2021 Illumination Design Problem

The Curse of Prince Lambert

By Sir William Castley, Sir Julius Music des Week, Sir Henning the Wren, and Sir John of Kosh

The Illumination Guild

Abstract

Prince Lambert was a good ruler, though being tired and old, he was cursed by the evil sorcerer Étendue.  All that is left is two million scrambled pieces of this fair ruler that must be put back into his bed. With the help of four members of The Illumination Guild, more clues were left to unscramble Prince Lambert so that he can return to his full, glorious power.  The people of Lambert are calling all illumination wizards and apprentices to assist in this epic quest.  Can you help the Lambertians?

1.    INTRODUCTION

Prince Lambert was meeting with Duke Radiance on a matter of utmost importance to the Principality of Lambert – the impending marriage to Queen Candela of the Empire of Light.  Prince Lambert had his four retainers from The Illumination Guild in attendance: Sir William Castley, Sir Julius Music des Week, Sir Henning the Wren, and Sir John of Kosh.  Sir William is a famous wizard of The Illumination Guild who has won a many illumination spell challenges and developed many new illumination spells.  Sir Julius des Week is a noted wizard of The Illumination Guild who has traveled the known world imparting his sagely advice to all who ask.  Sir Henning the Wren is the leading wizard of The Illumination Guild and has developed many illumination spells in his labs.  Sir John is the last wizard of The Illumination Guild in the employ of Prince Lambert, and he is known for being a jester with some skill while turning foremost to an administrative role.

Unknown to all in attendance was that Duke Radiance had two personalities, both thought to be tales, but still known by all.  First and foremost was the godly Mage of Power, who could luminate all with his illumination spells.  Second and more dire was the evil sorcerer Étendue.  The Duke of Radiance was a brilliant and stern man, but he was also these two others.  Fortunately, his persona of the Mage of Power held over the persona of the evil sorcerer Étendue – thankfully keeping it in check.  However, on this fateful day nearing autumn, the evil sorcerer Étendue took control, and froze him and his retainers while saying to all with a steely yet vague look in his eyes:

“Prince Lambert, I grow tiresome of your ways.  You think you can control me, Étendue, by removing all direction in my Radiance.  I say nay!  Your powers and those of your magicians are weak compared to mine.  You are a mere reflection of meekness.  Your power is diluted by those who surround you.  I will not let you marry the wise and great Queen Candela for she cannot love you, but rather she should be betrothed to me.  Thus, I will twist, bend, and cast you through my Curse of Entropy.  You will be conserved in a state that can be reversed, but I think there are none worthy to succeed.  They must first return your two million pieces to your small bed, and if they can rise to the challenge, unscramble you from my Curse.”

An explosion of light did then surround Prince Lambert, breaking asunder the royal Diamond Scepter and Orb of Noble Glass.  The staff and crystal ball, as they are more commonly known, did shatter, crack, and crash into the black box of the Entropy Curse.  Prince Lambert sitting in his small bed did then dissolve into many green rays of light cascading into the same black box.  The evil sorcerer did then laugh, “You are no more than a bundle of rays within the black box of Entropy.”  Upon the conclusion of his evil cackle, the evil sorcerer started to dissolve into an unbelievable pinpoint of light, but before he could fully disappear, Queen Candela, sensing Prince Lambert’s agony from afar, hurried to the scene, and using her strong wizardly talents released the wizards from The Illumination Guild from their holds.  They sprang into action, with one saying, “Hurry! We need a way to decipher this evil spell.”  Sir William did leap forward, grabbing at Étendue, to hold him with the help of the upstanding half of Radiance, the Mage of Power. Queen Candela cast an arcane beam of bluish light towards Étendue, who escaped by diluting the Queen’s Light into the void. However, the goodly Mage of Power half redirected a small amount of the Queen’s light towards William. With this light scattered from the visage of Duke Radiance and now held in William’s hands, William did say, “Help me, let us put this power in a controlled way through the box so that we may decipher the Curse.”  Julius and Henning set themselves to work with each respectively controlling the horizontal and vertical directions as William released rays of power in a controlled manner into the black box.  Sir John did watch the other side of the box, upon which he saw two visages – one was the two million greenish, pulsing, and scrambled rays of their Prince, while the other pulsed in a bluish color.  Before Étendue did vanish, his last utterance was, “You puny magicians who think you know me – enough of your interference.  You have a few minutes till you are mute, deaf, and blind!”  The four wizards along with Queen Candela did then set down to relay all their knowledge about the Curse of Entropy and what had befallen their well-loved Prince Lambert.  You are now reading their story notated in the way of all spells of The Illumination Guild.  Their last words to the courtiers who rushed into the throne room were,

