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Advanced Lab: Acceptance Angle and Numerical Aperture - INSTRUCTORAdapted from Lowe M, Donaldson N, Spiro A, Gosselin C, Nadeau M. Fiber Optics in Medicine Module. MedEdPORTAL Publications; 2015. Available from: https://www.mededportal.org/publication/10292 http://dx.doi.org/10.15766/mep_2374-8265.10292_______________________________

A. Acceptance Angle

Notes: For light to illuminate the tissue inside the body, it first needs to enter the fiber from the light source. We have learned that in order to undergo TIR down a fiber, a ray of light traveling through an optical fiber must strike the core-cladding boundary at an angle greater than the critical angle. Therefore, light entering the fiber must meet a geometric test to be guided down the fiber.

There is a specific angle with respect to the normal or fiber axis at which the fiber will accept light. This angle is called the acceptance angle and, for reasons we will investigate, this angle is dependent on the difference in the indices of refraction of the outside medium and the fiber medium. The acceptance angle is a measure of the fiber’s light gathering ability, and it is measured as the half-angle of the acceptance cone (the maximum cone of light that can enter or exit the fiber.) Note below:

The acceptance cone is three dimensional. The acceptance angle is ½ angle of the maximum cone of light that can enter or exit the fiber. The incident angle is greater than the critical angle so that the light undergoes TIR down the

fiber.

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What do you think happens to light that enters the fiber outside of the acceptance cone? Study the figure below and explain in words using the principles of TIR.Broken total internal reflection. The incident angle at the core/cladding interface is less than c. Light is refracted.

Experiment: Measurement of the acceptance angle of an acrylic rod

We will investigate the parameters that affect the acceptance angle of an “optical fiber.” For this experiment, we will use an acrylic rod to represent a macroscopic version of an optical fiber. This will allow you to easily see what is happening in a real optical fiber.

Materials: Acrylic Rod/Laser ApparatusGraduated Cylinder – 500 mlLaser pointer (green)Beaker with water(Optional) Pine Sol

Warning: The laser beam may be intense enough to burn the cornea of your eye. Never look directly into a laser or let the laser beam shine directly or reflect into anyone’s eye. Be very careful with the laser!!

Adapted from http://upload.wikimedia.org/wikipedia/commons/thumb/4/46/Optical-fibre.svg/550px-Optical-fibre.svg.png

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Place your acrylic rod/laser apparatus inside the graduated cylinder so that the protractor rests on

top. Make sure that the laser is aligned so that it will shine along the line of the apparatus that goes straight into the graduated cylinder.

To begin, we will do a qualitative investigation. Using the laser, investigate what you see when shining the laser light straight into the small face of the acrylic rod. Start with the laser aligned perpendicular to the end face, as in the above picture (call this 0), and then slowly increase the angle of the laser with respect to the normal of the end face (be sure to keep the laser light out of your eyes and the eyes of others.) At this point, the acrylic rod is surrounded by air.

Do you see TIR in the rod? If so, at what angle with the normal do you first see TIR? Students will answer 4, 5, 6 deg. The answer will depend on the exact alignment of the beam with the end face. A calculation shows that if nacrylic = 1.49, length of rod = 19.5 cm, width of rod = 1 cm, then the angle at which you first see TIR is 4.4 deg.

If you keep increasing the angle of the laser away from the rod normal, what happens?More reflections

Is there an angle between the laser light and the end of rod normal that won’t produce TIR? If so, what is it?No TIR at 0 deg, no visible TIR at 90 deg.

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What does this tell you about the acceptance cone and acceptance angle of an acrylic rod surrounded by air?Large acceptance cone with an acceptance angle of 90 deg. Acrylic in air accepts everything.

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In order to measure the acceptance cone and acceptance angle for this rod, we need to restrict the cone to a smaller angle. Let’s try filling the graduated cylinder apparatus with water until the water almost covers the straight acrylic rod. (Optional: Add a little bit of Pine Sol to the water to help see the beam.) Now the acrylic rod is surrounded by water except at the end face, which is in contact with air.

Again start with the laser at 0 and slowly increase the angle of the laser with respect to the normal of the end face.

