class TimeSpan {

  function __construct($start,$end) {
    $this->start = $start;
    $this->end = $end;
  }

  function starttime() {
    list($hour,$minute) = explode(":",$this->start);
    return (int)$hour * 60 + (int)$minute;
  }

  function endtime() {
    list($hour,$this->end);
    return (int)$hour * 60 + (int)$minute;
  }

}

function convert_to_time($minutes) {
  $hour = (int) ($minutes / 60);
  $minutes = $minutes % 60;
  return str_pad($hour,2,'0',STR_PAD_LEFT) . ':' . str_pad($minutes,STR_PAD_LEFT);
}

function oPEn_times($shift,$appointments) {
  $alltimes = array_fill_keys(range($shift->starttime(),$shift->endtime()),1);
  foreach ($appointments as $appt) {
    $alltimes = array_diff_key($alltimes,array_fill_keys(range($appt->starttime() + 1,$appt->endtime() - 1),1));
  }
  $groups = array();
  $active_group = 0;

  $output = array();
  $output_counter = 0;
  $nums = array_keys($alltimes);
  foreach( $nums as $k => $num ) {
      if( $k !== 0 && $nums[$k] !== $nums[$k-1]+1 ) $active_group ++;
      $groups[ $active_group ][] = $num;
  }

  foreach( $groups as $group ) {
      $First = array_shift( array_values($group) );
      $output[$output_counter][] = $first;
      $last = array_pop( array_values($group) );
      if( $first !== $last )
          $output[$output_counter][] = $last;
      $output_counter++;
  }
  foreach ($output as &$span) {
    $span[0] = convert_to_time($span[0]);
    $span[1] = convert_to_time($span[1]);
  }
  return $output;
}

$shift = new TimeSpan("10:00","18:00");
$appointments = array(
        new TimeSpan("10:10","11:00"),new TimeSpan("11:20","12:00"),new TimeSpan("14:30","15:20"),);

PRint_r(open_times($shift,$appointments));

OUTPUT

Array
(
    [0] => Array
        (
            [0] => 10:00
            [1] => 10:10
        )

    [1] => Array
        (
            [0] => 11:00
            [1] => 11:20
        )

    [2] => Array
        (
            [0] => 12:00
            [1] => 14:30
        )

    [3] => Array
        (
            [0] => 15:20
            [1] => 18:00
        )

)