PHP AJAX Issue

Well first of all, there is no DOM object with id="results" in your HTML, that could be from your cropping.

Also, your PHP looks ... not so great.  At the very least, you need to remove the semicolon from the line:

->from('exp_matrix_data');

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There may be other issues, hard to tell from that snippet.

Take care of those things, then let us know how it goes.

If you need more help, we will need to see more of the PHP, like what you're including, etc, as well as the relevant HTML also, thanks!

I did that and it cleared up most of the issue.  Now I'm getting an error; undefined index cw undefined index pulsed

Also, I guess I need to add formData to the data type.  How do I do that?

This is pretty much all of the code as it stands right now.
HTML

<div class="select-input">
		<form action="#" method="get" id="laser_finder">
  <!--  Operating Mode  -->
  	<div class="form-control main title">
  		Operating Mode
  	</div>
 <!--  START CW  -->
  	<div class="form-control">
  			<input type="checkbox" name="cw" class="laser_operating_mode"  id="cw" value="CW (Continuous Wave)" >
  			<label for="cw">
  				CW (Continuous Wave)
  		</label>
  	</div>
<!--  START Pulsed  -->
  	<div class="form-control">
  				<input type="checkbox" class="laser_operating_mode" name="pulsed" id="pulsed" value="Pulsed" >
  				<label for="pulsed">
  				Pulsed
  		</label>
  	</div>
</form>
</div>
<div id="results"></div>

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jQuery at the top of the HTML

<script>
$(document).ready(function(){
$( "#laser_finder" ).change(function (event) {
event.preventDefault();
function laser_finder() {
	$.ajax({
		type: 'GET',
				url: '{site_url}/ajax/laser_finder.php',
				data: {action:"laser_finder"},
				dataType: 'html',
				success:function(result) {
					$('#results').html(result)
        		}
			});
		}
  			laser_finder();
		});
	});
</script>

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ajax/laser_finder.php

<?php
error_reporting(E_ALL);
ini_set('display_errors', 1);

$cw ='';
$pulsed = '';
$action=$_GET["action"];
	if($action=="laser_finder") {
		$cw = $_GET['cw'];
		$pulsed = $_GET['pulsed'];
		
		$laser_query = ee()->db->select('entry_id')
		->from('exp_matrix_data')
		->where('col_id_6401', $cw)
		->get();
		
		$laser_finder = $laser_query->result_array();
	
		foreach($laser_query->result_array() as $row){
   			echo $row['entry_id'].'<br />;
   		}
   		 
	}  
?>

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You are missing a closing ' after your <br/> in your php file
As Marco pointed out you are not passing your cw and pulsed values to the AJAX function.

Some pointers - POST your data to the script - include the action in the URL
In the code below I have taken the laser_finder function out of the event handler but you can choose.

Updated JavaScript - posts cw and pulsed data to laser_finder

<script>
$(function(){
  $( "#laser_finder" ).change(function (event) {
    event.preventDefault();

    var data = $(this).serialize();
    
    laser_finder(data);
  });
});
// PASS DATA TO FUNCTION
// AND POST IT
function laser_finder(data) {
  $.ajax({
    type: 'POST',
    // ADD ACTION AS GET PARAMETER
    url: 'laser_finder.php?action=laser_finder',
    data: data,
    dataType: 'html',
    success:function(result) {
      $('#results').html(result)
          }
  });
}
</script>

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PHP - modified to not interface to CI - you will need to put that back in

<?php
error_reporting(E_ALL);
ini_set('display_errors', 1);

$cw ='';
$pulsed = '';
$action=isset($_GET["action"]) ? $_GET["action"] : '';
if($action=="laser_finder") {
  $cw = isset($_POST['cw']) ? $_POST['cw'] : '';
  $pulsed = isset($_POST['pulsed']) ? $_POST['pulsed'] : '';
  // REPLACE WITH CI CODE
  $laser_query = new stdClass;
  $laser_query = array (
    array('entry_id' => 1),
    array('entry_id' => 3),
    array('entry_id' => 4)
  );

  foreach($laser_query as $row){
    // MISSING ' INSERTED
    echo $row['entry_id'].'<br />';
  }
  // END REPLACE
}

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Working sample here