Does the water have an effect on the angle at which you first see TIR as you rotate the laser? If so, what is the effect? The students will get the same angle as before, i.e. 4-6 deg.

Describe what happens as the angle is increased. Do you see TIR? Is there any light coming out of the sides of the rod?There is TIR. No light comes out of the sides of the rod.

What happens when approaches 90? If there is broken TIR, measure the angle at which this occurs. Describe other phenomena that you see. (For a more accurate measurement, swing the laser beam so that it is parallel to the axis of the rod. Ideally the protractor should say 0, but it’s possible there is an offset. If there is a nonzero offset, you will need to “correct” your measurements by the offset value.)As is increased, there are more reflections. After about 42, TIR is broken, and a refracted beam will be visible in the water/Pine Sol. There will also be a weak reflected beam.

When TIR is occurring, what is the maximum number of reflections?5 reflections. Students can answer 4 because they miss the bottom one. (If quality of construction is slightly off, there might truly be only 4 reflections.) There is a homework problem to calculate this value.

Do you see a reflection at all angles? Is the reflection equally intense at all angles? Describe carefully.There is a reflection at all angles , even ones greater than the acceptance angle. When TIR occurs, the reflection is intense. When TIR is broken, there are both reflected and refracted beams. As increases (i.e. as the angle of incidence decreases at the core/cladding interface), the reflection becomes less intense (see Fig 4.41 in Hecht (1998)).

Even when is within the acceptance angle, the students may comment on a decrease in intensity in the reflections as is increased. This is due to scattering by the Al particles within the acrylic rod. As the pathlength increases, there is more scattering, leading to a reduction in intensity of the beam.

For an acrylic core surrounded by cladding made of water, state the acceptance angle:

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Measured acceptance angle (θa) = ___________. This angle is equal to ½ of the maximum cone of light (the acceptance cone) that can enter the acrylic rod while all but the end is submerged in water.42 deg

For the purposes of comparing the experimental results to those obtained from calculations, measure the width and length of the acrylic rod:

Measured length of the acrylic rod: _________________

Measured width of the acrylic rod: _________________19.5 cm length, 2 cm width

C. Numerical Aperture

In our first experiment, the acrylic rod was a macroscopic representation of the core of an optical fiber and the water was a macroscopic representation of the cladding of an optical fiber. Please answer the following questions as you extend your thinking from the experiment to the use of fiber optics.

For the acrylic rod (core), was the acceptance angle (and cone) smaller when you used air as the outside medium (cladding) or when you used water as the cladding? Water

Does this indicate that the acceptance angle of the rod (optical fiber core) increases or decreases as the indices of refraction of the two interacting media are numerically closer (i.e. when the core and cladding have more similar refractive indices)?The acceptance angle decreases as the indices of refraction are numerically closer.

To check your reasoning, describe what would happen if the two media had identical indices of refraction?No TIR, no refraction, no reflection. Light passes into the cladding undeflected.

Can you think of any reason why it might be unbeneficial for the fiber to have a smaller acceptance angle? Harder to get light into fiber and be totally internally reflected.If light is launched at an angle greater than the acceptance angle, overheating of the jacket may occur leading to early failure of the fiber optic cable.

Instructor: Discuss students’ thoughts on these questions before moving on to the next section.

Our previous work defined the acceptance angle of a fiber so that we could determine the maximum angle with the normal to direct light into the fiber so as to transfer light down the fiber by TIR. Manufacturers of optical fiber usually give the numerical aperture (NA) of the fiber instead of the

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acceptance angle. The NA is a dimensionless number that characterizes the range of angles over which the optical system can accept or emit light.

Geometrically, the numerical aperture is defined by the relationship NA = no sin a where a is the acceptance angle of the maximum cone of light that can enter the fiber and no is the refractive index of the medium outside the fiber.

In the figure below, θc is the critical angle, nf is the index of refraction of the fiber and nc is the index of refraction of the cladding.