“Call out to all wizards and apprentices of the Magic of Illumination.  Undo the Entropy Curse that has fallen onto our Prince Lambert and bring him back to the love of grieving Queen Candela.  In these words and what follows are the tools to unwind this most evil Curse.” 

Thereupon they did fall mute, deaf, and blind – unhelpful in matters henceforth.  The Queen is available to answer some questions, but she knows little of what transpired before she entered the room.

2.    Knowledge to reverse
the curse

2.1   Prince Lambert

We counted two million rays that comprised our fair Prince – we saw two million enter the black box, and we counted two million at its exit face.  The Curse of the black box allows you access to the exiting rays, but we have little knowledge of how they entered.  We only know the rays came from Prince Lambert sitting in his bed of 1 mm x 1 mm in size, located at the center of the left face of the black box.  They were green in color, and our spectral sight indicates they are all of a wavelength of 550 nm. Video 1 shows what happened to Prince Lambert!

Video 1. Depiction of the effect on Prince Lambert from the Curse of Entropy by the evil sorcerer Étendue.

2.2   Staff and Crystal Ball

The staff is 5 mm in diameter and 200 mm in length.  It is made of perfect diamond with an index of refraction of 2.417 with no dispersion.  It is perfectly transmissive and free of absorption.  The crystal ball is a sphere of 200 mm in diameter with an index of refraction of 2.0 with no dispersion.  It is also perfectly transmissive and free of absorption.  We counted no more than a total of 10,000 pieces from the staff and orb, but we cannot tell you the total for the actions were too fast.  Some pieces may have been left out of the box, but their whereabouts are unknown. Due to the perfection of the staff and crystal ball, the shards will also be perfect with no dispersion and absorption, while remaining perfectly transmissive.

2.3   Black Box

All Curse black boxes are initially filled with vacuum of an index of refraction of 1.0 with no dispersion, absorption, or for that matter any other optical properties. Upon light rays entering the box they are trapped within. The black box is 220 mm x 220 mm x 220 mm.

2.4   Duke Radiance

With the pieces of the Queen’s blue beam, scattered by Étendue, and captured by William with the Mage of Power’s help, we created 20,000 rays in a 100×100 uniformly spaced grid in the horizontal by vertical directions.  For each grid point we were able to release two rays per position. Each was at a wavelength close to 500 nm.  We have provided you the input rays at the center of the left face of box and the exiting rays on the right face.  Use this to reverse the Curse.

3.    Components to reverse the curse

Figure 1. Depiction of the Diamond Scepter (staff) and Noble Glass Orb (crystal ball).

Here lies a list of the components you can use to reverse the curse:

Tools to use in your laboratory to reverse the curse (see Fig. 1):

  • A staff made of pure diamond
    • Cylinder 5-mm diameter and 200-mm length,
    • n = 2.417, and
    • Free of absorption and dispersion;
  • A crystal ball, made of expensive glass
    • Sphere 200-mm diameter
    • n = 2.000, and
    • Free of absorption and dispersion;
  • As a wizard, you can cut and polish and assemble to sub nanometer precision, with perfect optical surfaces.

To reverse the curse, you have to:

Figure 2. Rays (reverse in this case) traced through an empty black box.
  • Determine the optical system inside the box by
    • Tracing the provided rayfiles through the black box (see Fig. 2),
    • Backwards for rays at the (right) exit face, and
    • Forwards for rays at the (left) entrance face;
  • There are two steps to reverse the curse:
    • Step 1: Collect all Prince Lambert rays in his 1-mm2 bed, which maximizes the collected ray source power and
    • Step 2: Unscramble all of the blue Duke Radiance rays to their original positions.

