The Unicorn’s Horn

By Sir William Castley, Sir Julius Music des Week, Sir Henning the Wren,
Sir John of Kosh, and Lady Lien of Stars
The Illumination Guild

ABSTRACT

The evil sorcerer Étendue has returned. He has teleported the unicorn to the Land of Unity and separated the unicorn’s horn. The Kingdom of Lambert will wither without the unicorn. Five members of The Illumination Guild obtained clues for how to reattach the unicorn’s horn. The people of Lambert are calling all illumination wizards and apprentices to assist in this epic quest. Can you help the Lambertians?

1. INTRODUCTION

Upon completing the spells to return Prince Lambert to his bed at IODC 2021, the frozen Illumination Guild mages shook themselves to consciousness. Queen Candela reached down to hold her prince and said to the mages, “Thanks to your efforts you have returned my dear Prince to me. Let us hope that is the last we see of Étendue and his evil ways.” Sir William Castley responded, “I fear that is not the last of him, for he is always there causing so much dismay to all those practicing illumination”.

Time did pass in the kingdom. The Queen and the Prince were married, the kingdom prospered, and there was much joy. All had pushed from their mind the problems that the evil sorcerer Étendue could do.

Two years later at the festival of Illumination design, the newly guild mage master Lady Lien of Stars came running into the Guild Hall. She is the watcher of magical beasts. Catching her breath and with a look of uncertainty she stated, “The unicorn is missing. All that I could find is this wax-sealed scroll.” Sir William said, “Please open it and read it to us.” Lady Lien broke the seal upon which there was a seeming gust of wind with a cackling sound at the limit of the hearing of all those in the hall. Lady Lien did read, “You thought you had defeated me. I am merely playing you, for you amuse me. I have taken the unicorn to the place where up is down, left is right, and forward is backward. For I be in the Land of Unity. Meet me at the break of your next day, for I challenge you where light can be both fast and slow. In this place the unicorn will wither
and along with it your kingdom.”

Upon concluding the reading of the scroll, a thunderclap did emanate from it, and onto the table a small, glimmering white spike did fall. As it rattled to rest, Sir Julius said, “The evil Sorcerer Étendue has returned and threatens our fair kingdom once again.” As Lady Lien picked up the white spike she scried into it, and the other mages turned to look at her. Sir John of Kosh asked, “What does this mean?” Lady Lien answered with dread in her voice, “This is the shrunken horn of the unicorn. Without reattaching it, the unicorn will perish; taking our illumination magics and the kingdom will fall.” Sir William said, “All five of us must go to meet Étendue in the Land of Unity for we will be weak while he will be indomitable conserving all of his powers.” With Sir Henning responding, “I will notify the Queen and the Prince, but note that the time of meeting
is nigh.” As he hastened from the Hall, Sir Julius did respond, “Sir John and I will convey our knowledge of the Land of Unity so that other wizards and apprentices can lend their support. For in the Land of Unity, an illumination master cannot cast much power, but they can channel illumination spells from those outside it.” Sir John, nodding his head, “Yes, we need all the support we can muster from our fellow illumination wizards and apprentices to restore the unicorn.”

Sir William and Lady Lien did then prepare the propagation spell to the Land of Unity, while the Sir Henning spoke to the Queen and Prince. Sir Julius and Sir John incanted the requirements for the illumination spells needed from their fellow wizards and apprentices.

In merely moments, the five all returned to the table in the Guild Hall. The first bell of the new day struck, “Bong,” as they all looked at each other. The Queen and Prince Lambert did enter the hall as the second bell run, “Bong.” The Queen did then say, as many more bongs of the clock pealed, “Go forth my Illumination Mages and defeat Étendue in the Land of Unity. Our wizards and apprentices will send their illumination spells to you to rejoin the horn to the unicorn. We know that we can only send their magical designs to you and there will be no other means of communication. I will ensure that all those able and willing receive the knowledge that you have incanted here.” Upon the last word of the Queen, the clock struck the last bell, “Bong,” and the spell of propagation to the Land of Unity was cast.

The five mages did vanish from the hall, reappearing in the Land of Unity. The hornless unicorn stood motionless, frozen in front of them. The Sorcerer Étendue also stood beyond it upon some pedestal, with his hood hanging, covering most of his face. All that could be seen were his chin and lips. An evil grin rose upon his lips, and he said,

“I must say, I am surprised to see you accept my challenge for I know you to be weak, but I now know that you are also foolhardy. For you know your own magics are insubstantial here. Light in the ether here works as expected or it is inverted. All is still conserved in my name. Thus, this should be enjoyable watching you flounder.”