Adapted from http://assets.newport.com/web600w-EN/images/1381509.gif

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Using geometry and trigonometry, it is possible to derive the following for large core fibers (Saleh, 2007):

NA = no sin a = √nf2−nc2

Use this equation to check your mathematical reasoning with your conceptual reasoning in previous questions:

What happens when nf = nc?a = 0. All rays enter the cladding.

What happens when nc > nf?No possible a. All rays refract into the cladding.

We have seen that the media (core, cladding, outside) make a difference in the numerical aperture. Explain this difference carefully in terms of the indices of refraction, the acceptance angle, the angle of refraction at the end face, and the critical angle..The numerical aperture is determined by the core and cladding. The acceptance angle is determined by core, cladding, and the outside. The greater the difference in the index of refraction between the core and cladding, the smaller the critical angle and the larger the acceptance angle.

D. Summary: Critical Angle, Acceptance Angle & Numerical Aperture

In fiber optics transmission, there is interplay of optics principles in order to achieve the best possible result. Desired results will differ depending on the purpose of the use of fiber optics in medicine. Sometimes we only want to illuminate and view the inside of the body as in a colonoscopy or appendectomy, but there are cases where we want to cut or irradiate tissue as in laser surgery. To move forward we will sum up what we have learned so far:

1. The incident angle needed for TIR to occur depends on the critical angle. TIR will only occur when the incident angle at the core/cladding interface is greater than the critical angle. Smaller critical angles will allow TIR over a larger range of incident light angles at the end face of the fiber where the light comes in.

2. The differences in the indices of refraction of the media (core, cladding, and outside) determine the critical angle, the numerical aperture, and the acceptance angle. The greater the difference in index of refraction between the core and cladding, the smaller the critical angle and the larger the NA.

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(Adapted from Cerussi, 2013)

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Homework for Section 2

Problem 1: (Solution is attached.)Refer to the experiment in which an acrylic rod was surrounded by water except at the end face. Based on the measured length of the acrylic rod (19.5 cm) and the width of the rod (2.0 cm), calculate the number of reflections when the laser is entering the rod in the middle of the end face at an angle of 40 , which is slightly less than the acceptance angle. Compare your calculated value with the observed value. Draw a picture showing the path of the rays. Let the index of refraction of the rod be 1.49.

Problem 2: (Solution is attached.)Justify the statements in the Summary.

a. The incident angle needed for TIR to occur depends on the critical angle. TIR will only occur when the incident angle at the core/cladding interface is greater than the critical angle. Explain why smaller critical angles will allow TIR over a larger range of incident light angles at the end face of the fiber where the light comes in.

b. The differences in the indices of refraction of the media (core, cladding, and outside) define the critical angle, the numerical aperture, and the acceptance angle. The greater the difference in index of refraction between the core and cladding, the smaller the critical angle and the larger the NA.

Problem 3: (Solution is attached.)The geometry of the 3-dim acceptance cone is dependent on the indices of refraction of the outside medium, the core medium, and the cladding medium.

Assume that a particular fiber optic cable is surrounded by water (n = 1.33) and the fiber core and cladding are plastic (n = 1.45 and 1.40, respectively), thus defining a certain acceptance cone and angle.

a. What is the acceptance angle?

b. Predict what would happen to the following fiber properties if the index of refraction of the plastic fiber core is increased to a more refractive material such as flint glass (n = 1.7):

The critical angle at the core/cladding interface? Would it get bigger, smaller, or stay the same? Why?

The acceptance cone? Would it get bigger, smaller, or stay the same? Why?

What is the acceptance angle of this new type of fiber?

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Problem 4: (Solution is attached.)Using Snell’s Law, geometric identities and your experimental data on acceptance angle (see Experiment 1), complete the following: (given: nair = 1.0003, nwater = 1.33, nacrylic = 1.49)

a. Carefully draw the path of light as it travels through the air into an acrylic rod surrounded by water except at the end face. Include the following on your drawing:

1. Label all indices of refraction on the diagram2. An incident light ray that is at the boundary of the acceptance cone. Label the angle a .3. The refracted angle t at the air/acrylic interface. Calculate t.4. The critical angle c at the core/cladding interface. Calculate c.

b. Calculate the acceptance angle for this experiment and compare your calculated value to the measured acceptance angle. Show all work.

c. Will light with an incident angle greater than the acceptance angle enter the core of the fiber and exhibit TIR down the fiber? If not, what will it do? Explain your answer by relating it to your experimental results.

d. Will light with an incident angle less than the acceptance angle enter the core of the fiber and exhibit TIR down the fiber? If not, what will it do? Explain your answer by relating it to your experimental results.

