 20 mm area with a iameter of = 0.50 mm when looking through the eyepieces. M objective Problem 4. a) The maximum moulation we can get in a moiré pattern is given by the MTF value at the Nyquist frequency (which epens on the -value). For a sensor, the MTF-values will be higher for a small pixel with, a (Imaging Physics, chap. 15). Low a ratios will therefore give high moiré contrast. b) For a vertical or horizontal pattern orientation, we get a Nyquist frequency of 1 2. With a = 0.75 we get a = MTF at the Nyquist frequency is then given by sin(π ) π = This is the maximum moulation we can theoretically get in the image when moiré effects are beginning to occur (but in reality it will always be lower ue to the MTF of the optics an non-ieal sensor properties). For a iagonal (45 ) pattern the maximum frequencies in the x- an y-irections are both 1 2 if we are to avoi moiré. The pattern frequency (measure perpenicular to the lines) is then 1 ( 2 higher than for vertical or horizontal orientation). The sensor MTF is then given by sin(π ) 2 π = This means that the contrast of the moiré patterns is expecte to be lower for iagonal pattern orientation compare with horizontal or vertical orientation. 2 Problem 5. Laser wavelength 458 nm will provie the most efficient excitation of both fluorophores, an it oesn t overlap with any of the emission spectra. Dichroic 458 shoul be use. Beamsplitter will reflect the maximum amount of green emission to channel 2 with very small interference from the re emission. The barrier filter in channel 2 shoul be This will block any reflecte laser light present, an will pass the maximum amount of green emission. It also blocks any re emission that may have been reflecte by beamsplitter In channel 1 the barrier filter LP630 shoul be use. It will transmit nearly all of the re emission, yet block laser light an green emission light. Problem 6 Uner perfect conitions the FWHM optical section thickness is given by (eq. 18 in the 8πnsin 2 ( α ) 2 microscopy compenium). A etector aperture size of 0.1 Airy units is very small, so it shoul not influence the results (sect. 2.3 in the compenium). The average wavelength λ 506 nm, an we get the angle α from 1.33 sin α = 1.1. This yiels a FWHM = = 5.9 8π 1.33 sin 2 (28 ) 10 7 m = 0.59 m. From the figure we get a FWHM > 2 m which is much larger than the theoretical value. The conclusion is therefore that the performance is not so goo. 8.5λ