Lady Lien stepped forward holding the shrunken unicorn horn. On its own it did rise from her hand to hover in front of her face, glowing in a pure whiteness while a disc on the forehead of where it was severed from the unicorn was utmost black possible. The four other mages stood behind her with Sir William opening himself to the normal realm. With Sir Henning being the conduit to Lady Lien. Sir Julius and Sir John pushed what little powers they retained in the Land of Unity to defend against Étendue’s machinations.

Étendue then broke into a large grin showing all his teeth, “Yes, this will be very entertaining and as they say, let the games begin for your very lives will depend on it…”

You are now reading their story, notated in the way of all spells of The Illumination Guild. Their last words were,

“Call out to all wizards and apprentices of the Magic of Illumination. Reattach the unicorn’s horn so the Illumination Guild may return the unicorn to the Kingdom of Lambert.”

2. KNOWLEDGE TO REATTACH THE HORN

2.1 Land of Unity

To reattach the shrunken unicorn’s horn, magic must flow from the detached unicorn’s horn to the base of the horn on the unicorn’s head. In the Land of Unity, magic propagates using the laws of geometrical optics, but only at wavelengths of 486nm and 656nm. The power at 486nm and 656nm must be balanced to produce white light. Other than air with an index of refraction of 1.0, there exists only a single optical material featuring an index of refraction of 2.0 at 486nm and n=0.5 at 656nm.

2.2 Unicorn’s Horn

The original base of the horn, on the unicorn’s head, has a radius of 25mm and shows 100% light absorption. The shrunken unicorn’s horn is a tapered cylinder with an 8:1 length to radius aspect ratio. The area of the shrunken unicorn’s horn equals the 25mm radius disk. Light rays at 486nm and 656nm are emitted outwardly with constant luminance (uniform spatial with a cosine angular distribution) from all points on the surface of the tapered horn. There is no emission from the interior surface of the horn, nor the circular base of the tapered cylinder.

2.3 Illustrative Example

An example of a problem solution is shown in Figure 1. 486nm and 656nm rays are traced from the 8:1 aspect ratio horn to the 25mm absorbing disk. Rays which pass through the magical material obey the laws of geometric optics using either refraction or TIR.

3. SPECIFICATIONS TO REATTACH THE HORN

Here lies a list of the specifications to reattach the horn:

• The surrounding media is air with an index of refraction of 1.0
• Optical Materials must use n=2 at 486nm and n=0.5 at 656nm
• All surfaces: 100% transmission and/or 100% TIR
• No reflective coatings, diffractive elements, or scattering
• Disk and Horn must be in air.
• Disk and Horn are absorbing.
• Disk only collects light from one side
• Horn area (only the cone, not the base) equals original Disk area
• Original Disk has a 25mm radius
• The Horn emits white light using wavelengths of 486nm and 656nm
• Shrunken Horn is a cone shaped surface with 8:1 length to radius aspect ratio
• Merit function
  • Step 1: maximize the power that transfers light at 468nm and at 656nm (light % from Horn to Disk).
  • Step 2: choose the wavelength which propagates the lower efficiency and report that efficiency.

4. SUBMISSION

The Illumination Guild is looking for the following by the submission date

• Your Submission should include:
  • Layout – figure of your design with a prescription, and, if possible,
    • STEP or IGES file and/or
    • System file from a commercial software package (indicate which software package);
  • Tables and drawings
    • Explain what you did and why you did it, to allow the Illumination Guild to properly judge your submission, and to possibly use this material when presenting the results.
• Email submissions to: illuminationguild@iodc.org
• Submission date is: No later than April 17, 2023.

5. WINNING SUBMISSION

Designs will be modeled in a single commercial software package. Your optical system will be compared to all other systems with the following criteria:

• Step 1: maximize the power that transfers light at 468nm and at 656nm (light % from Horn to Disk).
• Step 2: choose the wavelength which propagates the lower efficiency and report that efficiency.
• The submission with the highest value in step 2 will be the winner.

There will be at least two separate prizes:

• Wizard: practicing illumination engineers
• Apprentice: students in illumination engineering training

Additional prizes for innovative concepts may be awarded at the discretion of The Illumination Guild. All decisions by The Illumination Guild are final.