Problem 5: (Solution is attached.)Using the diagram below and trigonometric identities, show that the relationship between the

numerical aperture and the indices of refraction for large core fibers is given by NA=√nf 2−nc2 .

Recall sin2 t + cos2 t = 1.

Adapted from http://www.thorlabs.com/images/TabImages/NumericalApertureAngles1.jpg

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Problem 6: (Solution is attached.)1. a. Suppose the indices of refraction of the outside, core, and cladding are 1, 1.5, 1.33, respectively. Construct a ray tracing diagram showing the paths of two rays (dotted lines): one that starts inside the acceptance cone (gray triangle), and the other that starts outside the acceptance cone. Make sure the ray tracing at the end face is correct. You do not need to compute the exact angles.

b. Look carefully at the ray tracing shown below. Which aspects are correct and which are not correct?

Problem 7: (Solution is attached.)Show that the acceptance angle for an acrylic rod in air is 90. The index of refraction of the acrylic rod is 1.49.

Adapted from http://upload.wikimedia.org/wikipedia/commons/thumb/4/46/Optical-fibre.svg/550px-Optical-fibre.svg.png

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Solution to Problem 1:

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Solution to Problem 2:(a)

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Solution to Problem 3:

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Solution to Problem 4:

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Solution to Problem 5:

Solution to Problem 6:

a.

b. The incorrect aspect is that the figure does not show refraction at the end face as the ray travels from air into the core.

At each interface, there is refraction (black line) and reflection (gray line).

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Solution to Problem 7:

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Building the acrylic rod/goniometer apparatus:

Goniometer: Wards Natural Science 14-6603 Leg goniometer, 360 degree, 12 inch.

Acrylic rod: You can order either Nada Scientific N99-D20-1614 (you get two rods with a square cross section: one curved, one straight) or Nada N99-D20-1614-S (one rod straight rod).

Graduated cylinder – 1000 ml. Make sure the diameter is greater than the width of the goniometer.

Laser pointer: Educational Innovations or Arbor Scientific. Red or green laser pointers work, but the green laser pointer is clearer for three reasons. (1) Based on our experience, the green laser has a narrower beam diameter and less divergence than the red laser pointer. (2) Our eyes are more sensitive to green. (3) Rayleigh scattering is proportional to 1/ 4 so green will scatter better from the particles in the acrylic rod. So overall, if the red and green lasers had the same power, the green laser will be brighter and clearer to see.

Channel: 5/16” – 3/8” inner width, about 4” in length, metal or plastic. This is used to align the laser pointer.

Velcro: Several inches. The type that is used to hold computer cables together is good (i.e., doublesided).

Beaker or flask: This will be used by the students to fill the graduated cylinder with water.

Glue: Silicone glue works for a while, but does not hold the acrylic rod for a long time. We have recently tried a clear, 2-part epoxy from a standard hardware store; the results are good so far.

To build apparatus: Cut the goniometer as shown in the photos. You will need to cut the protractor so that it has a

straight edge across the diameter. Look at the reverse side of the goniometer. The pivot has a smaller diameter on this side. The

acrylic rod will be mounted on this side. Cut a notch in the straight acrylic rod so that the end face is aligned with the straight edge of the protractor. The notch is needed to avoid hitting the pivot of the goniometer.

Position two pieces of Velcro under the channel, and mount the channel and Velcro onto the goniometer with a couple of screws, nuts, and washers. Make sure the channel is aligned with the goniometer axis. The Velcro will be used to hold the laser pointer in place.

Glue the acrylic rod onto the goniometer so that the endface is aligned with the straight edge of the goniometer, and the axis of the rod is aligned with the axis of the goniometer.