Specifications:

  • The three rayfiles have the following origins and wavelengths
    • Prince Lambert: right face of the black box with l = 550 nm,
    • Duke Radiance exiting rays: right face of the black box with l » 500 nm, and
    • Duke Radiance entering rays: left face of the black box with l » 500 nm;
  • Medium inside the black box is
    • Vacuum with n = 1.0 with no dispersion and absorption, except pieces of the staff and crystal ball.
    • In your laboratory you will use pieces of the staff and crystal ball to reverse engineer the Curse;
  • All pieces of the staff and crystal ball are perfect meaning
    • 100% transmission or 100% total internal reflection (TIR) as the propagation of light dictates,
    • There is no absorption, no scatter, no reflective coatings, no diffraction, no wavelength conversion, and no gradient index materials,
    • No more than 10,000 pieces cut out of the crystal ball and staff,
    • All of the pieces must fit strictly inside the black box (220 mm)3, not touching its boundaries, and
    • Not all of the crystal ball and staff need be used, but you cannot use more than the volume of each component.
  • Geometry
    • Black box: 220 mm per side so the volume is (220 mm)3,
    • Prince Lambert’s bed: 1x1mm2 is located at the center of the left face of the black box, and
    • The origin of the coordinate system at the center of Prince Lambert’s bed, with x and y coordinate axes tangent to the bed, and the z axis perpendicular to the bed, toward the right face of the box
    • Exiting rays: right face of the black box, at z = 220 mm;
  • Merit functions
    • Step 1: maximize power returned to Prince Lambert’s bed (1×1 mm2 receiver at the entrance (left) face of the black box) and
    • Step 2: minimize the total error of Duke Radiance exiting rays that are reverse traced to the entrance (left) face of the black box;
  • Notes
    • Prince Lambert rays: two million rays entered, two million exited – you only have the exiting rays,
    • Duke Radiance rays: 100 x 100 uniformly spaced rays entered with two ray angles per ray, and 20,000 rays exited – you have entering and exiting ray sets, in matching sequence, and

4.    Submission

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

  • Your Reverse Curse to include:
    • Layout – figure of your design and at minimum one of the following,
      • 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;
    • Optional: Provide Duke Radiance exiting rayset traced backwards through the black box to the entrance face
      • Rayfile with X, Y, Z, L, M, N, power and wavelength values for the rays traced to the entrance face, in IES TM25 format or in plain text format with one line per ray, eight numbers per line
    • Calculation of two merit functions:
      • Step 1: power collected in Prince Lambert’s bed (1×1 mm2 receiver) as fraction of the total power of the original Prince Lambert rays, and
      • Step 2 (Optional): Total error of Duke Radiance rays at the receiver (i.e., Prince Lambert’s bed); the figure of merit (FOM) is the sum of individual contributions of each of the 20,000 rays, to be computed for the i -th ray as
Equation (1)

when ray #i is present in the Duke Radiance rays at the receiver and

Equation (2)

when ray #i is NOT present in the Duke Radiance rays at the receiver.  The data (uppercase) , , , and  are the ray coordinates and direction cosines for the Duke Radiance entering ray set that is provided to you by The Illumination Guild (desired ray data).  The data (lowercase) , , , and  are the ray coordinates and direction cosines for the entering ray set that you traced backward from the supplied Duke Radiance exiting ray set  (actual ray data); 

5.    Winning submission

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

  • Step 1: maximize power collected in Prince Lambert’s bed (1×1 mm2 receiver) using the Prince Lambert rayset and
  • Step 2: minimize the total error of the Duke Radiance exiting rays traced backwards to the receiver (i.e., the plane containing Prince Lambert’s bed) compared to the Duke Radiance entering rays.

Step 1 will be explored first to determine which submissions did the best job at returning all Prince Lambert rays to his bed.  The Step 2 merit function will be used to determine the winner in the event there is a tie from the Step 1 results, where a tie is assumed when Step 1 results are within one percent of the highest value.  There will be at least two prizes:

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

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

1 thought on “2021 Illumination Design Problem”

  1. In specifications you have “Duke Radiance exiting rays: right face of the black box with l » 500 nm,”, however based on problem description the Duke Radiance rays are blue. Would it be more in line with description to have I << 500 nm?

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