BMED 4501 Biophotonics

(Semester 2 – Year 2024 – 2025)

Homework 1 (Full mark 50%) (Due Date: April 30, 2025)

Before you work on this homework, please read the following:

Here we define three parameters:

D – Level of difficulty of the course content, from 1 (low) to 6 (high).

E – Level of effort you have spent on this course, from 1 (low) to 6 (high)

L – Level of understanding you have so far in the course, from 1 (low) to 6 (high).

**round () is to round to the closest integer;

• If you are in either region I or II, please work on all the questions

• If you are in either region III or IV, please work on all the questions, too.

**Remember to write down your coordinate in your submitted homework.

Question 1: (35%) Optical setup for laser pulse characterization

(a) (5%) A titanium sapphire (Ti:S) mode-lock laser delivers ultrashort laser pulses into a 200- meter long silica fiber. This laser has a center wavelength at 900 nm and a repetition rate offrep =80 MHz. The input laser pulse (hyperbolic secant pulse) being coupled into the fiber has a temporal width (FWHM) of Dt = 17 fs. Using the dispersion curve in Fig. 1, estimate the broadened pulse width at the fiber output.

Figure 1: Group velocity dispersion curve of the fiber

(b) (15%) The broadened laser pulse can be compressed back to a transform-limited pulse based  on a pair of identical diffraction gratings. The two gratings are parallel to each other and are  separated by a distance d (Fig. 2). To understand the working principle, we first consider two  overlapped monochromatic light beams (one at λ1=875 nm, another at λ2=925 nm) incident (at  an incident angle of 60o) onto the first diffraction grating with a groove density of 1800 lines/mm.

(i) Show that the m = -1 diffracted beams from the second grating at both λ1  and λ2 are separated and parallel to each other. And both beams are also parallel to the orginal incident overlapped beam. (Hint: straightforward geometry).

(ii) In this grating-pair system, a mirror is added such that the parallel diffracted beams are back-reflected to the original beam path (Fig. 2). L = 10 cm. Therefore, by calculating the round- trip path lengths of each of the two beams (i.e. at λ1  and λ2), i.e. from A to mirror, and back to A, explain why this system functions as a “pulse compressor”.

(iii) What is the required separation d in order to compress the broadened pulse in (a) back to a transform-limited pulse?

Figure 2: A diffraction grating pair

(c) (10%) Consider that this Ti:S laser delivers a laser beam output with the following characteristics:

•    Beam radius: 1 mm

•    Pulse width: 50 fs (FWHM of a Gaussian temporal intensity pulse)

•    Repetition rate: 80 MHz

•    Time average power: 1.5 W

Calculate (i) the duty cycle, (ii) the pulse energy (in mJ), (iii) the peak power of the pulse (in kW), (iv) the bandwidth (in nm), (v) the pulse energy density (J/cm2), and (vi) peak intensity of the pulse (W/cm2).

(d)  (5%) Consider you  have no photodetector which is faster enough to measure the compressed temporal pulse width (FWHM) of the Ti:S laser, one viable approach is to employ interferometry to extract  such  information.  Explain  the working  principle  of such  measurement  based  on  Michelson interferometer. (Hints (Fig. 3): you would need a movable mirror in one of the interferometer arms whereas another is fixed mirror.) Bonus of 5% will be given if you can quantify your explanation)

Figure 3 Interferometer for temporal pulse width measurement

Question 3: Hair thickness measurement (10%)

A student performed an experiment to study the light diffraction from a strand human hair. In her experiment, she launched a laser beam (with a beam diameter of 5 mm) passing through a hair fixed vertically. She observed the diffracted intensity pattern on a screen positioned at a distance far away from the hair (few meters away) (Fig. 4a).

Fig. 4a

Question 3 (Con’t)

(a) (5%) She observed that the diffraction pattern is very familiar to that generated by a single slit. Her classmate argued that the diffraction patterns generated in these two cases (i.e. hair and single slit) are indeed the same. Her classmate further explained the phenomenon by stating that the transmission functions in these two cases are “complementary” (i.e. one of them is perfectly   transparent at regions where the other is totally opaque (e.g. the slit and strip of the same width  (Fig. 4b). Explain if her classmate’s argument is correct (Hint: You might use the concept of Fourier transform).

Fig. 4b

(b) (5%) Both students further planned to make use of the diffraction pattern to measure the human hair thickness. Explain the working principle behind this measurement.

Question 4: OCT system (20%)

(a) Consider a spectral-domain OCT system using a image-sensor-based spectrometer as detection (i.e. CCD or CMOS sensors). And you are now given four different light sources(see Table 1).

(i)    (5%) From Table 1, calculate the bandwidth of each source in terms of wavelength (in nm).

(ii) (10%) Two of the light sources in Table 1 can be used for OCT imaging of skin (A1 and A2 in Fig.

5) and colon (B1 and B2 in Fig. 5). Identify the light source in each figure and explain why.

(*Note: The lines in A1 and A2 are the A-scan profiles) Hints: look carefully at the image quality differences and explain why quantitatively.

Table 1

Question 4 (Con’t)

Fig. 5

(d) (5%) It is also required to achieve real-time 3D imaging (512(x) × 512(y) × 1024(z) voxels) at a speed of 1 frame. per second, i.e. 512 x 512 A-scans in 1 second, choose the best line camera from the list shown in Table 2 and Fig. 6. You should also make your choice based on the source you choose in  (c). Explain your choice. (**x   andy are along the lateral direction whereas z is along the axial direction.)

Table 2 Four different line-cameras

Fig. 6

Question 5 (DIC and Phase contrast microscopy) (10%)

(a) (5%) Consider DIC, phase contrast and dark-field microscopy, which imaging modality can be used for visualizing neuronal network of a living rat in-vivo?

(b)  (5%) In DIC microscopy, why does the image contrast of the DIC image vary with the specimen orientation (see the figure below)? Does this effect appear in phase contrast microscopy? Why?